Improved bounds on the number of ternary square-free words
Uwe Grimm
备注：
17 pages, AMS LaTeX. Paper has been completely rewritten and comprises new results on both lower and upper bounds. The Mathematica program mentioned in the article can be downloaded at http://mcs.open.ac.uk/ugg2/wordcomb/brinkhuistriples.m
发表信息：
Journal of Integer Sequences, Vol. 4 (2001), Ar ticle 01.2.7
Improved upper and lower bounds on the number of square-free ternary words are obtained. The upper bound is based on the enumeration of square-free ternary words up to length 110. The lower bound is derived by constructing generalised Brinkhuis triples. The problem of finding such triples can essentially be reduced to a combinatorial problem, which can efficiently be treated by computer. In particular, it is shown that the number of square-free ternary words of length n grows at least as 65^(n/40), replacing the previous best lower bound of 2^(n/17).
We revisit the question of classification of balanced circular words and focus on the case of a ternary alphabet. We propose a $3$-dimensional generalisation of the discrete approximation representation of Christoffel words. By considering the minimal bound $3$ for abelian complexity of balanced circular words over a ternary alphabet, we provide a classification of all circular words over a ternary alphabet with abelian complexity subject to this bound. This result also allows us to construct an uncountable set of bi-infinite aperiodic words with abelian complexity equal to $3$.
A word is level if each letter appears in it the same number of times, plus or minus 1. We give a complete characterization of the lengths for which level ternary circular square-free words exist. Key words: combinatorics on words, circular words, necklaces, square-free words, non-repetitive sequences
We prove that the `connective constant' for ternary square-free words is at least $2^{1/17} = 1.0416 ... $, improving on Brinkhuis and Brandenburg's lower bounds of $2^{1/24}=1.0293 ...$ and $2^{1/22}=1.032 ...$ respectively. This is the first improvement since 1983.
We study infinite words u over an alphabet A satisfying the property P : P(n)+ P(n+1) = 1+ #A for any n in N, where P(n) denotes the number of palindromic factors of length n occurring in the language of u. We study also infinite words satisfying a stronger property PE: every palindrome of u has exactly one palindromic extension in u. For binary words, the properties P and PE coincide and these properties characterize Sturmian words, i.e., words with the complexity C(n)=n+1 for any n in N. In this paper, we focus on ternary infinite words with the language closed under reversal. For such words…
We investigate combinatorial properties of a kind of insets we defined in an earlier paper, interpreting them now in terms of restricted ternary words. This allows us to give new combinatorial interpretations of a number of known integer sequences, namely the coefficients of Chebyshev polynomials of both kinds, Fibonacci numbers, Delannoy numbers, asymmetric Delannoy numbers, Sulanke numbers, coordinating sequences for some cubic lattices, crystal ball sequences for some cubic lattices, and others. We also obtain several new properties of said insets. In particular, we derive three generating …
This paper focuses on generalized pseudostandard words, defined by de Luca and De Luca in 2006. In every step of the construction, the involutory antimorphism to be applied for the pseudopalindromic closure changes and is given by a so called directive bi-sequence. The concept of a normalized form of directive bi-sequences was introduced by Blondin-Massé et al. in 2013 and an algorithm for finding the normalized directive bi-sequence over a binary alphabet was provided. In this paper, we present an algorithm to find the normalized form of any directive bi-sequence over a ternary alphabet. More…
We answer a question of Harju: An infinite square-free ternary word with an $n$-stem factorization exists for any $n\ge 13$. We show that there are uniform ternary morphisms of length $k$ for every $k\ge 23$. This resolves almost completely a problem of the author and Rampersad.
Let $D^+$ be the first octant of the Euclidean space and consider the integral cube grid $G$ in $D^+$. The intersections of each line with $G$ form an infinite sequence of three letters which can be considered as an extension of well-known Sturmian words. A classification of such linear ternary sequences is presented and a family of examples is constructed from a notable sequence $S^M$ which could be viewed as an analogue of the Fibonacci word in the family of Sturmian words. The factor complexity and the palindromic complexity of these linear ternary sequences are also studied. The last resul…
An efficient, when compared to exhaustive enumeration, algorithm for computing the number of square-free words of length $n$ over the alphabet $\{a, b, c\}$ is presented.
