1 » » » » » »’ Paralipomènes à l’algèbre des fonctions caractéristiques et simples 14

L’Ours et les Abeilles (The Bear and the Bees). Origine : Fable nord-italienne de Laurentius Abstemius (De urso et apibus, Hecatomythium, 1495), largement diffusée et souvent attribuée à Ésope dans les compilations des XVIe–XIXe siècles.

A Bear roaming the woods in search of berries happened on a fallen tree in which a swarm of Bees had stored their honey. The Bear began to nose around the log very carefully to find out if the Bees were at home. Just then one of the swarm came home from the clover field with a load of sweets. Guessing what the Bear was after, the Bee flew at him, stung him sharply and then disappeared into the hollow log. The Bear lost his temper in an instant, and sprang upon the log tooth and claw, to destroy the nest. But this only brought out the whole swarm. The poor Bear had to take to his heels, and he was able to save himself only by diving into a pool of water.

Moral : It is wiser to bear a single injury in silence than to provoke a thousand by flying into a rage.

Tout le monde sait que les ours adorent le miel. Un jour, Monsieur Ours regarde dans son placard
et il ne trouve pas de miel. « Oh non ! Il n’y a plus de miel ! » dit-il. Il part donc en chercher dans la
forêt.
Il voit une ruche d’abeilles dans un arbre. Il sent le miel. « Miam ! Du miel ! » Il met sa patte dans
la ruche et… BZZZ ! Une abeille en sort. « Hé, ours ! Va-t-en ! » dit l’abeille. « Ce n’est pas ton
miel ! »
Monsieur Ours s’énerve. « Je suis plus gros que toi ! » dit-il. « Et je suis plus fort que toi aussi. Je
peux faire ce que je veux ! Maintenant… dégage ! » Il remet sa patte dans la ruche et… ZING ! Aïe
! L’abeille le pique sur le nez. « Mon nez ! Mon nez ! Je suis tellement en colère ! » grogne
Monsieur Ours. Il trouve un gros bâton. Il essaie de frapper l’abeille. Il balance et balance, mais
l’abeille est trop rapide ! Monsieur Ours devient de plus en plus en colère. « GRRRRR ! JE VAIS
T’ÉCRASER ! VIENS ICI ! » L’abeille rentre dans la ruche. « AH AH ! Maintenant tu es piégée ! »
crie Monsieur Ours. Il balance son bâton contre la ruche. Smash ! Smash ! Smash ! Il la réduit en
miettes !
Puis il entend un bruit étrange. Il devient de plus en plus fort… ZZZZ ! ZZZZ ! ZZZ ! Des milliers
d’abeilles sortent de la ruche cassée. Elles volent autour de la tête de Monsieur Ours…
Moral : Une colère chaude vous met toujours dans le pétrin. / Il vaut mieux supporter une seule
blessure en silence que d’en provoquer mille par la rage.

Abeille – Buzz, Bzzz

General formulas

• L(w) = symbol count after tokenisation
• Bord(w) = max_k<n k·δ(pref_k, suff_k)
• p = n − Bord(w), Root(w) = pref_p(w) if p ∣ n, else w
• Cyclic root = arg min_lex {σ^k(w)}

Buzz

• L(w) = symbol count after tokenisation = 4
• Bord(w) = max_k<n k·δ(pref_k, suff_k) = 0
• p = n − Bord(w), Root(w) = pref_p(w) if p ∣ n, else w = 4, Root(w) → BUZZ
• Cyclic root = arg min_lex {σ^k(w)} = BUZZ
• Lyndon factorization via Duval (non-increasing Lyndon words) = (BUZZ)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 2 (attained by ZZ)

Bzzz

• L(w) = symbol count after tokenisation = 4
• Bord(w) = max_k<n k·δ(pref_k, suff_k) = 0
• p = n − Bord(w), Root(w) = pref_p(w) if p ∣ n, else w = 4, Root(w) → BZZZ
• Cyclic root = arg min_lex {σ^k(w)} = BZZZ
• Lyndon factorization via Duval (non-increasing Lyndon words) = (BZZZ)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 3 (attained by ZZZ)

Property	BUZZ	BZZZ
Alphabet	{B, U, Z}	{B, Z}
Length	4	4
Border	ε	ε
Border length	0	0
Minimal period	4	4
Period set	{4}	{4}
Critical exponent	2	3
Primitive root	BUZZ	BZZZ
Root(w)	BUZZ	BZZZ
Cyclic root	BUZZ	BZZZ
Suffix array	[1, 2, 4, 3]	[1, 4, 3, 2]
Minimal absent words	BB, BZ, UB, UU, ZB, ZU, ZZZ	BB, ZB, ZZZZ
Lyndon factorization	(BUZZ)	(BZZZ)
Biroot	(BUZZ, ε), (ε, BUZZ)	(BZZZ, ε), (ε, BZZZ)
Internal factors	U, Z, UZ	Z, ZZ

This tale comes from the oral tradition of the Lur people of Luristan (Zagros Mountains, western Iran). It was recorded by Sekandar Amanolahi and W. M. Thackston in their collection of Luri folktales and appears as “The Unlucky Bear” (A/Th8). The story belongs to a wider Iranian / Persian animal-tale tradition in which the strong but naïve bear is repeatedly tricked by weaker, more cunning domestic animals.

Once a very hungry bear said, “O God, I am dying of hunger. Do something for me.”
His prayer was answered with a lame goat trailing behind a herd. The bear went near it, and the goat thought of a trick. The goat greeted the bear and offered itself to be eaten “to the last drop of my blood,” claiming its owner beat it mercilessly and that it wished to die after one final bleat. The bear believed this and waited for the goat to bleat, but the shepherd arrived with his dogs and beat the bear half to death.
The bear, still faint with hunger, wept and again asked God for prey. He saw a camel standing by itself. The camel greeted him, wished him well, and offered itself to be eaten, complaining that its owner took it on long trips without food or water and beat it. The camel’s last wish was that the bear should lift its foot and read what was written under it. The bear did so, and the camel crushed him.
The bear, bruised and still hungry, prayed once more. This time God sent a donkey standing by itself.
The bear drew close. The donkey pricked its ears, greeted the bear, and told him to eat it because its godless owner prodded it with a stick to go faster all the time. The bear said, “Thank goodness I have never been subject to human beings.”
The donkey’s last wish was that the bear should ride him. The bear, flattered and curious, climbed onto the donkey. At once the donkey ran straight to the human camp. An old woman saw them and cried, “Look, people, a bear is riding a donkey!” The people rushed out and beat the bear until he was half dead.
The bear fled and hid in a cave. There he wailed, “When has my father told me to meet a lame goat? Or sent me to school to learn how to read? Or taught me how to ride a donkey?”
A wood-collector heard him. When the bear said, “I hope nobody is around to grab me by my balls and fling me down the slope,” the man did exactly that. The unlucky bear died.
Thus the cunning donkey escaped unharmed and returned to its easy life, while the powerful bear was thoroughly defeated by wit, flattery, and a simple ride into danger.
Moral
Brute strength and hunger are no match for cunning and quick thinking. The donkey wins by playing on the bear’s pride and inexperience, turning the bear’s own curiosity against him. In the wider Iranian tradition this tale reminds listeners that cleverness, not size, decides who survives.

Ane – Hi-han

• L(w) = 5 (symbol count)
• Bord(w) = 0
• p = 5 − Bord(w) = 5
• Root(w) = w (since p ∣ n)
• Cyclic root = arg min_lex {σ^k(w)} = ANHIH
• Lyndon factorization: non-increasing sequence of Lyndon words (Duval) = (HI)(H)(AN)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 3/2 (attained by HIH)

Property	HIHAN
Alphabet	{A, H, I, N}
Length	5
Border	ε
Border length	0
Minimal period	5
Period set	{5}
Critical exponent	3/2
Primitive root	HIHAN
Root(w)	HIHAN
Cyclic root	ANHIH
Suffix array	[4, 3, 1, 2, 5]
Minimal absent words	AA, AH, AI, HH, HN, IA, II, IN, NA, NH, NI, NN, IHI
Lyndon factorization	(HI)(H)(AN)
Biroot	(HIHAN, ε), (ε, HIHAN)
Internal factors	A, H, I, HA, IH, IHA

Tale of the Bélier (Ram) and the Bear: The Frightened Bear and the Wolves
(Russian folk tale, collection of A. N. Afanasyev. )

In a certain kingdom, in a certain state, there lived a peasant who had a goat and a ram. The peasant was too lazy to cut hay; winter came, and the goat and the ram had nothing to eat. They began to bleat all over the yard, and the peasant seized a whip and started beating them. Then the goat said to the ram: “Come, brother, let’s go into the forest; we’ll find a haystack and live there.” “Let’s go, brother Kozma Mikitich! It won’t be worse.”

The goat stole the master’s gun, and the ram took a sack, and off they went. They walked along the road and found an old wolf’s head. “Brother ram!” said the goat, “take this head and put it in the sack.” “What the devil do we need it for? It’s heavy enough already!” “Take it! When we get there we’ll cook ourselves some headcheese.” The ram picked up the wolf’s head, put it in the sack, and carried it. They walked and walked and finally reached the forest. “I’m completely frozen,” said the ram. The goat saw a fire glowing in the distance and said: “There’s a fire burning over there; let’s go there!” They went toward the fire and ran straight into some wolves sitting around the fire warming themselves.

The ram was frightened, barely holding onto his soul. But the goat said to him: “Don’t be afraid, ram!” and himself went up to the wolves: “Hello, fellows!” “Hello, Kozma Mikitich!” The wolves thought: this will be fine prey—the goat and the ram have come of their own accord, ready to jump into our mouths. But the goat was clever. “Come now, brother ram! Bring me the wolf’s head; we’ll cook it and make headcheese. And make sure you pick an old wolf’s head!” The ram took the wolf’s head out of the sack and brought it to the goat. “Not that one!” said the goat. “There’s another head in the sack, from the oldest wolf—bring that one.”

The ram rummaged in his sack, rummaged and rummaged, and brought the same head again. “Ah, you fool!” shouted the goat and stamped his feet, “Not that one! Look, it’s lying right at the bottom.” The ram rummaged again and brought the same head. “Now that’s the one!” said the goat. “This is the head I need.” The wolves looked on and thought: “Look how many of our kind he’s killed! A whole sack of heads.” “Haven’t you got anything, brothers,” asked the goat, “in which we can prepare our supper?” At that the wolves jumped up and ran—some for firewood, some for water, some for pots—while thinking only of how to get away safely.

They ran and met a bear. “Where are you going, gray wolves?” “Ah, Mikhailo Ivanovich! You don’t know our trouble: a goat and a ram came to us, brought a whole sack of wolf heads, and want to cook headcheese; we were afraid they would get to us too, and we ran away.” “Ah, you fools!” said the bear. “The goat and the ram came to you of their own accord—just take them and eat them; and you got scared! Come with me.” “Let’s go!” The goat and the ram saw the wolves coming back, got busy, and started running. The goat climbed a tree, managed to seat himself, but the ram climbed and climbed and could not get high; he grabbed a branch with his front legs somehow and hung there. Then the bear came with the wolves and looked: where had the goat and the ram gone? Nowhere to be seen. “Well, brothers,” said the bear to the wolves, “gather some acorns; I’ll start divining: where have the goat and the ram disappeared to?” The wolves gathered acorns; the bear sat under the tree and began throwing acorns and divining the way women divine with beans.

The ram said to the goat: “Ah, goat, I’m going to fall; my legs hurt, I can’t hold on!” “Hold on,” answered the goat, “or we’ll be lost for nothing; they’ll eat us!” The ram held on and held on, then fell to the ground! The goat saw inevitable trouble, fired the gun at that moment and shouted at the top of his voice: “Grab the diviner, hold him!” The bear got frightened, dashed off running without looking back, and the wolves after him. So they all scattered. Then the goat climbed down from the tree and did not want to stay in the forest. He returned home together with the ram, and they began to live and prosper and do no harm.

Bélier – Bé, Bê, Bè

• L(w) = 2
• Bord(w) = 0
• p = 2, Root(w) = w
• Cyclic root = arg min_lex {σ^k(w)} = min_lex {BE, EB} = BE (under B < E)
• Lyndon factorization: (BE)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 1

(The three orthographic variants Bé, Bê, Bè encode to the same symbol word BE.)

Property	BE
Alphabet	{B, E}
Length	2
Border	ε
Border length	0
Minimal period	2
Period set	{2}
Critical exponent	1
Primitive root	BE
Root(w)	BE
Cyclic root	BE
Suffix array	[1, 2]
Minimal absent words	BB, EB, EE
Lyndon factorization	(BE)
Biroot	(BE, ε), (ε, BE)
Internal factors	none (∅)

The Clever Billy Goat (Traditional animal fable)

A Billy goat had strayed away from his village towards the forest. He was grazing contentedly when he heard some movement. Looking up he saw a large bear peering through the trees at him.
The goat was frightened. But he realised that the bear would attack and kill him in a flash if he showed any fear. So, he decided to pretend that he was very brave and strong. He continued to chew the grass ignoring the bear.
‘That’s funny!’ thought the bear, ‘He doesn’t seem afraid of me, even though he looked at me. He’s busy munching the grass!’
He came stealthily closer to get a better look at this strange creature. It seemed to have two sharp dagger-like horns on its head. It had a short tail that was flicking constantly. And there were wens (growths on the skin) on either side of its neck. ‘I wonder what those things are for!’ thought the bear, ‘I had better ask him.’
“Hello!” he called out. The goat looked up at him enquiringly. “What’s that you have on your head?”
“Oh, the knives? They are for killing my enemies,” said the goat casually, before returning his attention to the grass.
“And what’s that behind you?” asked the bear, “That flat thing which keeps moving?”
The goat said, “That’s a kind of an axe. I use it to chop off my enemy’s head.” And back it went to munch the grass.
“And the white things hanging from either side of your neck?”
“I clean my mouth with those,” replied the goat, “especially after I have feasted on bear meat.”
This really terrified the bear, who ran right back into the forest. A cunning little fox, who had seen him talking to the goat and then running away, stopped him to ask what the matter was.
“Why did you run from that goat?” asked the fox.
“That’s no goat!” exclaimed the bear, “It’s a fierce animal with knives and an axe for killing bears!”
“That’s a goat! Come; we’ll go and kill him and have a feast,” said the fox.
When the goat saw the bear returning with a fox, he was even more terrified. But he went on grazing calmly. When they were close, he called out to the fox, “Hello, my friend! I see that you have brought the bear for our feast as you said you would. Come; let’s kill him.”
The terrified bear ran into the forest. The fox saw the bear disappear among the trees and ran away as fast as he could!

Bouc – Bé, Bê, Bè

• L(w) = 2
• Bord(w) = 0
• p = 2, Root(w) = w
• Cyclic root = arg min_lex {σ^k(w)} = min_lex {BE, EB} = BE (under B < E)
• Lyndon factorization: (BE)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 1

(The three orthographic variants Bé, Bê, Bè encode to the same symbol word BE.)

