Squarefree Word

In combinatorics, a squarefree word is a word (a sequence of symbols) that does not contain any squares. A square is a word of the form , where is not empty. Thus, a squarefree word can also be defined as a word that avoids the pattern .

Finite squarefree words

Binary alphabet

Over a binary alphabet $\$, the only squarefree words are the empty word $\epsilon,0,1,01,10,010$, and $101$.

Ternary alphabet

Over a ternary alphabet ''$\$'', there are infinitely many squarefree words. It is possible to count the number $c(n)$ of ternary squarefree words of length .

This number is bounded by $c(n) = \Theta(\alpha^n)$, where $1.3017597 < \alpha < 1.3017619$. The upper bound on $\alpha$ can be found via Fekete's Lemma and approximation by automata. The lower bound can be found by finding a substitution that preserves squarefreeness.

Alphabet with more than three letters

Since there are infinitely many squarefree words over three-letter alphabets, this implies there are also infinitely many squarefree words over an alphabet with more than three letters.

The following table shows the exact growth rate of the -ary squarefree words:

2-dimensional words

Consider a map $\textbf$ from $\mathbb^2$ to , where is an alphabet and $\textbf$ is called a 2-dimensional word. Let $w_$ be the entry $\textbf(m,n)$. A word $\textbf$ is a line of $\textbf$ if there exists $i_1,i_2,j_1, j_2$such that $\text(j_1, j_2) = 1$, and for $t \ge 0, x_t = w_$.

Carpi proves that there exists a 2-dimensional word $\textbf$ over a 16-letter alphabet such that every line of $\textbf$ is squarefree. A computer search shows that there are no 2-dimensional words $\textbf$over a 7-letter alphabet, such that every line of $\textbf$ is squarefree.

Generating finite squarefree words

Shur proposes an algorithm called R2F (random-t(w)o-free) that can generate a squarefree word of length over any alphabet with three or more letters. This algorithm is based on a modification of entropy compression: it randomly selects letters from a k-letter alphabet to generate a -ary squarefree word.
algorithm R2F is
input: alphabet size $k \ge 2$'',''
word length ''$n > 1$''
output: a -ary squarefree word of length .

choose w /math> in $\Sigma_$ uniformly at random
set $\chi_w$ to ''w /math>'' followed by all other letters of $\Sigma_$ in increasing order
set the number of iterations to 0

while $, w, < n$ do
choose in $\Sigma_$ uniformly at random
append a = \chi_w +1/math> to the end of
update $\chi_w$ shifting the first elements to the right and setting \chi_w = a
increment by
if ends with a square of rank then
delete the last letters of

return
Every (k+1)-ary squarefree word can be the output of Algorithm R2F, because on each iteration it can append any letter except for the last letter of .

The expected number of random k-ary letters used by Algorithm R2F to construct a -ary squarefree word of length is$N=n\left(1 + \frac 2 + \frac 1 + \frac 4 + O\left(\frac 1 \right)\right)+O(1).$Note that there exists an algorithm that can verify the squarefreeness of a word of length in $O(n \log n)$ time. Apostolico and Preparata give an algorithm using suffix trees. Crochemore uses partitioning in his algorithm. Main and Lorentz provide an algorithm based on the divide-and-conquer method. A naive implementation may require ''$O(n^2)$'' time to verify the squarefreeness of a word of length .

Infinite squarefree words

There exist arbitrarily long squarefree words in any alphabet with three or more letters, as proved by Axel Thue.

Examples

First difference of the

Thue–Morse sequence

One example of an infinite squarefree word over an alphabet of size 3 is the word over the alphabet $\$ obtained by taking the first difference of the Thue–Morse sequence. That is, from the Thue–Morse sequence

: $0, 1 ,1, 0 ,1 ,0 ,0 ,1, 1 ,0 ,0 ,1, 0 ,1 ,1, 0 ...$

one forms a new sequence in which each term is the difference of two consecutive terms of the Thue–Morse sequence. The resulting squarefree word is

: $1,0,-1,1,-1,0,1,0,-1,0,1,-1,1,0,-1,...$.

Leech's morphism

Another example found by John Leech is defined recursively over the alphabet $\$. Let be any squarefree word starting with the letter . Define the words $\$ recursively as follows: the word $w_$ is obtained from by replacing each in with , each with , and each with . It is possible to prove that the sequence converges to the infinite squarefree word
:

Generating infinite squarefree words

Infinite squarefree words can be generated by squarefree morphism. A morphism is called squarefree if the image of every squarefree word is squarefree. A morphism is called k–squarefree if the image of every squarefree word of length k is squarefree.

Crochemore proves that a uniform morphism is squarefree if and only if it is 3-squarefree. In other words, is squarefree if and only if $h(w)$ is squarefree for all squarefree of length 3. It is possible to find a squarefree morphism by brute-force search.
algorithm squarefree_morphism is
output: a squarefree morphism with the lowest possible rank .

set $k = 3$
while True do
set ''k_sf_words'' to the list of all squarefree words of length over a ternary alphabet
for each $h(0)$ in ''k_sf_words'' do
for each $h(1)$ in ''k_sf_words'' do
for each $h(2)$ in ''k_sf_words'' do
if $h(1) = h(2)$ then
break from the current loop (advance to next $h(1)$)
if $h(0) \ne h(1)$ and $h(2) \ne h(0)$ then
if $h(w)$ is squarefree for all squarefree of length then
return $h(0), h(1), h(2)$
increment by
Over a ternary alphabet, there are exactly 144 uniform squarefree morphisms of rank 11 and no uniform squarefree morphisms with a lower rank than 11.

To obtain an infinite squarefree words, start with any squarefree word such as , and successively apply a squarefree morphism to it. The resulting words preserve the property of squarefreeness. For example, let be a squarefree morphism, then as $w \to \infty$, $h^(0)$ is an infinite squarefree word.

Note that, if a morphism over a ternary alphabet is not uniform, then this morphism is squarefree if and only if it is 5-squarefree.

Letter combinations in squarefree words

Avoid two-letter combinations

Over a ternary alphabet, a squarefree word of length more than 13 contains all the squarefree two-letter combinations.

This can be proved by constructing a squarefree word without the two-letter combination . As a result, ' is the longest squarefree word without the combination and its length is equal to 13.

Note that over a more than three-letter alphabet there are squarefree words of any length without an arbitrary two-letter combination.

Avoid three-letter combinations

Over a ternary alphabet, a squarefree word of length more than 36 contains all the squarefree three-letter combinations.

However, there are squarefree words of any length without the three-letter combination .

Note that over a more than three-letter alphabet there are squarefree words of any length without an arbitrary three-letter combination.

Density of a letter

The density of a letter in a finite word is defined as $\frac$ where $, w, _a$ is the number of occurrences of in $w$ and $, w,$ is the length of the word. The density of a letter in an infinite word is $\liminf_\frac$ where $w_l$ is the prefix of the word of length .

The minimal density of a letter in an infinite ternary squarefree word is equal to $\frac$.

The maximum density of a letter in an infinite ternary squarefree word is equal to $\frac$.

Notes

References

*
* .
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* {{cite book , last=Pytheas Fogg , first=N. , editor1=Berthé, Valérie, editor1-link=Valérie Berthé, editor2=Ferenczi, Sébastien, editor3=Mauduit, Christian, editor4=Siegel, Anne , title=Substitutions in dynamics, arithmetics and combinatorics , series=Lecture Notes in Mathematics , volume=1794 , location=Berlin , publisher=Springer-Verlag , year=2002 , isbn=978-3-540-44141-0 , zbl=1014.11015

Formal languages
Combinatorics on words