A \emph{square} is a finite non-empty word consisting of two identical adjacent blocks. A word is \emph{square-free} if it does not contain a square as a factor. In any finite word one may delete the repeated block of a square, obtaining thereby a shorter word. By repeating this process, a square-free word is eventually reached, which we call a \emph{reduct} of the original word. How many different reducts a single word may have? It is not hard to prove that any binary word has exactly one reduct. We prove that there exist ternary words with arbitrarily many reducts. Moreover, the function cou…
We enumerate all ternary length-l square-free words, which are words avoiding squares of words up to length l, for l<=24. We analyse the singular behaviour of the corresponding generating functions. This leads to new upper entropy bounds for ternary square-free words. We then consider ternary square-free words with fixed letter densities, thereby proving exponential growth for certain ensembles with various letter densities. We derive consequences for the free energy and entropy of ternary square-free words.
A square-free word $w$ over a fixed alphabet $Σ$ is extremal if every word obtained from $w$ by inserting a single letter from $Σ$ (at any position) contains a square. Grytczuk et al. recently introduced the concept of extremal square-free word, and demonstrated that there are arbitrarily long extremal square-free ternary words. We find all lengths which admit an extremal square-free ternary word. In particular, we show that there is an extremal square-free ternary word of every sufficiently large length. We also solve the analogous problem for circular words.
A word $u$ defined over an alphabet $\mathcal{A}$ is $c$-balanced ($c\in\mathbb{N}$) if for all pairs of factors $v$, $w$ of $u$ of the same length and for all letters $a\in\mathcal{A}$, the difference between the number of letters $a$ in $v$ and $w$ is less or equal to $c$. In this paper we consider a ternary alphabet $\mathcal{A}=\{L,S,M\}$ and a class of substitutions $ϕ_p$ defined by $ϕ_p(L)=L^pS$, $ϕ_p(S)=M$, $ϕ_p(M)=L^{p-1}S$ where $p>1$. We prove that the fixed point of $ϕ_p$, formally written as $ϕ_p^\infty(L)$, is 3-balanced and that its Abelian complexity is bounded above by the valu…
Any amicable pair ϕ, ψ of Sturmian morphisms enables a construction of a ternary morphism η which preserves the set of infinite words coding 3-interval exchange. We determine the number of amicable pairs with the same incidence matrix in $SL^\pm(2,N)$ and we study incidence matrices associated with the corresponding ternary morphisms η.
Circular words are cyclically ordered finite sequences of letters. We give a computer-free proof of the following result by Currie: square-free circular words over the ternary alphabet exist for all lengths $l$ except for 5, 7, 9, 10, 14, and 17. Our proof reveals an interesting connection between ternary square-free circular words and closed walks in the $K_{3{,}3}$ graph. In addition, our proof implies an exponential lower bound on the number of such circular words of length $l$ and allows one to list all lengths $l$ for which such a circular word is unique up to isomorphism.
We study the properties of the ternary infinite word p = 012102101021012101021012 ... , that is, the fixed point of the map h:0->01, 1->21, 2->0. We determine its factor complexity, critical exponent, and prove that it is 2-balanced. We compute its abelian complexity and determine the lengths of its bispecial factors. Finally, we give a characterization of p in terms of avoided factors.
We introduce the notion of general prints of a word, which is substantialized by certain canonical decompositions, to study repetition in words. These associated decompositions, when applied recursively on a word, result in what we term as core prints of the word. The length of the path to attain a core print of a general word is scrutinized. This paper also studies the class of square-free ternary words with respect to the Parikh matrix mapping, which is an extension of the classical Parikh mapping. It is shown that there are only finitely many matrix-equivalence classes of ternary words such…
Let s(n) be the number of representations of n as the sum of three squares. We prove a remarkable new identity for s(p^2n)- ps(n) with p being an odd prime. This identity makes nontrivial use of ternary quadratic forms with discriminants p^2 and 16p^2. These forms are related by Watson's transformations. To prove this identity we employ the Siegel--Weil and the Smith--Minkowski product formulas.
We begin a systematic study of the relations between subword complexity of infinite words and their power avoidance. Among other things, we show that -- the Thue-Morse word has the minimum possible subword complexity over all overlap-free binary words and all $(\frac 73)$-power-free binary words, but not over all $(\frac 73)^+$-power-free binary words; -- the twisted Thue-Morse word has the maximum possible subword complexity over all overlap-free binary words, but no word has the maximum subword complexity over all $(\frac 73)$-power-free binary words; -- if some word attains the minimum poss…
We review the recent progress in the investigation of powerfree words, with particular emphasis on binary cubefree and ternary squarefree words. Besides various bounds on the entropy, we provide bounds on letter frequencies and consider their empirical distribution obtained by an enumeration of binary cubefree words up to length 80.