Property	BE
Alphabet	{B, E}
Length	2
Border	ε
Border length	0
Minimal period	2
Period set	{2}
Critical exponent	1
Primitive root	BE
Root(w)	BE
Cyclic root	BE
Suffix array	[1, 2]
Minimal absent words	BB, EB, EE
Lyndon factorization	(BE)
Biroot	(BE, ε), (ε, BE)
Internal factors	none (∅)

Bourdon – Buzz, Bzzz

General formulas

• L(w) = ∣w∣ = n = symbol count after tokenisation
• Bord(w) = max_k<n k·δ(pref_k, suff_k)
• p = n − Bord(w), Root(w) = pref_p(w) if p ∣ n, else w
• Prim(w) = ⌊1 / (1 + ∑_k=1ⁿ⁻¹ δ(σ^k(w), w))⌋
• Cyclic root = arg min_lex {σ^k(w)}
• Lyndon factorization via Duval (non-increasing Lyndon words)

Buzz

• L(w) = symbol count after tokenisation = 4
• Bord(w) = max_k<n k·δ(pref_k, suff_k) = 0
• p = n − Bord(w) = 4, Root(w) = pref_p(w) = BUZZ
• Cyclic root = arg min_lex {σ^k(w)} = BUZZ
• Lyndon factorization via Duval (non-increasing Lyndon words) = (BUZZ)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 2 (attained by ZZ)

Bzzz

• L(w) = symbol count after tokenisation = 4
• Bord(w) = max_k<n k·δ(pref_k, suff_k) = 0
• p = n − Bord(w) = 4, Root(w) = pref_p(w) = BZZZ
• Cyclic root = arg min_lex {σ^k(w)} = BZZZ
• Lyndon factorization via Duval (non-increasing Lyndon words) = (BZZZ)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 3 (attained by ZZZ)

Property	BUZZ	BZZZ
Alphabet	{B, U, Z}	{B, Z}
Length	4	4
Border	ε	ε
Border length	0	0
Minimal period	4	4
Period set	{4}	{4}
Critical exponent	2	3
Primitive root	BUZZ	BZZZ
Root(w)	BUZZ	BZZZ
Cyclic root	BUZZ	BZZZ
Suffix array	[1, 2, 4, 3]	[1, 4, 3, 2]
Minimal absent words	BB, BZ, UB, UU, ZB, ZU, ZZZ	BB, ZB, ZZZZ
Lyndon factorization	(BUZZ)	(BZZZ)
Biroot	(BUZZ, ε), (ε, BUZZ)	(BZZZ, ε), (ε, BZZZ)
Internal factors	U, Z, UZ	Z, ZZ

Once upon a time, in a deep forest by a wide pond, lived a proud and hungry Bear. He was the strongest of all the animals and liked to take whatever he wanted.
One day the Bear saw a fine fat Duck swimming on the pond. “You will make a good meal,” growled the Bear, and he waded into the water to catch her.
The Duck was not frightened. She swam a little farther out and called, “Mighty Bear, you are strong, but you cannot catch me in the water. Come deeper if you dare!”
The Bear, proud of his strength, went deeper. The mud sucked at his heavy paws. The Duck swam in circles around him, always just out of reach.
“You are slow,” laughed the Duck. “Perhaps you would rather catch fish? Put your head under the water and open your mouth wide. The fish will swim right in!”
The Bear, already angry and stuck in the soft mud, did as she said. He thrust his head under the water. At once the Duck flew up and flapped her wings hard against the water, sending a great splash into the Bear’s face and ears. Blinded and choking, the Bear floundered, lost his footing, and fell headlong into the deepest part of the pond.
When he finally dragged himself out, soaked and coughing, the Duck was already far away on the opposite bank, preening her feathers.
“Remember, great Bear,” she called, “strength is nothing when the ground gives way beneath you and the clever one stays free.”
And from that day the Bear never again tried to catch a Duck on the water.
Moral: The strong may fall where the clever keep their footing.

Canard – Coin-coin

• L(w) = 8
• Bord(w) = 4
• p = 8 − 4 = 4, Root(w) = pref₄(w) = COIN (since 4 ∣ 8)
• Cyclic root = arg min_lex {σ^k(w)} = COINCOIN
• Lyndon factorization: (COIN)(COIN)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 2

Property	COINCOIN
Alphabet	{C, I, N, O}
Length	8
Border	COIN
Border length	4
Minimal period	4
Period set	{4, 8}
Critical exponent	2
Primitive root	COIN
Root(w)	COIN
Cyclic root	COINCOIN
Suffix array	[5, 1, 7, 3, 8, 4, 6, 2]
Minimal absent words	CC, CI, CN, IC, II, IO, NI, NN, NO, OC, ON, OO, NCOINC
Lyndon factorization	(COIN)(COIN)
Biroot	(COIN, COIN), (COINCOIN, ε), (ε, COINCOIN)
Internal factors	C, I, N, O, CO, IN, NC, OI, COI, INC, NCO, OIN, INCO, NCOI, OINC, INCOI, OINCO, OINCOI

Hungarian folk tale: The Bear and the Cat (Magyar népmesék).

Many many years ago, in a land far far away, there lived a bear.
And one day the bear met a cat and said: Let us work together cat, and build a house.
So the two of them worked together and they built a house in the forest.
When the autumn came the leaves began to fall, and the bear said: The winter will soon be here. Let us work together cat and collect some food.
So the two of them bought a bucket, went deep into the forest and filled the bucket with wild honey. Then they got a second bucket and filled it with fat, and carried the buckets up to the attic.
The winter that came was a very bleak one indeed, with plenty of wind, lots of snow and very cold days. So the bear laid down to go to sleep. While the cat watched over the fire and sat comfy in the warmth.
But then the cat slowly became hungry and wondered how he could get out of the house, so he could climb up into the attic to eat from the buckets of honey and fat.
Where are you going, cat?
You will never believe me but a mouse just came past, and invited me to a baptism. The mouse has just had a baby and says it wants me to be the baby’s godfather.
Very well, then you had better go – said the bear.
So then the cat sneaked up to the attic and ate a quarter of the honey, and the quarter of the fat and sneaked back down again.
Oh, you’re home from the baptism! What’s the little one’s name?
A quarter.
How about that? I’ve never heard of a name like that.
Time went by and the cat got hungry again.
Hey cat, where are you going?
Well, you will never believe me but another little mouse came by, and invited me to another baptism.
Very well, then you had better go.
You’re home again soon. That’s right. – said the cat.
And what’s the little one’s name?
A half.
After a few days the cat got hungry again and quietly opened the door.
Where are you going, cat?
You will never believe me but I’ve been invited to another baptism; another baby mouse was born, and I’m to be its godfather.
Very well, then you had better go.
So what’s the little one’s name?
Nothing.
Oh my, that’s the strangest name of all.
When the spring came at last, the bear awoke from its winter sleep thin and hungry, as it had not eaten since the autumn. But everything is fine because we have plenty of honey and fat in the attic. Cat, please fetch some and we’ll make delicious pancakes with honey. Go up to the attic, cat, and fetch the two buckets down.
Oh, but I’m so dizzy! So very very dizzy! And we all know that cats don’t get dizzy, don’t we? But the cat was pretending.
Very well, I’ll go and get it myself – said the bear.
So the bear went up to the attic and found both buckets empty.
I’m going to teach that bad cat a lesson! Cat, you ate everything up and that is why you said a quarter, a half and nothing.
The cat denied the whole thing and said to the bear: Let’s make a test! We’ll both lie in the Sun for a while, and whoever ate the honey and the fat will sweat it out, and then we’ll know.
So they laid down in the Sun and the cat ran up to the attic and scraped what was left of the fat and the honey from the buckets and smeared it on the bear’s fur.
Then the bear woke up and looked at its sticky fur and said: Oh dear! I must have eaten it all! But when did I do it?
When the cat saw the bear it couldn’t stop laughing. And it laughed, and it laughed, and it laughed and it ran off into the forest.
And the poor foolish bear lived unhappily ever after.

Chat – Miaou, Maou (miaulement), Ron-ron (ronronnement), Ksss, Sshhh (chat qui souffle), Lap-lap (chat qui boit du lait/de l’eau)

General formulas

• L(w) = ∣w∣ = n = symbol count after tokenisation
• Bord(w) = max_k<n k·δ(pref_k, suff_k)
• p = n − Bord(w), Root(w) = pref_p(w) if p ∣ n, else w
• Cyclic root = arg min_lex {σ^k(w)}
• Lyndon factorization via Duval (non-increasing Lyndon words)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε}

Miaou

• L(w) = 5
• Bord(w) = 0
• p = 5 − 0 = 5, Root(w) = MIAOU
• Cyclic root = arg min_lex {σ^k(w)} = AOUMI
• Lyndon factorization = (M)(I)(AOU)
• ce(w) = 1

Maou

• L(w) = 4
• Bord(w) = 0
• p = 4 − 0 = 4, Root(w) = MAOU
• Cyclic root = arg min_lex {σ^k(w)} = AOUM
• Lyndon factorization = (M)(AOU)
• ce(w) = 1

Ron-ron (tokenised Ronron)

• L(w) = 6
• Bord(w) = 3
• p = 6 − 3 = 3, Root(w) = RON (since p ∣ n)
• Cyclic root = arg min_lex {σ^k(w)} = NRONRO
• Lyndon factorization = (R)(O)(NRO)(N)
• ce(w) = 2

Ksss

• L(w) = 4
• Bord(w) = 0
• p = 4 − 0 = 4, Root(w) = KSSS
• Cyclic root = arg min_lex {σ^k(w)} = KSSS
• Lyndon factorization = (KSSS)
• ce(w) = 3

Sshhh

• L(w) = 5
• Bord(w) = 0
• p = 5 − 0 = 5, Root(w) = SSHHH
• Cyclic root = arg min_lex {σ^k(w)} = HHHSS
• Lyndon factorization = (S)(S)(H)(H)(H)
• ce(w) = 3

Lap-lap (tokenised Laplap)

• L(w) = 6
• Bord(w) = 3
• p = 6 − 3 = 3, Root(w) = LAP (since p ∣ n)
• Cyclic root = arg min_lex {σ^k(w)} = APLAPL
• Lyndon factorization = (L)(APL)(AP)
• ce(w) = 2

Property	MIAOU	MAOU	RONRON	KSSS	SSHHH	LAPLAP
Alphabet	{A, I, M, O, U}	{A, M, O, U}	{N, O, R}	{K, S}	{H, S}	{A, L, P}
Length	5	4	6	4	5	6
Border	ε	ε	RON	ε	ε	LAP
Border length	0	0	3	0	0	3
Minimal period	5	4	3	4	5	3
Period set	{5}	{4}	{3, 6}	{4}	{5}	{3, 6}
Critical exponent	1	1	2	3	3	2
Primitive root	MIAOU	MAOU	RON	KSSS	SSHHH	LAP
Root(w)	MIAOU	MAOU	RON	KSSS	SSHHH	LAP
Cyclic root	AOUMI	AOUM	NRONRO	KSSS	HHHSS	APLAPL
Suffix array	[3, 2, 1, 4, 5]	[2, 1, 3, 4]	[6, 3, 5, 2, 4, 1]	[1, 4, 3, 2]	[5, 4, 3, 2, 1]	[5, 2, 4, 1, 6, 3]
Minimal absent words	AA, AI, AM, AU, II, IM, IO, IU, MA, MM, MO, MU, OA, OI, OM, OO, UA, UI, UM, UO, UU	AA, AM, AU, MM, MO, MU, OA, OM, OO, UA, UM, UO, UU	NN, NO, OO, OR, RN, RR, NRONR	KK, SK, SSSS	HS, SSS, HHHH	AA, AL, LL, LP, PA, PP, PLAPL
Lyndon factorization	(M)(I)(AOU)	(M)(AOU)	(R)(O)(NRO)(N)	(KSSS)	(S)(S)(H)(H)(H)	(L)(APL)(AP)
Biroot	(MIAOU, ε), (ε, MIAOU)	(MAOU, ε), (ε, MAOU)	(RON, RON), (RONRON, ε), (ε, RONRON)	(KSSS, ε), (ε, KSSS)	(SSHHH, ε), (ε, SSHHH)	(LAP, LAP), (LAPLAP, ε), (ε, LAPLAP)
Internal factors	A, I, O, AO, IA, IAO	A, O, AO	N, O, R, NR, ON, RO, NRO, ONR, ONRO	S, SS	H, S, HH, SH, SHH	A, L, P, AP, LA, PL, APL, PLA, APLA

Moldavian (Moldovan) folk tale: The Horse and the Bear.

Once upon a time, there was a miller who kept a horse at his mill. The miller had ridden the horse so much and carried so many loads that the poor animal was reduced to skin and bones. Seeing that the horse was no longer fit for work, the miller tied the reins around its neck and set it free, saying, “Go wherever your eyes take you.” The horse wandered down the road and eventually reached a forest. There, it found lush grass, cool shade, and clear spring water. It grazed, rested, and regained its strength: its coat became glossy, its sides rounded out, and its ribs no longer showed through its skin. The horse lived happily and freely, even frolicking in the mornings and evenings, prancing, rearing up, and rolling in the green grass. But one day, hard times came for the horse too—a bear began following it, always trying to catch and eat it. The horse neighed:
“Get that foolish idea out of your head, bear, or you might curse the moment we met.”
“What, do you think my teeth have worn down, and I can’t skin you and eat your flesh?”
“Are you really that strong?”
“Of course I am.”
“Then let’s test our strength. Show me what you’re made of,” said the horse. The bear looked around, saw a fallen rock, went up to it, and squeezed it in its embrace—crumbling it into sand and scattering it in the wind.
“Is that all?” asked the horse. “When I strike a stone with my hoof, I don’t just make dust—I create fire.”
“I see you’re quite the braggart,” the bear growled angrily.
“Then let’s go out to the road, and you’ll see for yourself.”
They went. The bear stood on its hind legs, watching as the horse backed away, then charged forward like a whirlwind, sparks flying from the stones beneath its hooves.
The bear sat down in awe and fear.
“Well, what do you say?”
“Indeed, you have far more strength than I do,” replied the bear.
“How should I punish you for daring to bare your teeth at me? Killing you wouldn’t even be enough!”
“Don’t kill me, noble horse. I’ll serve you in any way you wish, just spare my life.”
“Very well, as punishment, you will lead me by the reins and find the tastiest grass for me until I fall asleep.”
The bear didn’t dare object, fearing the horse might raise its hoof and trample him. So the bear began grazing the horse, waiting for it to fall asleep. But the cursed horse wouldn’t sleep! The bear grew exhausted, grinding its teeth and cursing under its breath, careful not to let the horse hear. One day, the bear gathered its courage and timidly said to the horse:
“Your equine grace, perhaps it’s time for you to rest.”
“I already rest too much.”
“When do you rest? I’ve never seen you lie down, close your eyes, or even doze off.”
“Don’t you know that a horse sleeps with one ear and listens with the other?”
“But when do you sleep?”
“Well, I sleep when I’m not grazing, just standing and shifting from one leg to the other.”
The bear took note of these words and began observing when the horse grazed, drank water, or stood still. One day, the bear noticed the horse standing motionless, staring at one spot and shifting its weight. It tried to let go of the reins. The horse stood as if nothing had happened. The bear took a step back, trembling, then another, sweating profusely. The horse didn’t move. So the bear threw the reins over its shoulders and bolted. It ran as fast as it could, not caring where it was going, until it ran into a wolf.
“Why are you running like that, friend, as if you’re fleeing death?”
“Because I am.”
“Really?”
“Truly! There’s a horse sleeping nearby.”
“A horse?!”
“Yes.”
“So what? Surely you’re not afraid of it.”
“That’s exactly who I’m afraid of. If it wakes up and kicks, it’ll be the end of me.”
“You’re such a coward and a fool, bear! Where’s this horse?”
“Look over there, where I came from.”
“I don’t see anything.”
“Climb higher.”
“Still nothing.”
The bear turned around, stood on its tiptoes, but still saw nothing. The horse was hidden behind a hill.
“Where is it?” the wolf couldn’t resist asking.
“Let me lift you up, then you’ll see.”
As the bear wrapped its powerful paws around the wolf to lift it higher, the wolf just opened its mouth and died on the spot, without even a squeak. But the bear didn’t even realize it had squeezed the life out of the wolf. Seeing the wolf dead, the bear sighed and said:
“Ah, poor thing, you only saw the horse and died. Imagine what it’s like for me, having to graze it all this time?”
And, overcome with terror, the bear took off running even faster than before—perhaps it’s still running to this day, never looking back.

Cheval – Tagada-tagada, Cataclop cataclop (bruit des sabots), Hiiii (qui hennit)

General formulas

• L(w) = ∣w∣ = n = symbol count after tokenisation
• Bord(w) = max_k<n k·δ(pref_k, suff_k)
• p = n − Bord(w), Root(w) = pref_p(w) if p ∣ n, else w
• Cyclic root = arg min_lex {σ^k(w)}
• Lyndon factorization via Duval (non-increasing Lyndon words)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε}

Tagada-tagada (tokenised Tagadatagada)

• L(w) = 12
• Bord(w) = 6
• p = 12 − 6 = 6, Root(w) = TAGADA (since p ∣ n)
• Cyclic root = arg min_lex {σ^k(w)} = ADATAGADATAG
• Lyndon factorization = (T)(AG)(ADATAG)(AD)(A)
• ce(w) = 2

Cataclop cataclop (tokenised Cataclopcataclop)

• L(w) = 16
• Bord(w) = 8
• p = 16 − 8 = 8, Root(w) = CATACLOP (since p ∣ n)
• Cyclic root = arg min_lex {σ^k(w)} = ACLOPCATACLOPCAT
• Lyndon factorization = (C)(AT)(ACLOPCAT)(ACLOP)
• ce(w) = 2

Hiiii

• L(w) = 5
• Bord(w) = 0
• p = 5 − 0 = 5, Root(w) = HIIII
• Cyclic root = arg min_lex {σ^k(w)} = HIIII
• Lyndon factorization = (HIIII)
• ce(w) = 4

Property	TAGADATAGADA	CATACLOPCATACLOP	HIIII
Alphabet	{A, D, G, T}	{A, C, L, O, P, T}	{H, I}
Length	12	16	5
Border	TAGADA	CATACLOP	ε
Border length	6	8	0
Minimal period	6	8	5
Period set	{6, 12}	{8, 16}	{5}
Critical exponent	2	2	4
Primitive root	TAGADA	CATACLOP	HIIII
Root(w)	TAGADA	CATACLOP	HIIII
Cyclic root	ADATAGADATAG	ACLOPCATACLOPCAT	HIIII
Suffix array	[12, 10, 4, 8, 2, 6, 11, 5, 9, 3, 7, 1]	[12, 4, 10, 2, 9, 1, 13, 5, 14, 6, 15, 7, 16, 8, 11, 3]	[1, 5, 4, 3, 2]
Minimal absent words	AA, DD, DG, DT, GD, GG, GT, TD, TG, TT, DAD, DAG, GAG, GAT, TAD, TAT, ATAGADAT	AA, AL, AO, AP, CC, CO, CP, CT, LA, LC, LL, LP, LT, OA, OC, OL, OO, OT, PA, PL, PO, PP, PT, TC, TL, TO, TP, TT, ACA, CAC, PCL, TAT, PCATACLOPC	HH, IH, IIIII
Lyndon factorization	(T)(AG)(ADATAG)(AD)(A)	(C)(AT)(ACLOPCAT)(ACLOP)	(HIIII)
Biroot	(TAGADA, TAGADA), (TAGADATAGADA, ε), (ε, TAGADATAGADA)	(CATACLOP, CATACLOP), (CATACLOPCATACLOP, ε), (ε, CATACLOPCATACLOP)	(HIIII, ε), (ε, HIIII)
Internal factors	A, D, G, T, AD, AG, AT, DA, GA, TA, ADA, AGA, ATA, DAT, GAD, TAG, ADAT, AGAD, ATAG, DATA, GADA, TAGA, ADATA, AGADA, ATAGA, DATAG, GADAT, TAGAD, ADATAG, AGADAT, ATAGAD, DATAGA, GADATA, ADATAGA, AGADATA, DATAGAD, GADATAG, ADATAGAD, AGADATAG, GADATAGA, AGADATAGA, GADATAGAD, AGADATAGAD	A, C, L, O, P, T, AC, AT, CA, CL, LO, OP, PC, TA, ACL, ATA, CAT, CLO, LOP, OPC, PCA, TAC, ACLO, ATAC, CATA, CLOP, LOPC, OPCA, PCAT, TACL, ACLOP, ATACL, CATAC, CLOPC, LOPCA, OPCAT, PCATA, TACLO, ACLOPC, ATACLO, CATACL, CLOPCA, LOPCAT, OPCATA, PCATAC, TACLOP, ACLOPCA, ATACLOP, CATACLO, CLOPCAT, LOPCATA, OPCATAC, PCATACL, TACLOPC, ACLOPCAT, ATACLOPC, CLOPCATA, LOPCATAC, OPCATACL, PCATACLO, TACLOPCA, ACLOPCATA, ATACLOPCA, CLOPCATAC, LOPCATACL, OPCATACLO, TACLOPCAT, ACLOPCATAC, ATACLOPCAT, CLOPCATACL, LOPCATACLO, TACLOPCATA, ACLOPCATACL, ATACLOPCATA, CLOPCATACLO, TACLOPCATAC, ACLOPCATACLO, ATACLOPCATAC, TACLOPCATACL, ATACLOPCATACL, TACLOPCATACLO, ATACLOPCATACLO	I, II, III

The Tale
Once upon a time a proud Bear lived in the high mountains. He liked to come down into the lower pastures and take whatever he wanted from the smaller animals.
One day he met a sturdy white Chèvre grazing on a steep rocky slope. “You will make a fine meal,” growled the Bear, and he charged up the path to catch her.
The Chèvre did not run. She stood her ground, lowered her head, and waited. When the Bear came close, she suddenly leapt sideways and butted him hard under the chin with her horns. The Bear staggered.
Before he could recover, the Chèvre turned and kicked with both hind legs, striking him full in the belly. The Bear rolled down the slope, roaring, and crashed into a pile of loose stones.
He tried to climb back up, but the Chèvre met him again. She lowered her horns once more and drove him back with a second strong butt, then another kick that sent him tumbling farther down.
At last the Bear gave up. Bruised and angry, he limped away into the forest, muttering that he would never again try to catch a goat on the rocks.
The Chèvre went back to her grass, and from that day the other animals of the pasture said that even a Bear must respect a goat who knows how to use her horns and her hooves.
Moral: Strength on the wrong ground is no strength at all.

Chèvre – Bê, Mê (bêlement)

General formulas

• L(w) = ∣w∣ = n = symbol count after tokenisation
• Bord(w) = max_k<n k·δ(pref_k, suff_k)
• p = n − Bord(w), Root(w) = pref_p(w) if p ∣ n, else w
• Cyclic root = arg min_lex {σ^k(w)}
• Lyndon factorization via Duval (non-increasing Lyndon words)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε}

Bê

• L(w) = 2
• Bord(w) = 0
• p = 2 − 0 = 2, Root(w) = BE
• Cyclic root = arg min_lex {σ^k(w)} = BE
• Lyndon factorization = (BE)
• ce(w) = 1

Mê

• L(w) = 2
• Bord(w) = 0
• p = 2 − 0 = 2, Root(w) = ME
• Cyclic root = arg min_lex {σ^k(w)} = EM
• Lyndon factorization = (M)(E)
• ce(w) = 1

Property	BE	ME
Alphabet	{B, E}	{E, M}
Length	2	2
Border	ε	ε
Border length	0	0
Minimal period	2	2
Period set	{2}	{2}
Critical exponent	1	1
Primitive root	BE	ME
Root(w)	BE	ME
Cyclic root	BE	EM
Suffix array	[1, 2]	[2, 1]
Minimal absent words	BB, EB, EE	EE, EM, MM
Lyndon factorization	(BE)	(M)(E)
Biroot	(BE, ε), (ε, BE)	(ME, ε), (ε, ME)
Internal factors	none (∅)	none (∅)

Once upon a time a large Bear lived in the forest and took what he wanted from the smaller animals. One day he came upon a farmyard where a lean but quick Chien was guarding the chickens.
“Go away, little dog,” growled the Bear. “I will eat what I please.”
The Chien did not run. He barked sharply, then darted between the Bear’s legs and bit hard at one hind paw. The Bear turned to swat him, but the Chien had already leapt aside. He circled, barked again, and nipped the other leg.
The Bear grew angry and charged. The Chien waited until the last moment, then sprang straight at the Bear’s face and clamped his jaws on the soft nose. The Bear roared, shook his head, and staggered backward, unable to see or strike cleanly.
The Chien held on for a moment, then dropped, dashed away, and barked from a distance. The Bear, bloodied and confused, turned and lumbered back into the trees.
From that day the Bear stayed clear of the farmyard, and the other animals said that a small dog who is fast and knows where to bite can drive even a Bear away.
Moral: Speed and a well-placed bite can overcome size.

Chien – Ouaf, Aou, Ahouahou, Wouf, Wuf wuf, Wouaf Wouaf, Wuah, Whaf, Whouaf (aboiement), Slurp (chien qui lèche), Grrr (chien qui grogne)

General formulas

• L(w) = ∣w∣ = n = symbol count after tokenisation
• Bord(w) = max_k<n k·δ(pref_k, suff_k)
• p = n − Bord(w), Root(w) = pref_p(w) if p ∣ n, else w
• Cyclic root = arg min_lex {σ^k(w)}
• Lyndon factorization via Duval (non-increasing Lyndon words)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε}

Ouaf

• L(w) = 4
• Bord(w) = 0
• p = 4 − 0 = 4, Root(w) = OUAF
• Cyclic root = arg min_lex {σ^k(w)} = AFOU
• Lyndon factorization = (OU)(AF)
• ce(w) = 1

Aou

• L(w) = 3
• Bord(w) = 0
• p = 3 − 0 = 3, Root(w) = AOU
• Cyclic root = arg min_lex {σ^k(w)} = AOU
• Lyndon factorization = (AOU)
• ce(w) = 1

Ahouahou

• L(w) = 8
• Bord(w) = 4
• p = 8 − 4 = 4, Root(w) = AHOU (since p ∣ n)
• Cyclic root = arg min_lex {σ^k(w)} = AHOUAHOU
• Lyndon factorization = (AHOU)(AHOU)
• ce(w) = 2

Wouf

• L(w) = 4
• Bord(w) = 0
• p = 4 − 0 = 4, Root(w) = WOUF
• Cyclic root = arg min_lex {σ^k(w)} = FWOU
• Lyndon factorization = (W)(OU)(F)
• ce(w) = 1

Wuf wuf (tokenised Wufwuf)

• L(w) = 6
• Bord(w) = 3
• p = 6 − 3 = 3, Root(w) = WUF (since p ∣ n)
• Cyclic root = arg min_lex {σ^k(w)} = FWUFWU
• Lyndon factorization = (W)(U)(FWU)(F)
• ce(w) = 2

Wouaf Wouaf (tokenised Wouafwouaf)

• L(w) = 10
• Bord(w) = 5
• p = 10 − 5 = 5, Root(w) = WOUAF (since p ∣ n)
• Cyclic root = arg min_lex {σ^k(w)} = AFWOUAFWOU
• Lyndon factorization = (W)(OU)(AFWOU)(AF)
• ce(w) = 2

Wuah

• L(w) = 4
• Bord(w) = 0
• p = 4 − 0 = 4, Root(w) = WUAH
• Cyclic root = arg min_lex {σ^k(w)} = AHWU
• Lyndon factorization = (W)(U)(AH)
• ce(w) = 1

Whaf

• L(w) = 4
• Bord(w) = 0
• p = 4 − 0 = 4, Root(w) = WHAF
• Cyclic root = arg min_lex {σ^k(w)} = AFWH
• Lyndon factorization = (W)(H)(AF)
• ce(w) = 1

Whouaf

• L(w) = 6
• Bord(w) = 0
• p = 6 − 0 = 6, Root(w) = WHOUAF
• Cyclic root = arg min_lex {σ^k(w)} = AFWHOU
• Lyndon factorization = (W)(HOU)(AF)
• ce(w) = 1

Slurp

• L(w) = 5
• Bord(w) = 0
• p = 5 − 0 = 5, Root(w) = SLURP
• Cyclic root = arg min_lex {σ^k(w)} = LURPS
• Lyndon factorization = (S)(LURP)
• ce(w) = 1

Grrr

• L(w) = 4
• Bord(w) = 0
• p = 4 − 0 = 4, Root(w) = GRRR
• Cyclic root = arg min_lex {σ^k(w)} = GRRR
• Lyndon factorization = (GRRR)
• ce(w) = 3

Property	OUAF	AOU	AHOUAHOU	WOUF	WUFWUF	WOUAFWOUAF	WUAH	WHAF	WHOUAF	SLURP	GRRR
Alphabet	{A, F, O, U}	{A, O, U}	{A, H, O, U}	{F, O, U, W}	{F, U, W}	{A, F, O, U, W}	{A, H, U, W}	{A, F, H, W}	{A, F, H, O, U, W}	{L, P, R, S, U}	{G, R}
Length	4	3	8	4	6	10	4	4	6	5	4
Border	ε	ε	AHOU	ε	WUF	WOUAF	ε	ε	ε	ε	ε
Border length	0	0	4	0	3	5	0	0	0	0	0
Minimal period	4	3	4	4	3	5	4	4	6	5	4
Period set	{4}	{3}	{4, 8}	{4}	{3, 6}	{5, 10}	{4}	{4}	{6}	{5}	{4}
Critical exponent	1	1	2	1	2	2	1	1	1	1	3
Primitive root	OUAF	AOU	AHOU	WOUF	WUF	WOUAF	WUAH	WHAF	WHOUAF	SLURP	GRRR
Root(w)	OUAF	AOU	AHOU	WOUF	WUF	WOUAF	WUAH	WHAF	WHOUAF	SLURP	GRRR
Cyclic root	AFOU	AOU	AHOUAHOU	FWOU	FWUFWU	AFWOUAFWOU	AHWU	AFWH	AFWHOU	LURPS	GRRR
Suffix array	[3, 4, 1, 2]	[1, 2, 3]	[5, 1, 6, 2, 7, 3, 8, 4]	[4, 2, 3, 1]	[6, 3, 5, 2, 4, 1]	[9, 4, 10, 5, 7, 2, 8, 3, 6, 1]	[3, 4, 2, 1]	[3, 4, 2, 1]	[5, 6, 2, 3, 4, 1]	[2, 5, 4, 1, 3]	[1, 4, 3, 2]
Minimal absent words	AA, AO, AU, FA, FF, FO, FU, OA, OF, OO, UF, UO, UU	AA, AU, OA, OO, UA, UO, UU	AA, AO, AU, HA, HH, HU, OA, OH, OO, UH, UO, UU, UAHOUA	FF, FO, FU, FW, OF, OO, OW, UO, UU, UW, WF, WU, WW	FF, FU, UU, UW, WF, WW, FWUFW	AA, AO, AU, AW, FA, FF, FO, FU, OA, OF, OO, OW, UF, UO, UU, UW, WA, WF, WU, WW, FWOUAFW	AA, AU, AW, HA, HH, HU, HW, UH, UU, UW, WA, WH, WW	AA, AH, AW, FA, FF, FH, FW, HF, HH, HW, WA, WF, WW	AA, AH, AO, AU, AW, FA, FF, FH, FO, FU, FW, HA, HF, HH, HU, HW, OA, OF, OH, OO, OW, UF, UH, UO, UU, UW, WA, WF, WO, WU, WW	LL, LP, LR, LS, PL, PP, PR, PS, PU, RL, RR, RS, RU, SP, SR, SS, SU, UL, UP, US, UU	GG, RG, RRRR
Lyndon factorization	(OU)(AF)	(AOU)	(AHOU)(AHOU)	(W)(OU)(F)	(W)(U)(FWU)(F)	(W)(OU)(AFWOU)(AF)	(W)(U)(AH)	(W)(H)(AF)	(W)(HOU)(AF)	(S)(LURP)	(GRRR)
Biroot	(OUAF, ε), (ε, OUAF)	(AOU, ε), (ε, AOU)	(AHOU, AHOU), (AHOUAHOU, ε), (ε, AHOUAHOU)	(WOUF, ε), (ε, WOUF)	(WUF, WUF), (WUFWUF, ε), (ε, WUFWUF)	(WOUAF, WOUAF), (WOUAFWOUAF, ε), (ε, WOUAFWOUAF)	(WUAH, ε), (ε, WUAH)	(WHAF, ε), (ε, WHAF)	(WHOUAF, ε), (ε, WHOUAF)	(SLURP, ε), (ε, SLURP)	(GRRR, ε), (ε, GRRR)
Internal factors	A, U, UA	O	A, H, O, U, AH, HO, OU, UA, AHO, HOU, OUA, UAH, HOUA, OUAH, UAHO, HOUAH, OUAHO, HOUAHO	O, U, OU	F, U, W, FW, UF, WU, FWU, UFW, UFWU	A, F, O, U, W, AF, FW, OU, UA, WO, AFW, FWO, OUA, UAF, WOU, AFWO, FWOU, OUAF, UAFW, WOUA, AFWOU, FWOUA, OUAFW, UAFWO, AFWOUA, OUAFWO, UAFWOU, OUAFWOU, UAFWOUA, OUAFWOUA	A, U, UA	A, H, HA	A, H, O, U, HO, OU, UA, HOU, OUA, HOUA	L, R, U, LU, UR, LUR	R, RR

Once upon a time a large Bear ruled the forest and took what he wanted. One night he came upon a tall oak where a Chouette was resting on a high branch.

“Come down, little bird,” growled the Bear. “I am hungry.”

The Chouette did not move. She waited until the Bear stood directly under the tree, then called out in a sharp voice: “Look behind you, Bear! The hunters are coming with dogs and spears!”

The Bear turned in fear. While he was looking, the Chouette dropped a heavy pine cone straight onto his head. The Bear roared and staggered.

“Again!” cried the Chouette, and she dropped a second cone, then a third. The Bear, dizzy and bleeding, stumbled away into the dark, convinced that more danger waited above.

From that day the Bear avoided the tall oaks at night, and the other animals said that a Chouette who knows how to use height and noise can drive even a Bear away.

Moral: A high place and a well-timed blow can overcome size.

Chouette – Hou-hou, Uh-uh (hululement)

General formulas

• L(w) = ∣w∣ = n = symbol count after tokenisation
• Bord(w) = max_k<n k·δ(pref_k, suff_k)
• p = n − Bord(w), Root(w) = pref_p(w) if p ∣ n, else w
• Cyclic root = arg min_lex {σ^k(w)}
• Lyndon factorization via Duval (non-increasing Lyndon words)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε}

Hou-hou (tokenised Houhou)

• L(w) = 6
• Bord(w) = 3
• p = 6 − 3 = 3, Root(w) = HOU (since p ∣ n)
• Cyclic root = arg min_lex {σ^k(w)} = HOUHOU
• Lyndon factorization = (HOU)(HOU)
• ce(w) = 2

Uh-uh (tokenised Uhuh)

• L(w) = 4
• Bord(w) = 2
• p = 4 − 2 = 2, Root(w) = UH (since p ∣ n)
• Cyclic root = arg min_lex {σ^k(w)} = HUHU
• Lyndon factorization = (U)(HU)(H)
• ce(w) = 2

Property	HOUHOU	UHUH
Alphabet	{H, O, U}	{H, U}
Length	6	4
Border	HOU	UH
Border length	3	2
Minimal period	3	2
Period set	{3, 6}	{2, 4}
Critical exponent	2	2
Primitive root	HOU	UH
Root(w)	HOU	UH
Cyclic root	HOUHOU	HUHU
Suffix array	[4, 1, 5, 2, 6, 3]	[4, 2, 3, 1]
Minimal absent words	HH, HU, OH, OO, UO, UU, UHOUH	HH, UU, HUHU
Lyndon factorization	(HOU)(HOU)	(U)(HU)(H)
Biroot	(HOU, HOU), (HOUHOU, ε), (ε, HOUHOU)	(UH, UH), (UHUH, ε), (ε, UHUH)
Internal factors	H, O, U, HO, OU, UH, OUH, UHO, OUHO	H, U, HU

Once upon a time a large Bear lived in the woods and took what he wanted. One summer day he heard a loud, continuous song coming from a high tree.
“Who is making that noise?” he growled, and he climbed toward the sound.
A Cigale was singing on a thin branch far above. When the Bear drew near, the Cigale stopped for a moment, then began to sing even louder, right into the Bear’s ear.
The sound was sharp and endless. The Bear shook his head and tried to reach the Cigale, but the branch was too thin. He slipped, then tried again. Still the song went on, high and piercing.
At last the Bear could bear it no longer. He climbed down, covered his ears, and lumbered away into the forest, muttering that he would never again climb a tree where a Cigale was singing.
From that day the other animals said that even a Bear must yield to a small singer who knows how to use a high place and a loud, unending voice.
Moral: A persistent sound in the right place can drive away strength.

Cigale – Crii crii crii

• L(w) = 12
• Bord(w) = 8
• p = 12 − 8 = 4, Root(w) = pref₄(w) = CRII (since 4 ∣ 12)
• Cyclic root = arg min_lex {σ^k(w)} = CRIICRIICRII
• Lyndon factorization = (CRII)(CRII)(CRII)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 3

Property	CRIICRIICRII
Alphabet	{C, I, R}
Length	12
Border	CRIICRII
Border length	8
Minimal period	4
Period set	{4, 8, 12}
Critical exponent	3
Primitive root	CRII
Root(w)	CRII
Cyclic root	CRIICRIICRII
Suffix array	[9, 5, 1, 12, 8, 4, 11, 7, 3, 10, 6, 2]
Minimal absent words	CC, CI, IR, RC, RR, III, RIC, ICRIICRIIC
Lyndon factorization	(CRII)(CRII)(CRII)
Biroot	(CRII, CRII), (CRIICRII, CRII), (CRIICRIICRII, ε), (ε, CRIICRIICRII)
Internal factors	C, I, R, CR, IC, II, RI, CRI, ICR, IIC, RII, CRII, ICRI, IICR, RIIC, CRIIC, ICRII, IICRI, RIICR, CRIICR, ICRIIC, IICRII, RIICRI, CRIICRI, ICRIICR, IICRIIC, RIICRII, ICRIICRI, IICRIICR, RIICRIIC, IICRIICRI, RIICRIICR, RIICRIICRI

The Tale
Once upon a time a large Bear lived in the forest and took what he wanted. One day he came upon a muddy wallow where a sturdy Cochon was resting.
“Move, little pig,” growled the Bear. “I want this place.”
The Cochon did not move. He stood up, lowered his head, and waited. When the Bear came close, the Cochon charged with his tusks and struck hard under the Bear’s belly. The Bear roared and staggered.
Before the Bear could recover, the Cochon turned and kicked with both hind legs, sending mud and the Bear sliding backward into the soft ground. The Bear tried to rise, but the Cochon charged again, drove him deeper into the mud, and held him there with his weight until the Bear stopped struggling.
At last the Bear crawled out, muddy and bruised, and lumbered away into the trees, muttering that he would never again try to take a wallow from a Cochon.
From that day the other animals said that a Cochon who knows how to use his tusks and the soft ground can drive even a Bear away.
Moral: Strength on the wrong ground is no strength at all.

Cochon – Groin-groin, Gruîîî, Roiiirk, Gruik-gruik, Grouik grouik (grognement)

• $L(w)$L(w) = symbol count after tokenisation
• $\mathrm{B}\mathrm{o}\mathrm{r}\mathrm{d}(w)={\max }_{k<n}k\cdot \delta ({\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{f}}_k,{\mathrm{s}\mathrm{u}\mathrm{f}\mathrm{f}}_k)$Bord(w)=maxk<n​k⋅δ(prefk​,suffk​)
• $p=n-\mathrm{B}\mathrm{o}\mathrm{r}\mathrm{d}(w)$p=n−Bord(w), $\mathrm{R}\mathrm{o}\mathrm{o}\mathrm{t}(w)={\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{f}}_p(w)$Root(w)=prefp​(w) if $p\mid n$p∣n, else $w$w
• Cyclic root $=\arg {\min }_{\mathrm{l}\mathrm{e}\mathrm{x}}\{{\sigma }_k(w)\}$=argminlex​{σk​(w)}
• Lyndon factorization via Duval (non-increasing Lyndon words)

Groin-groin (tokenised Groingroin)

• $L(w)=$L(w)= symbol count after tokenisation $=10$=10
• $\mathrm{B}\mathrm{o}\mathrm{r}\mathrm{d}(w)={\max }_{k<n}k\cdot \delta ({\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{f}}_k,{\mathrm{s}\mathrm{u}\mathrm{f}\mathrm{f}}_k)=5$Bord(w)=maxk<n​k⋅δ(prefk​,suffk​)=5
• $p=n-\mathrm{B}\mathrm{o}\mathrm{r}\mathrm{d}(w)=5$p=n−Bord(w)=5, $\mathrm{R}\mathrm{o}\mathrm{o}\mathrm{t}(w)={\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{f}}_p(w)$Root(w)=prefp​(w) if $p\mid n$p∣n, else $w$w $\to$→ Groin
• Cyclic root $=\arg {\min }_{\mathrm{l}\mathrm{e}\mathrm{x}}\{{\sigma }^k(w)\}=$=argminlex​{σk(w)}= Groingroin
• Lyndon factorization via Duval (non-increasing Lyndon words) $=$= (Groin)(Groin)

Gruîîî

• $L(w)=$L(w)= symbol count after tokenisation $=6$=6
• $\mathrm{B}\mathrm{o}\mathrm{r}\mathrm{d}(w)={\max }_{k<n}k\cdot \delta ({\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{f}}_k,{\mathrm{s}\mathrm{u}\mathrm{f}\mathrm{f}}_k)=0$Bord(w)=maxk<n​k⋅δ(prefk​,suffk​)=0
• $p=n-\mathrm{B}\mathrm{o}\mathrm{r}\mathrm{d}(w)=6$p=n−Bord(w)=6, $\mathrm{R}\mathrm{o}\mathrm{o}\mathrm{t}(w)={\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{f}}_p(w)$Root(w)=prefp​(w) if $p\mid n$p∣n, else $w$w $\to$→ Gruîîî
• Cyclic root $=\arg {\min }_{\mathrm{l}\mathrm{e}\mathrm{x}}\{{\sigma }^k(w)\}=$=argminlex​{σk(w)}= Gruîîî
• Lyndon factorization via Duval (non-increasing Lyndon words) $=$= (Gruîîî)

Roiiirk

• $L(w)=$L(w)= symbol count after tokenisation $=7$=7
• $\mathrm{B}\mathrm{o}\mathrm{r}\mathrm{d}(w)={\max }_{k<n}k\cdot \delta ({\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{f}}_k,{\mathrm{s}\mathrm{u}\mathrm{f}\mathrm{f}}_k)=0$Bord(w)=maxk<n​k⋅δ(prefk​,suffk​)=0
• $p=n-\mathrm{B}\mathrm{o}\mathrm{r}\mathrm{d}(w)=7$p=n−Bord(w)=7, $\mathrm{R}\mathrm{o}\mathrm{o}\mathrm{t}(w)={\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{f}}_p(w)$Root(w)=prefp​(w) if $p\mid n$p∣n, else $w$w $\to$→ Roiiirk
• Cyclic root $=\arg {\min }_{\mathrm{l}\mathrm{e}\mathrm{x}}\{{\sigma }^k(w)\}=$=argminlex​{σk(w)}= Roiiirk
• Lyndon factorization via Duval (non-increasing Lyndon words) $=$= (Roiiirk)

Gruik-gruik (tokenised Gruikgruik)

• $L(w)=$L(w)= symbol count after tokenisation $=10$=10
• $\mathrm{B}\mathrm{o}\mathrm{r}\mathrm{d}(w)={\max }_{k<n}k\cdot \delta ({\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{f}}_k,{\mathrm{s}\mathrm{u}\mathrm{f}\mathrm{f}}_k)=5$Bord(w)=maxk<n​k⋅δ(prefk​,suffk​)=5
• $p=n-\mathrm{B}\mathrm{o}\mathrm{r}\mathrm{d}(w)=5$p=n−Bord(w)=5, $\mathrm{R}\mathrm{o}\mathrm{o}\mathrm{t}(w)={\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{f}}_p(w)$Root(w)=prefp​(w) if $p\mid n$p∣n, else $w$w $\to$→ Gruik
• Cyclic root $=\arg {\min }_{\mathrm{l}\mathrm{e}\mathrm{x}}\{{\sigma }^k(w)\}=$=argminlex​{σk(w)}= Gruikgruik
• Lyndon factorization via Duval (non-increasing Lyndon words) $=$= (Gruik)(Gruik)

Grouik grouik (tokenised Grouikgrouik)

• $L(w)=$L(w)= symbol count after tokenisation $=12$=12
• $\mathrm{B}\mathrm{o}\mathrm{r}\mathrm{d}(w)={\max }_{k<n}k\cdot \delta ({\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{f}}_k,{\mathrm{s}\mathrm{u}\mathrm{f}\mathrm{f}}_k)=6$Bord(w)=maxk<n​k⋅δ(prefk​,suffk​)=6
• $p=n-\mathrm{B}\mathrm{o}\mathrm{r}\mathrm{d}(w)=6$p=n−Bord(w)=6, $\mathrm{R}\mathrm{o}\mathrm{o}\mathrm{t}(w)={\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{f}}_p(w)$Root(w)=prefp​(w) if $p\mid n$p∣n, else $w$w $\to$→ Grouik
• Cyclic root $=\arg {\min }_{\mathrm{l}\mathrm{e}\mathrm{x}}\{{\sigma }^k(w)\}=$=argminlex​{σk(w)}= Grouikgrouik
• Lyndon factorization via Duval (non-increasing Lyndon words) $=$= (Grouik)(Grouik)

Property	GROINGROIN	GRUIII	ROIIIRK	GRUIKGRUIK	GROUIKGROUIK
Alphabet	{G, I, N, O, R}	{G, I, R, U}	{I, K, O, R}	{G, I, K, R, U}	{G, I, K, O, R, U}
Length	10	6	7	10	12
Border	GROIN	ε	ε	GRUIK	GROUIK
Border length	5	0	0	5	6
Minimal period	5	6	7	5	6
Period set	{5, 10}	{6}	{7}	{5, 10}	{6, 12}
Critical exponent	2	1	1	2	2
Primitive root	GROIN	GRUIII	ROIIIRK	GRUIK	GROUIK
Root(w)	GROIN	GRUIII	ROIIIRK	GRUIK	GROUIK
Cyclic root	GROINGROIN	GRUIII	IIRKROI	GRUIKGRUIK	GROUIKGROUIK
Suffix array	[6, 1, 7, 2, 8, 3, 9, 4, 10, 5]	[1, 4, 5, 6, 2, 3]	[2, 3, 4, 7, 1, 5, 6]	[6, 1, 7, 2, 8, 3, 9, 4, 10, 5]	[7, 1, 8, 2, 9, 3, 10, 4, 11, 5, 12, 6]
Minimal absent	GG, OO, RR, II, NN, GO, GR, GI, GN	GG, RR, UU, II, GR, GU, GI	OO, RR, II, KK, RO, RI, RK	GG, RR, UU, II, KK, GR, GU, GI, GK	GG, OO, RR, UU, II, KK, GR, GO, GU, GI, GK
Lyndon factorization	(GROIN)(GROIN)	(GRUIII)	(ROIIIRK)	(GRUIK)(GRUIK)	(GROUIK)(GROUIK)
Biroot	(GROIN, GROIN), (GROINGROIN, ε), (ε, GROINGROIN)	(GRUIII, ε), (ε, GRUIII)	(ROIIIRK, ε), (ε, ROIIIRK)	(GRUIK, GRUIK), (GRUIKGRUIK, ε), (ε, GRUIKGRUIK)	(GROUIK, GROUIK), (GROUIKGROUIK, ε), (ε, GROUIKGROUIK)
Internal factors	RO, OI, IN, NG, GRO, ROI, OIN, ING, GROI, ROIN, OING	RU, UI, II, RUI, UII, RIII	OI, II, II, IR, RK, OII, III, IIR, IRK	RU, UI, IK, KG, GRU, RUI, UIK, IKG, GRUI, RUIK	RO, OU, UI, IK, KG, GRO, ROU, OUI, UIK, IKG, GROU, ROUI, OUIK

Once upon a time a large Bear lived in the forest and took what he wanted. One morning he came to a farmyard where a proud Coq was crowing on the fence.

“Be quiet, little bird,” growled the Bear. “I am hungry.”

The Coq did not fly away. He stood tall, flapped his wings, and crowed as loud as he could, right into the Bear’s face. The sound was sharp and sudden. The Bear shook his head and stepped back.

Before the Bear could recover, the Coq flew down, struck with his spurs at the Bear’s nose, and crowed again. The Bear, half-blinded by the noise and the blows, turned and lumbered away into the trees, covering his ears.

From that day the Bear avoided the farmyard at dawn, and the other animals said that a Coq who knows how to use his voice and his spurs can drive even a Bear away.

Moral: A loud cry and a sharp spur in the right moment can overcome size.

Coq – Cocorico (chant du coq)

• L(w) = 8
• Bord(w) = 2
• p = 8 − 2 = 6, Root(w) = w (since 6 ∤ 8)
• Cyclic root = arg min_lex {σ^k(w)} = COCOCORI
• Lyndon factorization = (COCORI)(CO)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 2

Property	COCORICO
Alphabet	{C, I, O, R}
Length	8
Border	CO
Border length	2
Minimal period	6
Period set	{6, 8}
Critical exponent	2
Primitive root	COCORICO
Root(w)	COCORICO
Cyclic root	COCOCORI
Suffix array	[7, 1, 3, 6, 8, 2, 4, 5]
Minimal absent words	CC, CI, CR, II, IO, IR, OI, OO, RC, RO, RR, ICOC, ICOR, OCOC
Lyndon factorization	(COCORI)(CO)
Biroot	(COCORICO, ε), (ε, COCORICO)
Internal factors	C, I, O, R, CO, IC, OC, OR, RI, COR, OCO, ORI, RIC, CORI, OCOR, ORIC, CORIC, OCORI, OCORIC

Once upon a time a large Bear lived deep in the forest and took what he wanted from every animal he met. One autumn day he came upon a tall pine where a black Corbeau was sitting on a high branch, watching him.
“Come down, little bird,” growled the Bear. “I am hungry, and your meat will do.”
The Corbeau did not move. He tilted his head and called out in a harsh voice: “Why climb, Bear? There is better meat waiting for you in the clearing beyond the stream. A whole deer has fallen there, and no one has found it yet.”
The Bear’s eyes brightened. He turned and lumbered toward the stream, sniffing the air. The Corbeau flew ahead, silent, and settled on another tree. When the Bear reached the clearing, he found nothing but dry leaves and a few old bones. He roared in anger and looked up.
“You lied to me!” he shouted.
“I only said there was better meat,” answered the Corbeau from his branch. “I did not say it was still there. But look behind you, Bear — the hunters are coming with dogs.”
The Bear turned. There were no hunters, only the wind in the trees. While he looked, the Corbeau dropped a heavy pine cone onto the Bear’s head, then another, then a third. The cones struck hard. The Bear staggered, shook his head, and tried to climb the tree, but the branches were thin and broke under his weight. He fell back to the ground, bruised and dizzy.
The Corbeau flew to a still higher branch and called again: “Go home, Bear. The next time you come under a tall tree where a Corbeau sits, remember the cones and the empty clearing. A bird that sees farther than you can lead you into trouble before you even begin.”
The Bear, his head ringing, turned and walked away into the forest, muttering that he would never again trust a black bird on a high branch.
From that day the other animals said that a Corbeau who knows how to use words, height, and a well-timed blow can drive even a Bear away without ever leaving his perch.

Moral: A clear view and a careful word can turn strength into a trap.

Corbeau – Croâ

• L(w) = 4
• Bord(w) = 0
• p = 4, Root(w) = w
• Cyclic root = arg min_lex {σ^k(w)} = ACRO
• Lyndon factorization = (CRO)(A)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 1

Property	CROA
Alphabet	{A, C, O, R}
Length	4
Border	ε
Border length	0
Minimal period	4
Period set	{4}
Critical exponent	1
Primitive root	CROA
Root(w)	CROA
Cyclic root	ACRO
Suffix array	[4, 1, 3, 2]
Minimal absent words	AA, AC, AO, AR, CA, CC, CO, OC, OO, OR, RA, RC, RR
Lyndon factorization	(CRO)(A)
Biroot	(CROA, ε), (ε, CROA)
Internal factors	O, R, RO

Once upon a time a large Bear lived in the forest and took what he wanted. One spring morning he heard a clear, repeated call coming from a thicket: “Coucou… coucou… coucou.”

He followed the sound and found a Coucou perched on a low branch, singing the same two notes over and over.

“Stop that noise,” growled the Bear. “I am looking for food.”

The Coucou did not fly away. Instead it answered, “I am counting the days until the hunters come with their dogs. Every time I say coucou, another hunter is on the path. Listen.”

The Coucou began again, louder and faster: “Coucou… coucou… coucou…”

The Bear grew uneasy. He turned his head toward the path, then back to the bird. The sound seemed to come from everywhere at once. He took a step forward, then another step back. Each call made him hesitate.

At last the Coucou flew a short distance and called once more from a new place, as if leading the way deeper into the thicket. The Bear, confused by the constant sound and the thought of dogs, turned and lumbered away in the opposite direction, covering his ears.

From that day the Bear avoided the thickets in spring, and the other animals said that a Coucou who knows how to use a single repeated note at the right moment can drive even a Bear away.

Moral: A steady, well-timed sound can turn strength into hesitation.

Coucou – Coucou

• L(w) = 6
• Bord(w) = 3
• p = 6 − 3 = 3, Root(w) = pref₃(w) = COU (since 3 ∣ 6)
• Cyclic root = arg min_lex {σ^k(w)} = COUCOU
• Lyndon factorization = (COU)(COU)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 2

Property	COUCOU
Alphabet	{C, O, U}
Length	6
Border	COU
Border length	3
Minimal period	3
Period set	{3, 6}
Critical exponent	2
Primitive root	COU
Root(w)	COU
Cyclic root	COUCOU
Suffix array	[4, 1, 5, 2, 6, 3]
Minimal absent words	CC, CU, OC, OO, UO, UU, UCOUC
Lyndon factorization	(COU)(COU)
Biroot	(COU, COU), (COUCOU, ε), (ε, COUCOU)
Internal factors	C, O, U, CO, OU, UC, OUC, UCO, OUCO

Once upon a time a large Bear lived near a wide river and took what he wanted from the bank. One hot day he came down to drink and saw a Crocodile lying half in the water, half on the mud.

“Move,” growled the Bear. “This is my place.”

The Crocodile did not move. It opened one eye and said, “The water is deeper than it looks, Bear. Come closer and see for yourself.”

The Bear stepped into the shallows. The mud was soft under his feet. He took another step, then another. The Crocodile waited, still and silent. When the Bear was deep enough that the water reached his chest, the Crocodile moved for the first time. It slid sideways, turned its body, and struck once with its tail against the Bear’s legs. The Bear lost his footing and sank into the soft bottom.

He tried to rise, but the mud held him. The Crocodile stayed just out of reach, watching. Each time the Bear struggled, the Crocodile shifted the water with its tail, keeping the mud loose and the current against the Bear’s side. The Bear grew tired. He sank lower. The Crocodile moved closer, opened its jaws, and took the Bear by the neck. The Bear thrashed once more, then went still.

The Crocodile dragged him into deeper water and ate him there.

From that day the other animals drank only from the rocks and said that a Crocodile who knows how to wait and use the water can finish even a Bear.

Moral: Strength on the wrong ground is no strength at all.

Crocodile – Clap (claquement de la mâchoire)

• L(w) = 4
• Bord(w) = 0
• p = 4, Root(w) = w
• Cyclic root = arg min_lex {σ^k(w)} = APCL
• Lyndon factorization = (CL)(AP)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 1

Property	CLAP
Alphabet	{A, C, L, P}
Length	4
Border	ε
Border length	0
Minimal period	4
Period set	{4}
Critical exponent	1
Primitive root	CLAP
Root(w)	CLAP
Cyclic root	APCL
Suffix array	[3, 1, 2, 4]
Minimal absent words	AA, AC, AL, CA, CC, CP, LC, LL, LP, PA, PC, PL, PP
Lyndon factorization	(CL)(AP)
Biroot	(CLAP, ε), (ε, CLAP)
Internal factors	A, L, LA

Once upon a time a large Bear lived in the woods and took what he wanted. One autumn day he came upon a clearing where a Dindon was walking, its feathers trailing.

“You look fat,” growled the Bear. “I will eat you.”

The Dindon did not run. It stopped, raised its head, and began to gobble — a deep, rolling sound that filled the clearing. At the same time it spread its tail and lowered its wings until it looked twice its size.

The Bear stopped. He had never seen a bird do this. The noise continued, low and steady, and the feathers kept the Dindon looking large and close. The Bear took one step forward, then hesitated. Each gobble made him pause. The Dindon turned slowly, always facing the Bear, the display never dropping.

At last the Bear grew uneasy. The sound did not stop, and the shape did not shrink. He turned and walked away into the trees, muttering that the bird was not worth the trouble.

From that day the Bear avoided the clearings in autumn, and the other animals said that a Dindon who knows how to use its voice and its feathers together can drive even a Bear away.

Moral: A steady display can turn strength into doubt.

Dindon (et Dinde) – Glouglou

• L(w) = 8
• Bord(w) = 4
• p = 8 − 4 = 4, Root(w) = pref₄(w) = GLOU (since 4 ∣ 8)
• Cyclic root = arg min_lex {σ^k(w)} = GLOUGLOU
• Lyndon factorization = (GLOU)(GLOU)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 2

Property	GLOUGLOU
Alphabet	{G, L, O, U}
Length	8
Border	GLOU
Border length	4
Minimal period	4
Period set	{4, 8}
Critical exponent	2
Primitive root	GLOU
Root(w)	GLOU
Cyclic root	GLOUGLOU
Suffix array	[5, 1, 6, 2, 7, 3, 8, 4]
Minimal absent words	GG, GO, GU, LG, LL, LU, OG, OL, OO, UL, UO, UU, UGLOUG
Lyndon factorization	(GLOU)(GLOU)
Biroot	(GLOU, GLOU), (GLOUGLOU, ε), (ε, GLOUGLOU)
Internal factors	G, L, O, U, GL, LO, OU, UG, GLO, LOU, OUG, UGL, LOUG, OUGL, UGLO, LOUGL, OUGLO, LOUGLO

Once upon a time a large Bear lived on an island and took what he wanted from the shore. One day he came upon a heavy, flightless bird standing on the sand — a Dodo, broad and low to the ground, with a large curved beak.

“You cannot fly,” said the Bear. “That makes you easy.”

The Dodo did not move. It stood where it was, feet planted wide, body low. When the Bear stepped closer, the Dodo lowered its head still further and stayed still. The Bear reached out. The Dodo shifted its weight, just a little, so that its body pressed against the Bear’s foreleg and pinned it against a low rock that stood behind. The rock was smooth on top and steep at the back. The Bear pushed. The Dodo did not push back; it simply stayed, its weight and the rock together holding the leg.

The Bear tried to lift the leg free. Each time he lifted, the Dodo settled lower, and the rock held the other side. The Bear grew tired of the effort. At last he pulled free, stepped back, and walked away along the shore, shaking his leg.

From that day the Bear avoided the low rocks on the beach, and the other animals said that a Dodo who knows how to use its own weight and the ground can stop even a Bear without moving far.

Moral: What cannot run can still hold its place.

Dodo – Blèctre

• L(w) = 7
• Bord(w) = 0
• p = 7, Root(w) = w
• Cyclic root = arg min_lex {σ^k(w)} = BLECTRE
• Lyndon factorization = (BLECTRE)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 5/4

Property	BLECTRE
Alphabet	{B, C, E, L, R, T}
Length	7
Border	ε
Border length	0
Minimal period	7
Period set	{7}
Critical exponent	5/4
Primitive root	BLECTRE
Root(w)	BLECTRE
Cyclic root	BLECTRE
Suffix array	[1, 4, 7, 3, 2, 6, 5]
Minimal absent words	BB, BC, BE, BR, BT, CB, CC, CE, CL, CR, EB, EE, EL, ER, ET, LB, LC, LL, LR, LT, RB, RC, RL, RR, RT, TB, TC, TE, TL, TT, REC
Lyndon factorization	(BLECTRE)
Biroot	(BLECTRE, ε), (ε, BLECTRE)
Internal factors	C, E, L, R, T, CT, EC, LE, TR, CTR, ECT, LEC, ECTR, LECT, LECTR

Once upon a time a large Bear lived near a pond and took what he wanted from the edge. One day he came upon a Grenouille sitting on a lily pad.

“You are small,” said the Bear. “I will eat you.”

The Grenouille did not move. It stayed on the pad and answered, “The water is deeper than it looks, Bear. Step in and see.”

The Bear put one paw into the shallows. The mud was soft. He put the other paw in. The Grenouille waited. When the Bear was deep enough that the water reached his chest, the Grenouille jumped — not away, but onto a floating log that lay between the Bear and the bank. The log shifted under the jump. The Bear turned to reach for it and lost his footing. The log drifted just far enough that the Bear could not catch it, and the soft bottom held him for a moment.

The Bear pulled free and staggered back to the dry edge, shaking water from his fur. The Grenouille stayed on the log, out of reach.

From that day the Bear drank only from the rocks and avoided the soft places by the pond. The other animals said that a Grenouille who knows how to use the water and a single well-timed leap can drive even a Bear away.

Moral: A small leap on the right ground can turn strength into a slip.

Grenouille – Coââ-coââ (coassement), Coa, Croa Croa

General formulas

• L(w) = ∣w∣ = n = symbol count after tokenisation
• Bord(w) = max_k<n k·δ(pref_k, suff_k)
• p = n − Bord(w), Root(w) = pref_p(w) if p ∣ n, else w
• Cyclic root = arg min_lex {σ^k(w)}
• Lyndon factorization via Duval (non-increasing Lyndon words)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε}

Coââ-coââ (tokenised Coââcoââ)

• L(w) = 8
• Bord(w) = 4
• p = 8 − 4 = 4, Root(w) = COAA (since p ∣ n)
• Cyclic root = arg min_lex {σ^k(w)} = AACOAACO
• Lyndon factorization = (CO)(AACO)(A)(A)
• ce(w) = 2

Coa

• L(w) = 3
• Bord(w) = 0
• p = 3 − 0 = 3, Root(w) = COA
• Cyclic root = arg min_lex {σ^k(w)} = ACO
• Lyndon factorization = (CO)(A)
• ce(w) = 1

Croa Croa (tokenised Croacroa)

• L(w) = 8
• Bord(w) = 4
• p = 8 − 4 = 4, Root(w) = CROA (since p ∣ n)
• Cyclic root = arg min_lex {σ^k(w)} = ACROACRO
• Lyndon factorization = (CRO)(ACRO)(A)
• ce(w) = 2

Property	COAACOAA	COA	CROACROA
Alphabet	{A, C, O}	{A, C, O}	{A, C, O, R}
Length	8	3	8
Border	COAA	ε	CROA
Border length	4	0	4
Minimal period	4	3	4
Period set	{4, 8}	{3}	{4, 8}
Critical exponent	2	1	2
Primitive root	COAA	COA	CROA
Root(w)	COAA	COA	CROA
Cyclic root	AACOAACO	ACO	ACROACRO
Suffix array	[8, 7, 3, 4, 5, 1, 6, 2]	[3, 1, 2]	[8, 4, 5, 1, 7, 3, 6, 2]
Minimal absent words	AO, CA, CC, OC, OO, AAA, OAC, ACOAAC	AA, AC, AO, CA, CC, OC, OO	AA, AO, AR, CA, CC, CO, OC, OO, OR, RA, RC, RR, ACROAC
Lyndon factorization	(CO)(AACO)(A)(A)	(CO)(A)	(CRO)(ACRO)(A)
Biroot	(COAA, COAA), (COAACOAA, ε), (ε, COAACOAA)	(COA, ε), (ε, COA)	(CROA, CROA), (CROACROA, ε), (ε, CROACROA)
Internal factors	A, C, O, AA, AC, CO, OA, AAC, ACO, COA, OAA, AACO, ACOA, OAAC, AACOA, OAACO, OAACOA	O	A, C, O, R, AC, CR, OA, RO, ACR, CRO, OAC, ROA, ACRO, OACR, ROAC, OACRO, ROACR, ROACRO

Once upon a time a large Bear lived in the woods and took what he wanted. One night he came to a clearing where a Hibou was perched on a low branch, silent and still.

“You are small and soft,” said the Bear. “I will eat you.”

The Hibou did not move. It stayed on the branch and answered in a low voice, “The dark is deeper than it looks, Bear. Step under the branch and see.”

The Bear stepped under the branch. The ground was uneven with roots. The Hibou waited. When the Bear was directly beneath, the Hibou turned its head once, fixed its eyes on a thick root that crossed the path, and gave a single sharp call. The call was not loud, but it was sudden. The Bear started, shifted his weight, and his foot caught the root. He stumbled. The branch was low enough that, when he rose, the Hibou was already gone into the higher dark, out of reach.

The Bear stood still for a moment, then walked away into the trees, shaking his head.

From that day the Bear avoided the low branches at night, and the other animals said that a Hibou who knows how to use silence and one well-timed call can drive even a Bear away.

Moral: A single sound in the right darkness can turn strength into a stumble.

Hibou – Hou-hou (hululement)

• L(w) = 6
• Bord(w) = 3
• p = 6 − 3 = 3, Root(w) = pref₃(w) = HOU (since 3 ∣ 6)
• Cyclic root = arg min_lex {σ^k(w)} = HOUHOU
• Lyndon factorization = (HOU)(HOU)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 2

Property	HOUHOU
Alphabet	{H, O, U}
Length	6
Border	HOU
Border length	3
Minimal period	3
Period set	{3, 6}
Critical exponent	2
Primitive root	HOU
Root(w)	HOU
Cyclic root	HOUHOU
Suffix array	[4, 1, 5, 2, 6, 3]
Minimal absent words	HH, HU, OH, OO, UO, UU, UHOUH
Lyndon factorization	(HOU)(HOU)
Biroot	(HOU, HOU), (HOUHOU, ε), (ε, HOUHOU)
Internal factors	H, O, U, HO, OU, UH, OUH, UHO, OUHO

Tale of the Lapin (Hare) and the Bear Traditional Ukrainian fairy tale.

Once upon a time, there was a bear in a forest. He was so strong and fierce! He would roam the forest, strangling and tearing apart everything he encountered.
The forest was vast, and there were many animals living in it, but fear spread among them all. For it seemed that not a year would pass before there would be no living soul left in the entire forest if the bear continued to wreak havoc like this.
After much deliberation, the animals decided to send emissaries to the bear and say to him: “Noble Lord Bear! Why do you torment us so? You eat one, and in your rage, you tear apart ten and leave them! If this continues, there will be no living soul left in the forest by the end of the year. Here’s what you should do: sit quietly in your den, and we will send you one of us every day to eat.”
The bear listened to this speech and said:
“Fine! But remember this: if you deceive me even for a single day, I will tear you all apart!”
From that day on, the animals began sending one of their own to the bear each day. They started drawing lots: whoever the lot fell upon had to go to the bear…
One day, the lot fell upon the hare.
The poor hare was so frightened, it’s hard to describe! But what could he do? Others had gone, so he must too. He didn’t resist. He only asked for an hour to say goodbye to his wife and children. But by the time he found his wife, gathered his family, said their goodbyes, cried, and hugged each other, the sun had already passed noon.
Finally, the hare had to set off on his journey.
The poor creature made his way to the bear’s den. But don’t think he was hopping along like a hare, racing against the wind! Oh no, the poor hare wasn’t in the mood for hopping now. He walked slowly, step by step, stopping often, wiping away his tears, and sighing so deeply that the sound echoed through the forest. Then he saw it: a stone well in the middle of the forest, lined with bricks, with deep water at the bottom.
The hare stood by the well, looked down, and his tears fell drop by drop into the water. Suddenly, he cheered up and even jumped for joy. A happy thought had come to him: how to save himself from death and all the animals from this fierce and senseless bear.
No longer crying or sighing, but running as fast as he could, he hurried to the bear’s den.
It was already evening. The bear had been sitting in his den all day, waiting for the animals to send him someone for dinner.
He waited and waited, but no one came. Hunger began to gnaw at him, and with hunger came anger.
“What is this?” roared the bear. “What are they thinking? Have they forgotten about me, or do they think I can be satisfied with one crow for two days? Oh, cursed beasts! If they don’t send me my meal right now, I swear by the beech and the birch that tomorrow at dawn I will go into the forest and strangle everything living in it! Not a single tail will remain!”
But moment after moment, hour after hour passed, and still no meal came.
By evening, the bear didn’t know what to do with his hunger and rage.
It was in this mood that the hare found him.
“Ah, you scoundrel, you little runt, you goose feather,” the bear shouted at him. “What do you think, coming so late? Do you think I, starving all day, should wait for a little gnat like you?”
The hare trembled at the bear’s roar and furious words, but he quickly composed himself and, standing on his hind legs before the bear, spoke as politely as he could:
“Noble Lord! It’s not my fault that I came so late. And you can’t blame the animals either. Today, on your name day, they gathered at dawn and sent four of us to you, and we all raced like the wind to bring you a grand feast, noble sir.”
“Well, and what? Why are you so late, and where are the other three?” asked the bear.
“A terrible misfortune befell us,” said the hare. “Thinking there was no other lord in this forest but you, we were walking peacefully along the path when a huge bear jumped out of a stone castle and shouted, ‘Stop!’ We stopped. ‘Where are you going?’ We told him the truth. ‘Ho, ho,’ he roared. ‘That won’t do! This is my forest, and I won’t allow you to feed some stray with your flesh, who has no right here! You are mine, and I will have you for dinner!’”
“We begged and pleaded, telling him it was your name day and it would be very wrong for you to go without a meal on such a day. But it was no use. He wouldn’t listen. ‘I am the lord here,’ he shouted, ‘and I alone have the right to you! No one dares interfere!’ And he took all four of us to his castle. I barely managed to persuade him to let me go to you. Now, noble lord, you decide whether we are to blame for your hunger and what you should do next.”
Hearing this, the bear bristled with rage. All his anger turned to this new rival who had so unexpectedly appeared in his path.
“What kind of scoundrel dares to intrude here!” he roared, clawing the ground with his paws. “Hey, hare, take me to him right now, and I’ll tear him to pieces!”
“Noble lord,” said the hare. “He is a very powerful lord, terribly fierce.”
“What? Do you think I’m afraid of him? Take me to him right now, and we’ll see who’s stronger!”
“Noble lord, but he lives in a stone castle.”
“Bah, what do I care about his castle! Take me to him, and I’ll get him, even if he hides at the top of the tallest tree.”
The hare led the enraged bear to the well and said:
“Your strength is great! See, your enemy, as soon as he saw you approaching, ran off and hid in his castle.”
“Where is he? Where is he?” shouted the bear, looking around and seeing no one.
“Come here and look down here!” said the hare, leading the bear to the well.
The bear stood over the well, looked down—and there indeed was a bear.
“Do you see your enemy,” said the hare, “peeking out from his fortress?”
“I won’t be myself if I don’t get him out of there!” said the bear, and he roared with all his might into the well.
And from the well, his voice echoed back twice as loud, as if from a huge trumpet!
“So that’s how it is?” shouted the bear. “You dare threaten me? Just wait, I’ll show you!”
And with that, the bear leaped into the well and drowned.
The hare ran back to the animals as fast as he could and told them how he had outwitted the bear and saved them all from a terrible fate.
Needless to say, joy spread throughout the entire forest.

Lapin – Tap-tap-tap (lapin qui tape le sol avec sa patte)

• L(w) = 9
• Bord(w) = 6
• p = 9 − 6 = 3, Root(w) = pref₃(w) = TAP (since 3 ∣ 9)
• Cyclic root = arg min_lex {σ^k(w)} = APTAPTAPT
• Lyndon factorization = (T)(APT)(APT)(AP)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 3

Property	TAPTAPTAP
Alphabet	{A, P, T}
Length	9
Border	TAPTAP
Border length	6
Minimal period	3
Period set	{3, 6, 9}
Critical exponent	3
Primitive root	TAP
Root(w)	TAP
Cyclic root	APTAPTAPT
Suffix array	[8, 5, 2, 9, 6, 3, 7, 4, 1]
Minimal absent words	AA, AT, PA, PP, TP, TT, PTAPTAPT
Lyndon factorization	(T)(APT)(APT)(AP)
Biroot	(TAPTAPTAP, ε), (ε, TAPTAPTAP)
Internal factors	A, P, T, AP, PT, TA, APT, PTA, TAP, APTA, PTAP, TAPT, APTAP, PTAPT, TAPTA, APTAPT, PTAPTA, APTAPTA

Traditional Cochiti (Native American) tale.

Bear and Lion had a contest to see who was the strongest. The queen of the animals wanted a man frightened who was always disobedient to his people. She wished him put back into the right road. She asked all of the animals which one of them would be willing to take the risk of frightening this man in order to put him back on the right road. Grizzly Bear said, “I will take the risk of frightening this man in order to put him back on the right road.” The queen said, “I am afraid of you because you are so fierce. I am afraid that you will do him harm, and not just put him on the right road.” But at last she consented.
Grizzly Bear set out for the mountains. He was not to do the man any harm but only frighten him. The bear found him, and he went against him, but instead of just frightening him he killed the man. He took word back to the queen. He said, “I have done as you told me.” “How did you frighten him?” “I killed him.” “What did I tell you! I told you that was what you would do, and I told you not to kill him. Now you will be punished for this.” “Well, if anyone is stronger than I am and can punish me, let it be so!” So the queen called the Lion and told him that he was to kill the Grizzly Bear. They started off for the mountains. The power the Bear had was taken from him and given to the Lion. They fought. The Lion roped the Bear with his tail and climbed up the pine tree with him. The Lion split the pine tree as he climbed up and put the Bear in the split tree to dry up. So the Lion killed the Bear. Lion took the dead Bear back to the queen and she was very thankful, because the Bear had no control over himself when he was angry.

Lion – Roah, Roaaar, Raaah! (Rugissement)

Roah

• L(w) = 4
• Bord(w) = 0
• p = 4 − 0 = 4, Root(w) = pref₄(w) = ROAH (since 4 ∣ 4)
• Cyclic root = arg min_lex {σ^k(w)} = AHRO
• Lyndon factorization = (R)(O)(AH)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 1

Roaaar

• L(w) = 6
• Bord(w) = 1
• p = 6 − 1 = 5, Root(w) = pref₅(w) = ROAAAR (since 5 ∤ 6, Root(w) = w)
• Cyclic root = arg min_lex {σ^k(w)} = AAARRO
• Lyndon factorization = (R)(O)(AAAR)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 3

Raaah

• L(w) = 5
• Bord(w) = 0
• p = 5 − 0 = 5, Root(w) = pref₅(w) = RAAAH (since 5 ∣ 5)
• Cyclic root = arg min_lex {σ^k(w)} = AAAHR
• Lyndon factorization = (R)(AAAH)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 3

Property	ROAH	ROAAAR	RAAAH
Alphabet	{A, H, O, R}	{A, O, R}	{A, H, R}
Length	4	6	5
Border	ε	R	ε
Border length	0	1	0
Minimal period	4	5	5
Period set	{4}	{5, 6}	{5}
Critical exponent	1	3	3
Primitive root	ROAH	ROAAAR	RAAAH
Root(w)	ROAH	ROAAAR	RAAAH
Cyclic root	AHRO	AAARRO	AAAHR
Suffix array	[3, 4, 2, 1]	[3, 4, 5, 2, 6, 1]	[2, 3, 4, 5, 1]
Minimal absent words	AA, AO, AR, HA, HH, HO, HR, OH, OO, OR, RA, RH, RR	AO, OO, OR, RA, RR, ARO, OAR, AAAA, OAAR	AR, HA, HH, HR, RH, RR, RAH, AAAA, RAAH
Lyndon factorization	(R)(O)(AH)	(R)(O)(AAAR)	(R)(AAAH)
Biroot	(ROAH, ε), (ε, ROAH)	(ROAAAR, ε), (ε, ROAAAR)	(RAAAH, ε), (ε, RAAAH)
Internal factors	A, O, OA	A, O, AA, OA, AAA, OAA, OAAA	A, AA, AAA

Once upon a time a large Bear lived in the woods and took what he wanted. One day he came upon a Loup standing alone on a narrow path between two rocks.
“You are thin,” said the Bear. “I will eat you.”
The Loup did not run. It stood still and answered, “The path is longer than it looks, Bear. Follow me and see.”
The Loup turned and walked slowly between the rocks. The Bear followed. After a short distance the Loup stopped and howled once — a short, sharp call. From the trees on both sides other wolves answered, not many, but enough that their voices came from left and right at the same time. The Bear stopped. The path was narrow. When he turned to leave, the Loup was no longer alone; two others stood at the far end of the rocks, blocking the way back.
The Bear took one step forward. The wolves did not attack. They simply stayed, one in front, two behind, and howled again, short and steady. The sound came from three places at once. The Bear stood still for a moment, then turned sideways, forced his way past the nearest rock, and left the path, shaking his head.
From that day the Bear avoided the narrow paths between rocks, and the other animals said that a Loup who knows how to use its voice and the place can drive even a Bear away without a fight.
Moral: A few voices in the right place can turn strength into a retreat.

Loup – Ouuuh, Aouuuuh

Ouuuh

• L(w) = 5
• Bord(w) = 0
• p = 5 − 0 = 5, Root(w) = pref₅(w) = OUUUH (since 5 ∣ 5)
• Cyclic root = arg min_lex {σ^k(w)} = HOUUU
• Lyndon factorization = (OUUU)(H)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 3

Aouuuuh

• L(w) = 7
• Bord(w) = 0
• p = 7 − 0 = 7, Root(w) = pref₇(w) = AOUUUUH (since 7 ∣ 7)
• Cyclic root = arg min_lex {σ^k(w)} = AOUUUUH
• Lyndon factorization = (AOUUUUH)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 4

Property	OUUUH	AOUUUUH
Alphabet	{H, O, U}	{A, H, O, U}
Length	5	7
Border	ε	ε
Border length	0	0
Minimal period	5	7
Period set	{5}	{7}
Critical exponent	3	4
Primitive root	OUUUH	AOUUUUH
Root(w)	OUUUH	AOUUUUH
Cyclic root	HOUUU	AOUUUUH
Suffix array	[5, 1, 4, 3, 2]	[1, 7, 2, 6, 5, 4, 3]
Minimal absent words	HH, HO, HU, OH, OO, UO, OUH, OUUH, UUUU	AA, AH, AU, HA, HH, HO, HU, OA, OH, OO, UA, UO, OUH, OUUH, OUUUH, UUUUU
Lyndon factorization	(OUUU)(H)	(AOUUUUH)
Biroot	(OUUUH, ε), (ε, OUUUH)	(AOUUUUH, ε), (ε, AOUUUUH)
Internal factors	U, UU, UUU	O, U, OU, UU, OUU, UUU, OUUU, UUUU, OUUUU

Once upon a time a large Bear sat in the shade of a tree, ready to sleep. A small Mouche settled on his ear and began to buzz, low and steady.

“Go away,” said the Bear, and he shook his head. The Mouche rose, circled once, and settled again, this time on the tip of his nose.

The Bear snorted and waved a paw. The Mouche flew up, waited, and came back to the same place. Each time the Bear moved, the Mouche left and returned, always buzzing close to his ear or his eye.

After a long while the Bear stood up. He walked to a stream, lowered his head to drink, and the Mouche landed between his eyes. The Bear lifted his head, water dripping, and the Mouche was still there. He turned and walked into the deeper woods, looking for a place with no open air. When he found a thicket of thorns, he pushed in. The Mouche did not follow; the thorns were too close for its wings.

From that day the Bear slept only where the branches were thick, and the other animals said that a Mouche who will not stop can drive even a Bear from his rest.

Moral: What will not leave can force a stronger one to leave.

Mouche – Zzz

• L(w) = 3
• Bord(w) = 0
• p = 3 − 0 = 3, Root(w) = pref₃(w) = Zzz (since 3 ∣ 3)
• Cyclic root = arg min_lex {σ^k(w)} = Zzz (with Z < z)
• Lyndon factorization = (Zzz)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 2

Property	Value for w = Zzz	Computable formula (compact)
Alphabet	{Z, z}	Σ(w) = {a ∣ a occurs in w}
Length	3	L(w) = ∣w∣
Border	ε	longest proper prefix of w that is also a suffix
Border length	0	Bord(w) = max{k < n : prefk = suffk}
Minimal period	3	p(w) = n − Bord(w)
Period set	{3}	{q ≤ n : wi = wi+q for all valid i} = {n − ∣b∣ : b border of w}
Critical exponent	2	ce(w) = max{∣x∣ / per(x) : x factor of w}
Primitive root	Zzz	prefp(w) if p ∣ n, else w
Root(w)	Zzz	prefp(w) if p ∣ n, else w
Cyclic root	Zzz	arg minlex {σk(w)} (Z < z)
Suffix array	[1, 3, 2]	starting positions of the suffixes in lexicographic order
Minimal absent words	ZZ, zZ, zzz	shortest words not occurring in w whose proper factors all occur
Lyndon factorization	(Zzz)	Duval: non-increasing sequence of Lyndon words
Biroot	(Zzz, ε), (ε, Zzz)	splits (u, v) of w = uv with u, v both powers of the same primitive word
Internal factors	z	factors starting at position ≥ 2 and ending at position ≤ n − 1

Slovak folk tale “Medveď a komár” (The Bear and the Mosquito), collected by Pavol Dobšinský.

A bear met a mosquito.

The bear said: “You, mosquito, you land on every creature and suck your fill. Tell me, which blood is the sweetest?”

The mosquito answered: “Truly, human blood is the sweetest.”

Good. The bear set out after that human blood.

He met a boy: “Stop! Are you a man?”

“I will be one yet!” the boy jumped proudly, as if he wanted to grow into a full man at once.

“Well, just keep jumping,” the bear muttered. “What is only going to be is of no use to me.”

He went farther and met a beggar: “Stop! Are you a man?”

“I already was!” the beggar coughed and hunched so as not to be seen.

“Well, just keep hunching,” the bear muttered. “What already was is of no use to me either.”

He went on and on and met a hussar on horseback: “Stop! Who are you?”

“A man with a head!” the hussar thundered and spurred his horse.

But the bear set off after him, right at him. The hussar turned and with his sabre struck the bear full in the living flesh so that the blood poured from him. The bear took no joke and showed the hussar his heels. The hussar, not slow, tore the carbine from his shoulder and fired after him.

The bear met the mosquito again: “You, mosquito, you may be right that human blood is the sweetest. But do not send me after it any more!”

“And why?” the mosquito laughed.

“Because a man does not joke. As soon as I set upon him he stuck out his tongue at me — but such a big one that it reached a fathom in front of him, and so sharp that it went into my bone. Then I left him, but he turned and spat after me and with a clear shot hit me in the side so that I almost died of the pain; even now it is festering under my skin!”

Moustique – Ziziziz

Ziziziz

• L(w) = 7
• Bord(w) = 0
• p = 7 − 0 = 7, Root(w) = w (since 7 ∣ 7, pref₇(w) = Ziziziz)
• Cyclic root = arg min_lex {σ^k(w)} = Ziziziz (with Z < i < z)
• Lyndon factorization = (Ziziziz)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 3

Property	Value for w = Ziziziz	Computable formula (compact)
Alphabet	{Z, i, z}	Σ(w) = {a ∣ a occurs in w}
Length	7	L(w) = ∣w∣
Border	ε	longest proper prefix of w that is also a suffix
Border length	0	Bord(w) = max{k < n : prefk = suffk}
Minimal period	7	p(w) = n − Bord(w)
Period set	{7}	{n − ∣b∣ : b border of w, including b = ε}
Critical exponent	3	ce(w) = max{∣x∣ / per(x) : x factor of w}
Primitive root	Ziziziz	prefp(w) if p ∣ n, else w
Root(w)	Ziziziz	prefp(w) if p ∣ n, else w
Cyclic root	Ziziziz	arg minlex {σk(w)} (Z < i < z)
Suffix array	[1, 6, 4, 2, 7, 5, 3]	starting positions of the suffixes in lexicographic order
Minimal absent words	ZZ, Zz, iZ, ii, zZ, zz, zizizi	shortest words not occurring in w whose proper factors all occur
Lyndon factorization	(Ziziziz)	Duval: non-increasing sequence of Lyndon words
Biroot	(Ziziziz, ε), (ε, Ziziziz)	splits (u, v) of w = uv with u, v both powers of the same primitive word
Internal factors	i, z, iz, zi, izi, ziz, iziz, zizi, izizi	factors starting at position ≥ 2 and ending at position ≤ n − 1

Once upon a time a large Bear came down from the mountain and found a flock of Moutons grazing on a steep slope. He stood above them and said, “I will take one of you.”

The oldest Mouton stepped forward. “Take me,” it said, “but first look at the ground where we stand.”

The Bear looked. The slope was covered with short grass and loose stones. The Mouton turned and walked slowly across the face of the slope, not up or down. The others followed, one behind the other, until they formed a long line that stretched from one side of the slope to the other. Their feet pressed the loose stones. The Bear took one step down to follow. The stones under his weight began to move. He stopped. The line of Moutons kept walking, slow and even, and more stones shifted. The Bear took another step and the ground gave. He slid a short way and caught himself against a rock. When he looked up, the Moutons were already higher on the slope, standing on firm ground, and the place where he stood was covered with moving stones.

From that day the Bear stayed above the slopes where the Moutons grazed, and the other animals said that a flock that knows the ground can turn a mountain against a Bear.

Moral: What walks together on the right ground can move what is heavier.

Mouton – Beeeee! (bêlement)

• L(w) = 6
• Bord(w) = 0
• p = 6 − 0 = 6, Root(w) = pref₆(w) = Beeeee (since 6 ∣ 6)
• Cyclic root = arg min_lex {σ^k(w)} = Beeeee (with B < e)
• Lyndon factorization = (Beeeee)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 5

Property	Value for w = Beeeee
Alphabet	{B, e}
Length	6
Border	ε
Border length	0
Minimal period	6
Period set	{6}
Critical exponent	5
Primitive root	Beeeee
Root(w)	Beeeee
Cyclic root	Beeeee
Suffix array	[1, 6, 5, 4, 3, 2]
Minimal absent words	BB, eB, eeeeee
Lyndon factorization	(Beeeee)
Biroot	(Beeeee, ε), (ε, Beeeee)
Internal factors	e, ee, eee, eeee

Once in summer-time the bear and the wolf were walking in the forest, and the bear heard a bird singing so beautifully that he said, “Brother wolf, what bird is it that sings so well?”

“That is the King of birds,” said the wolf, “before whom we must bow down.” It was, however, in reality the willow-wren.

The bear went to see the nest. When he saw the young birds he said, “Is that the royal palace? It is a wretched palace, and you are not King’s children, you are disreputable children!”

The young wrens were angry and called for war. The willow-wren declared war on the bear. All the birds and all the four-footed animals were summoned.

The fox was made standard-bearer of the land animals, with the order that if the tail stayed up, all was well; if it went down, they must flee. The birds sent a hornet to sting the fox’s tail. The fox put his tail down from the pain. When the land animals saw the tail down, they thought the battle was lost and fled into their holes. The birds had won the battle.

Then the King and Queen flew home to their children and cried, “Children, rejoice, eat and drink to your heart’s content, we have won the battle!”

Oiseau → Cui-cui (cri de petits oiseaux), Flap-flap-flap (battement d’ailes)

Cuicui

• L(w) = 6
• Bord(w) = 3
• p = 6 − 3 = 3, Root(w) = pref₃(w) = CUI (since 3 ∣ 6)
• Cyclic root = arg min_lex {σ^k(w)} = CUICUI
• Lyndon factorization = (CUI)(CUI)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 2

Flapflapflap

• L(w) = 12
• Bord(w) = 8
• p = 12 − 8 = 4, Root(w) = pref₄(w) = FLAP (since 4 ∣ 12)
• Cyclic root = arg min_lex {σ^k(w)} = APFLAPFLAPFL
• Lyndon factorization = (FL)(APFL)(APFL)(AP)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 3

Property	CUICUI	FLAPFLAPFLAP
Alphabet	{C, I, U}	{A, F, L, P}
Length	6	12
Border	CUI	FLAPFLAP
Border length	3	8
Minimal period	3	4
Period set	{3, 6}	{4, 8, 12}
Critical exponent	2	3
Primitive root	CUI	FLAP
Root(w)	CUI	FLAP
Cyclic root	CUICUI	APFLAPFLAPFL
Suffix array	[4, 1, 6, 3, 5, 2]	[11, 7, 3, 9, 5, 1, 10, 6, 2, 12, 8, 4]
Minimal absent words	CC, CI, II, IU, UC, UU, ICUIC	AA, AF, AL, FA, FF, FP, LF, LL, LP, PA, PL, PP, PFLAPFLAPF
Lyndon factorization	(CUI)(CUI)	(FL)(APFL)(APFL)(AP)
Biroot	(CUI, CUI), (CUICUI, ε), (ε, CUICUI)	(FLAPFLAPFLAP, ε), (ε, FLAPFLAPFLAP)
Internal factors	C, I, U, CU, IC, UI, ICU, UIC, UICU	A, F, L, P, AP, FL, LA, PF, APF, FLA, LAP, PFL, APFL, FLAP, LAPF, PFLA, APFLA, FLAPF, LAPFL, PFLAP, APFLAP, FLAPFL, LAPFLA, PFLAPF, APFLAPF, FLAPFLA, LAPFLAP, PFLAPFL, APFLAPFL, LAPFLAPF, PFLAPFLA, APFLAPFLA, LAPFLAPFL, LAPFLAPFLA

Once upon a time a large Bear came to a tree where a Perroquet lived. The Bear said, “I will take the fruit from this tree.”
The Perroquet answered, “The fruit is not yours.”
The Bear climbed the trunk. As he climbed, the Perroquet flew to the other side of the tree and called out in the exact voice of another Bear, “Who is climbing my tree?”
The Bear stopped and looked around. He saw no other Bear. He climbed higher. The Perroquet flew again and called, still in the other Bear’s voice, “Come down. This tree is claimed.”
The Bear looked again. He saw only the Perroquet. He took another step. The Perroquet called a third time, closer now, “If you stay, we fight.”
The Bear climbed down. He stood at the foot of the tree and said, “Where is the other Bear?”
The Perroquet answered in his own voice, “There is no other Bear. There is only the voice. The tree stays.”
From that day the Bear did not climb trees where the Perroquet called.
Moral: A voice that is not your own can still move a heavy body.

Perroquet – Coco

• L(w) = 4
• Bord(w) = 2
• p = 4 − 2 = 2, Root(w) = pref₂(w) = CO (since 2 ∣ 4)
• Cyclic root = arg min_lex {σ^k(w)} = COCO
• Lyndon factorization = (CO)(CO)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 2

Property	Value for w = COCO
Alphabet	{C, O}
Length	4
Border	CO
Border length	2
Minimal period	2
Period set	{2, 4}
Critical exponent	2
Primitive root	CO
Root(w)	CO
Cyclic root	COCO
Suffix array	[3, 1, 4, 2]
Minimal absent words	CC, OO, OCOC
Lyndon factorization	(CO)(CO)
Biroot	(CO, CO), (COCO, ε), (ε, COCO)
Internal factors	C, O, OC

Once upon a time a large Bear came to a tree where a Pic-vert lived. The Bear said, “I will take the hollow of this tree.”
The Pic-vert answered, “The hollow is not yours.”
The Bear put his paw on the trunk. The Pic-vert flew to the upper part of the tree and began to drum hard with his beak, toc-toc-toc, on a dead branch. The branch cracked. The Bear looked up. He put his other paw higher. The Pic-vert moved to another branch and drummed again. A second crack came. The Bear stopped. He stood with both paws on the trunk and listened. The Pic-vert drummed a third time, lower now, on a branch that still held. The branch gave. The Bear pulled his paws back and stepped away from the tree.
He stood on the ground and said, “The tree is speaking.”
The Pic-vert answered, “The tree speaks when it is struck. The hollow stays.”
From that day the Bear did not put his paws on trees where the Pic-vert drummed.
Moral: What strikes the wood can move what climbs the wood.

Pic-vert – Pic! Pic! Pic!

• L(w) = 9
• Bord(w) = 6
• p = 9 − 6 = 3, Root(w) = pref₃(w) = PIC (since 3 ∣ 9)
• Cyclic root = arg min_lex {σ^k(w)} = CPICPICPI
• Lyndon factorization = (P)(I)(CPI)(CPI)(C)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 3

Property	Value for w = PICPICPIC
Alphabet	{C, I, P}
Length	9
Border	PICPIC
Border length	6
Minimal period	3
Period set	{3, 6, 9}
Critical exponent	3
Primitive root	PIC
Root(w)	PIC
Cyclic root	CPICPICPI
Suffix array	[9, 6, 3, 8, 5, 2, 7, 4, 1]
Minimal absent words	CC, CI, II, IP, PC, PP, CPICPICP
Lyndon factorization	(P)(I)(CPI)(CPI)(C)
Biroot	(PICPICPIC, ε), (ε, PICPICPIC)
Internal factors	C, I, P, CP, IC, PI, CPI, ICP, PIC, CPIC, ICPI, PICP, CPICP, ICPIC, PICPI, CPICPI, ICPICP, ICPICPI

Once upon a time a large Bear came to a field where a flock of Pigeons was feeding. He said, “I will take this ground.”
The oldest Pigeon answered, “The ground is not yours.”
The Bear stepped onto the field. The Pigeons rose together. They flew low, one behind the other, in a long line that crossed the field in front of the Bear. The Bear stopped. He took another step. The line of Pigeons turned and crossed again, closer. Their wings beat the air in front of his face. He stopped. He took a third step. The line crossed once more, lower still. The Bear stepped back. He stood at the edge of the field and watched the Pigeons settle again on the ground.
He said, “The air is full.”
The oldest Pigeon answered, “The air is full when we fly together. The ground stays.”
From that day the Bear did not step onto fields where the Pigeons rose as one.
Moral: What flies as one line can turn a heavy step.

Pigeon – Roucoul, Rou-cou, Rou-rou

Roucoul

• L(w) = 7
• Bord(w) = 0
• p = 7 − 0 = 7, Root(w) = pref₇(w) = ROUCOUL (since 7 ∣ 7)
• Cyclic root = arg min_lex {σ^k(w)} = COULROU
• Lyndon factorization = (R)(OU)(COUL)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 5/3

Roucou

• L(w) = 6
• Bord(w) = 0
• p = 6 − 0 = 6, Root(w) = pref₆(w) = ROUCOU (since 6 ∣ 6)
• Cyclic root = arg min_lex {σ^k(w)} = COUROU
• Lyndon factorization = (R)(OU)(COU)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 5/3

Rourou

• L(w) = 6
• Bord(w) = 3
• p = 6 − 3 = 3, Root(w) = pref₃(w) = ROU (since 3 ∣ 6)
• Cyclic root = arg min_lex {σ^k(w)} = OUROUR
• Lyndon factorization = (R)(OUR)(OU)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 2

Property	ROUCOUL	ROUCOU	ROUROU
Alphabet	{C, L, O, R, U}	{C, O, R, U}	{O, R, U}
Length	7	6	6
Border	ε	ε	ROU
Border length	0	0	3
Minimal period	7	6	3
Period set	{7}	{6}	{3, 6}
Critical exponent	5/3	5/3	2
Primitive root	ROUCOUL	ROUCOU	ROU
Root(w)	ROUCOUL	ROUCOU	ROU
Cyclic root	COULROU	COUROU	OUROUR
Suffix array	[4, 7, 2, 5, 1, 3, 6]	[4, 5, 2, 1, 6, 3]	[5, 2, 4, 1, 6, 3]
Minimal absent words	CC, CL, CR, CU, LC, LL, LO, LR, LU, OC, OL, OO, OR, RC, RL, RR, RU, UO, UR, UU, COUC, ROUL	CC, CR, CU, OC, OO, OR, RC, RR, RU, UO, UR, UU, COUC	OO, OR, RR, RU, UO, UU, UROUR
Lyndon factorization	(R)(OU)(COUL)	(R)(OU)(COU)	(R)(OUR)(OU)
Biroot	(ROUCOUL, ε), (ε, ROUCOUL)	(ROUCOU, ε), (ε, ROUCOU)	(ROU, ROU), (ROUROU, ε), (ε, ROUROU)
Internal factors	C, O, U, CO, OU, UC, COU, OUC, UCO, OUCO, UCOU, OUCOU	C, O, U, CO, OU, UC, OUC, UCO, OUCO	O, R, U, OU, RO, UR, OUR, URO, OURO

Once upon a time a large Bear came to a yard where a Poule was scratching. He said, “I will take this ground.”
The Poule answered, “The ground is not yours.”
The Bear put his paw on the earth. The Poule began to peck the ground hard, in a small circle around the paw. She pecked once. The earth moved. She pecked twice. The circle closed. She pecked a third time. The paw sank a little. The Bear pulled the paw back. He stood and looked at the circle of loose earth.
He said, “The ground is open.”
The Poule answered, “The ground opens when it is pecked. The yard stays.”
From that day the Bear did not put his paws on ground where the Poule pecked in a circle.
Moral: What pecks the earth in a ring can shift a heavy paw.

Poule – Cot-cot, Cot-cot-codêêêt

Cot-cot (written COTCOT)

• L(w) = 6
• Bord(w) = 3
• p = 6 − 3 = 3, Root(w) = pref₃(w) = COT (since 3 ∣ 6)
• Cyclic root = arg min_lex {σ^k(w)} = COTCOT
• Lyndon factorization = (COT)(COT)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 2

Cot-cot-codêêêt (written COTCOTCODEEET)

• L(w) = 13
• Bord(w) = 0
• p = 13 − 0 = 13, Root(w) = pref₁₃(w) = COTCOTCODEEET (since 13 ∣ 13)
• Cyclic root = arg min_lex {σ^k(w)} = CODEEETCOTCOT
• Lyndon factorization = (COT)(COT)(CODEEET)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 3

Property	Cot-cot	Cot-cot-codêêêt
Alphabet	{C, O, T}	{C, D, E, O, T}
Length	6	13
Border	COT	ε
Border length	3	0
Minimal period	3	13
Period set	{3, 6}	{13}
Critical exponent	2	3
Primitive root	COT	COTCOTCODEEET
Root(w)	COT	COTCOTCODEEET
Cyclic root	COTCOT	CODEEETCOTCOT
Suffix array	[4, 1, 5, 2, 6, 3]	[7, 4, 1, 9, 10, 11, 12, 8, 5, 2, 13, 6, 3]
Minimal absent words	CC, CT, OC, OO, TO, TT, TCOTC	CC, CD, CE, CT, DC, DD, DO, DT, EC, ED, EO, OC, OE, OO, TD, TE, TO, TT, DET, ETC, DEET, EEEE, TCOTCOT
Lyndon factorization	(COT)(COT)	(COT)(COT)(CODEEET)
Biroot	(COT, COT), (COTCOT, ε), (ε, COTCOT)	(COTCOTCODEEET, ε), (ε, COTCOTCODEEET)
Internal factors	C, O, T, CO, OT, TC, OTC, TCO, OTCO	C, D, E, O, T, CO, DE, EE, OD, OT, TC, COD, COT, DEE, EEE, ODE, OTC, TCO, CODE, COTC, DEEE, ODEE, OTCO, TCOD, TCOT, CODEE, COTCO, ODEEE, OTCOD, OTCOT, TCODE, TCOTC, CODEEE, COTCOD, OTCODE, OTCOTC, TCODEE, TCOTCO, COTCODE, OTCODEE, OTCOTCO, TCODEEE, TCOTCOD, COTCODEE, OTCODEEE, OTCOTCOD, TCOTCODE, COTCODEEE, OTCOTCODE, TCOTCODEE, OTCOTCODEE, TCOTCODEEE, OTCOTCODEEE

Once upon a time a large Bear came to a farmyard where a small Poussin was walking near the chicken coop. The Bear said, “I will take this ground.”
The Poussin answered, “The ground is not yours. Follow me if you want the grain.”
The Poussin ran toward the low door of the coop, peeping. The Bear followed. The door was only as high as the Bear’s knee. The Poussin slipped inside. The Bear put his head down to follow. His shoulders struck the frame. He pushed. The door frame held. He pushed harder. His paw slipped on the threshold. He stopped. He tried once more. His head was inside, his body was outside. He could not go forward and he did not want to go back with the Poussin peeping from inside.
He pulled his head out and stood in the yard. The Poussin came to the doorway and peeped once more.
The Bear said, “The door is too small.”
The Poussin answered, “The door is small for a heavy body. The yard stays.”
From that day the Bear did not enter yards where a Poussin ran into a low coop.
Moral: What is small can lead a heavy body into a place it cannot enter.

Poussin – Pouic-pouic

• L(w) = 10
• Bord(w) = 5
• p = 10 − 5 = 5, Root(w) = pref₅(w) = POUIC (since 5 ∣ 10)
• Cyclic root = arg min_lex {σ^k(w)} = CPOUICPOUI
• Lyndon factorization = (P)(OU)(I)(CPOUI)(C)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 2

Property	Value for w = POUICPOUIC
Alphabet	{C, I, O, P, U}
Length	10
Border	POUIC
Border length	5
Minimal period	5
Period set	{5, 10}
Critical exponent	2
Primitive root	POUIC
Root(w)	POUIC
Cyclic root	CPOUICPOUI
Suffix array	[10, 5, 9, 4, 7, 2, 6, 1, 8, 3]
Minimal absent words	CC, CI, CO, CU, II, IO, IP, IU, OC, OI, OO, OP, PC, PI, PP, PU, UC, UO, UP, UU, CPOUICP
Lyndon factorization	(P)(OU)(I)(CPOUI)(C)
Biroot	(POUIC, POUIC), (POUICPOUIC, ε), (ε, POUICPOUIC)
Internal factors	C, I, O, P, U, CP, IC, OU, PO, UI, CPO, ICP, OUI, POU, UIC, CPOU, ICPO, OUIC, POUI, UICP, CPOUI, ICPOU, OUICP, UICPO, ICPOUI, OUICPO, UICPOU, OUICPOU, UICPOUI, OUICPOUI

Once upon a time a large Bear came to a clearing where a Sanglier was rooting. The Bear said, “I will take this ground.”
The Sanglier answered, “The ground is not yours.”
The Bear stepped forward. The Sanglier lowered his head and charged. His tusks struck the Bear’s front legs low, just above the ground. The Bear stopped. The Sanglier turned and charged again from the side. The tusks hit the same place. The Bear lifted one paw and stepped back. The Sanglier charged a third time, lower still. The Bear stepped fully out of the clearing.
He stood at the edge and said, “The charge is low.”
The Sanglier answered, “The charge is low where the legs stand. The clearing stays.”
From that day the Bear did not step onto ground where a Sanglier lowered his head and charged.
Moral: What charges low can move a heavy body that stands high.

Sanglier – Groin-groin

• L(w) = 10
• Bord(w) = 5
• p = 10 − 5 = 5, Root(w) = pref₅(w) = GROIN (since 5 ∣ 10)
• Cyclic root = arg min_lex {σ^k(w)} = GROINGROIN
• Lyndon factorization = (GROIN)(GROIN)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 2

Property	Value for w = GROINGROIN
Alphabet	{G, I, N, O, R}
Length	10
Border	GROIN
Border length	5
Minimal period	5
Period set	{5, 10}
Critical exponent	2
Primitive root	GROIN
Root(w)	GROIN
Cyclic root	GROINGROIN
Suffix array	[6, 1, 9, 4, 10, 5, 8, 3, 7, 2]
Minimal absent words	GG, GI, GN, GO, IG, II, IO, IR, NI, NN, NO, NR, OG, ON, OO, OR, RG, RI, RN, RR, NGROING
Lyndon factorization	(GROIN)(GROIN)
Biroot	(GROIN, GROIN), (GROINGROIN, ε), (ε, GROINGROIN)
Internal factors	G, I, N, O, R, GR, IN, NG, OI, RO, GRO, ING, NGR, OIN, ROI, GROI, INGR, NGRO, OING, ROIN, INGRO, NGROI, OINGR, ROING, INGROI, OINGRO, ROINGR, OINGROI, ROINGRO, ROINGROI

The Man who made a Pet of a Bear (also called The Man and the Bear or The Man with a Bear)
Source: Jalāl ad-Dīn Rūmī, Masnavi-i Maʿnavī, Book II, Story VIII

Full Original Story (E. H. Whinfield translation, 1898)

A kind man, seeing a serpent overcoming a bear, went to the bear’s assistance, and delivered him from the serpent. The bear was so sensible of the kindness the man had done him that he followed him about wherever he went, and became his faithful slave, guarding him from everything that might annoy him.

One day the man was lying asleep, and the bear, according to his custom, was sitting by him and driving off the flies. The flies became so persistent in their annoyances that the bear lost patience, and seizing the largest stone he could find, dashed it at them in order to crush them utterly; but unfortunately the flies escaped, and the stone lighted upon the sleeper’s face and crushed it.

The moral is, “Do not make friends with fools.”

Full Original Story (Coleman Barks version)

For the man who saved the bear from the dragon’s mouth, the bear became a sort of pet. When he would lie down to rest, the bear would stand guard. A certain friend passed by, “Brother how did this bear get connected to you?” He told the adventure with the dragon, and the friend responded, “Don’t forget what your companion is. This friend is not human! It would be better to choose one of your own kind.” “You’re just jealous of my unusual helper. Look at his sweet devotion. Ignore the bearishness!” But the friend was not convinced, “Don’t go into the forest with a comrade like this! Let me go with you.” “I’m tired. Leave me alone.” The man began imagining motives other than kindness for his friend’s concern. “He has made a bet with someone that he can separate me from my bear.” Or, “He will attack me when my bear is gone.” He had begun to think like a bear! So the human friends went different ways, the one with his bear into a forest, where he fell asleep again. The bear stood over him waving the flies away. But the flies kept coming back, which irritated the bear. He dislodged a stone from the mountainside and raised it over the sleeping man. When he saw that the flies had returned and settled comfortably on the man’s face, He slammed the stone down, crushing to powder the man’s face and skull. Which proves the old saying: IF YOU’RE FRIENDS WITH A BEAR, THE FRIENDSHIP WILL DESTROY YOU. WITH THAT ONE, IT’S BETTER TO BE ENEMIES.

Serpent – Kss kss, Ssshhh, ssss

Kss kss (written KSSKSS)

• L(w) = 6
• Bord(w) = 3
• p = 6 − 3 = 3, Root(w) = pref₃(w) = KSS (since 3 ∣ 6)
• Cyclic root = arg min_lex {σ^k(w)} = KSSKSS
• Lyndon factorization = (KSS)(KSS)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 2

Ssshhh (written SSSHHH)

• L(w) = 6
• Bord(w) = 0
• p = 6 − 0 = 6, Root(w) = pref₆(w) = SSSHHH (since 6 ∣ 6)
• Cyclic root = arg min_lex {σ^k(w)} = HHHSSS
• Lyndon factorization = (S)(S)(S)(H)(H)(H)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 3

ssss (written SSSS)

• L(w) = 4
• Bord(w) = 3
• p = 4 − 3 = 1, Root(w) = pref₁(w) = S (since 1 ∣ 4)
• Cyclic root = arg min_lex {σ^k(w)} = SSSS
• Lyndon factorization = (S)(S)(S)(S)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 4

Property	Kss kss	Ssshhh	ssss
Alphabet	{K, S}	{H, S}	{S}
Length	6	6	4
Border	KSS	ε	SSS
Border length	3	0	3
Minimal period	3	6	1
Period set	{3, 6}	{6}	{1, 2, 3, 4}
Critical exponent	2	3	4
Primitive root	KSS	SSSHHH	S
Root(w)	KSS	SSSHHH	S
Cyclic root	KSSKSS	HHHSSS	SSSS
Suffix array	[4, 1, 6, 3, 5, 2]	[6, 5, 4, 3, 2, 1]	[4, 3, 2, 1]
Minimal absent words	KK, KSK, SSS, SKSSK	HS, HHHH, SSSS	SSSSS
Lyndon factorization	(KSS)(KSS)	(S)(S)(S)(H)(H)(H)	(S)(S)(S)(S)
Biroot	(KSS, KSS), (KSSKSS, ε), (ε, KSSKSS)	(SSSHHH, ε), (ε, SSSHHH)	(SSSS, ε), (ε, SSSS)
Internal factors	K, S, KS, SK, SS, SKS, SSK, SSKS	H, S, HH, SH, SS, SHH, SSH, SSHH	S, SS

Korean Origin Myth: The Tiger and the Bear (Dangun Myth) Traditional Korean founding myth, recorded in the Samguk Yusa (13th century).

Long ago, a tiger and a bear lived together in a cave and prayed to the divine king Hwanung to be made human. Hwanung heard their prayers and gave them each twenty cloves of garlic and a bundle of mugwort, and ordered them to stay out of the sunlight and eat only this food for one hundred days. Due to hunger, the tiger left the cave after roughly twenty days, but the bear remained inside. After she endured the full period, she was transformed into a woman named Ungnyeo (Bear Woman). Ungnyeo made offerings to Hwanung and prayed for a child. Hwanung took her as his wife, and she gave birth to a son, Dangun, who founded the first Korean kingdom, Gojoseon. In this myth the bear succeeds through endurance while the tiger fails and remains an animal.

Morality: a bear eating garlic and a bundle of mugwort make a Korean forever Russian!

Tigre – Roar (rugissement), Ahrrfff ! (tigre qui feule)

Roar (written ROAR)

• L(w) = 4
• Bord(w) = 1
• p = 4 − 1 = 3, Root(w) = w = ROAR (since 3 ∤ 4)
• Cyclic root = arg min_lex {σ^k(w)} = ARRO
• Lyndon factorization = (R)(O)(AR)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 4/3

Ahrrfff (written AHRRFFF)

• L(w) = 7
• Bord(w) = 0
• p = 7 − 0 = 7, Root(w) = pref₇(w) = AHRRFFF (since 7 ∣ 7)
• Cyclic root = arg min_lex {σ^k(w)} = AHRRFFF
• Lyndon factorization = (AHRRFFF)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 3

Property	Roar	Ahrrfff
Alphabet	{A, O, R}	{A, F, H, R}
Length	4	7
Border	R	ε
Border length	1	0
Minimal period	3	7
Period set	{3, 4}	{7}
Critical exponent	4/3	3
Primitive root	ROAR	AHRRFFF
Root(w)	ROAR	AHRRFFF
Cyclic root	ARRO	AHRRFFF
Suffix array	[3, 2, 4, 1]	[1, 7, 6, 5, 2, 4, 3]
Minimal absent words	AA, AO, OO, OR, RA, RR, ARO	AA, AF, AR, FA, FH, FR, HA, HF, HH, RA, RH, HRF, RRR, FFFF
Lyndon factorization	(R)(O)(AR)	(AHRRFFF)
Biroot	(ROAR, ε), (ε, ROAR)	(AHRRFFF, ε), (ε, AHRRFFF)
Internal factors	A, O, OA	F, H, R, FF, HR, RF, RR, HRR, RFF, RRF, HRRF, RRFF, HRRFF

Once upon a time a Bear came into a meadow and said to the Vache, “I will eat you and then sleep here.”

The Vache answered, “Before you eat me, look at the ground.”

The Bear looked. Around the Vache the grass was short and the soil was bare in a wide circle. Farther out the grass was long and thick.

The Vache said, “I have eaten everything close to me. If you eat me now, you will have nothing left to eat after me, and you will have to walk far to find more food before you can sleep. If you leave me, I will keep eating the edge of the meadow. The grass will stay short near you, and you will not have to hunt when you wake.”

The Bear sat down. He watched the Vache graze for a while. She moved only a little and kept the bare circle growing slowly outward.

The Bear said, “If I eat you, the circle stops.”

The Vache answered, “Yes. Then the grass grows back and you must hunt again before you sleep.”

The Bear stood up and walked out of the meadow. He said, “The circle is useful while it grows.”

From that day the Bear did not eat a Vache who was slowly enlarging a bare circle in a meadow.

Moral: What keeps making a useful empty space around itself is harder to remove than what simply stands still.

Vache – Meuh (meuglement), Crunch (rumination)

Meuh (written MEUH)

• L(w) = 4
• Bord(w) = 0
• p = 4 − 0 = 4, Root(w) = pref₄(w) = MEUH (since 4 ∣ 4)
• Cyclic root = arg min_lex {σ^k(w)} = EUHM
• Lyndon factorization = (M)(EUH)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 1

Crunch (written CRUNCH)

• L(w) = 6
• Bord(w) = 0
• p = 6 − 0 = 6, Root(w) = pref₆(w) = CRUNCH (since 6 ∣ 6)
• Cyclic root = arg min_lex {σ^k(w)} = CHCRUN
• Lyndon factorization = (CRUN)(CH)
• Critical exponent ce(w) = max{∣x∣ / per(x) : x factor of w, x ≠ ε} = 5/4

Property	Meuh	Crunch
Alphabet	{E, H, M, U}	{C, H, N, R, U}
Length	4	6
Border	ε	ε
Border length	0	0
Minimal period	4	6
Period set	{4}	{6}
Critical exponent	1	5/4
Primitive root	MEUH	CRUNCH
Root(w)	MEUH	CRUNCH
Cyclic root	EUHM	CHCRUN
Suffix array	[2, 4, 1, 3]	[5, 1, 6, 4, 2, 3]
Minimal absent words	EE, EH, EM, HE, HH, HM, HU, MH, MM, MU, UE, UM, UU	CC, CN, CU, HC, HH, HN, HR, HU, NH, NN, NR, NU, RC, RH, RN, RR, UC, UH, UR, UU, NCR
Lyndon factorization	(M)(EUH)	(CRUN)(CH)
Biroot	(MEUH, ε), (ε, MEUH)	(CRUNCH, ε), (ε, CRUNCH)
Internal factors	E, U, EU	C, N, R, U, NC, RU, UN, RUN, UNC, RUNC