We study words that barely avoid repetitions, for several senses of "barely". A squarefree (respectively, overlap-free, cubefree) word is irreducible if removing any one of its interior letters creates a square (respectively, overlap, cube). A squarefree (respectively, overlap-free, cubefree) word is delicate if changing any one of its letters creates a square (respectively, overlap, cube). We classify the lengths of irreducible and delicate squarefree, overlap-free, and cubefree words over binary and ternary alphabets.
In combinatorics on words, a word $w$ over an alphabet $Σ$ is said to avoid a pattern $p$ over an alphabet $Δ$ if there is no factor $f$ of $w$ such that $f=h(p)$ where $h:Δ^*\toΣ^*$ is a non-erasing morphism. A pattern $p$ is said to be $k$-avoidable if there exists an infinite word over a $k$-letter alphabet that avoids $p$. A pattern is \emph{doubled} if every variable occurs at least twice. Doubled patterns are known to be $3$-avoidable. Currie, Mol, and Rampersad have considered a generalized notion which allows variable occurrences to be reversed. That is, $h(V^R)$ is the mirror image of…
We study the avoidability of long $k$-abelian-squares and $k$-abelian-cubes on binary and ternary alphabets. For $k=1$, these are Mäkelä's questions. We show that one cannot avoid abelian-cubes of abelian period at least $2$ in infinite binary words, and therefore answering negatively one question from Mäkelä. Then we show that one can avoid $3$-abelian-squares of period at least $3$ in infinite binary words and $2$-abelian-squares of period at least 2 in infinite ternary words. Finally we study the minimum number of distinct $k$-abelian-squares that must appear in an infinite binary word.
We introduce new avoidability problems for words by considering equivalence relations, k-abelian equivalences, which lie properly in between equality and commutative equality, i.e. abelian equality. For two k-abelian equivalent words the numbers of occurrences of different factors of length k coincide and the prefixies (resp. suffixies) of length k-1 are equal as well. The size of the smallest alphabet avoiding 2-repetitions of words, i.e. squares, is three and for abelian squares it is four. It follows that for 2-abelian squares this size has to be three or four. Similarly, the size of the sm…
We show that the set of binary words containing overlaps is not unambiguously context-free and that the set of ternary words containing overlaps is not context-free. We also show that the set of binary words that are not subwords of the Thue-Morse word is not unambiguously context-free.
We consider words $w$ over the alphabet $Σ=\{0,1,2\}$. It is shown that there are irreducibly square-free words of all lengths $n$ except 4,5,7 and 12. Such a word is square-free (i.e., it has no repetitions $uu$ as factors), but by removing any one internal letter creates a square in the word.
We characterize exactly the lengths of binary circular words containing no squares other than 00, 11, and 0101. Key words: combinatorics on words, circular words, necklaces, square-free words, non-repetitive sequences
A word is \emph{square-free} if it does not contain non-empty factors of the form $XX$. In 1906 Thue proved that there exist arbitrarily long square-free words over $3$-letter alphabet. We consider a new type of square-free words. A square-free word is \emph{extremal} if it cannot be extended to a new square-free word by inserting a single letter on arbitrary position. We prove that there exist infinitely many extremal words over $3$-letter alphabet. Some parts of our construction relies on computer verifications. We also pose some related open problems.
A square is a concatenation of two identical words, and a word $w$ is said to have a square $yy$ if $w$ can be written as $xyyz$ for some words $x$ and $z$. It is known that the ratio of the number of distinct squares in a word to its length is less than two and any location of a word could begin with at most two rightmost distinct squares. A square whose first location starts with the last occurrence of two distinct squares is an FS-double square. We explore and identify the conditions to generate a sequence of locations in a word that starts with FS-double squares. We first find the structur…
A word is square-free if it does not contain nonempty factors of the form $XX$. In 1906 Thue proved that there exist arbitrarily long square-free words over a $3$-letter alphabet. It was proved recently [7] that among these words there are infinitely many extremal ones, that is, having a square in every single-letter extension. We study diverse problems concerning extensions of words preserving the property of avoiding squares. Our main motivation is the conjecture stating that there are no extremal words over a $4$-letter alphabet. We also investigate a natural recursive procedure of generati…
A word is square-free if it does not contain a nonempty word of the form $XX$ as a factor. A famous 1906 result of Thue asserts that there exist arbitrarily long square-free words over a $3$-letter alphabet. We study square-free words with additional properties involving single-letter deletions and extensions of words. A square-free word is steady if it remains square-free after deletion of any single letter. We prove that there exist infinitely many steady words over a $4$-letter alphabet. We also demonstrate that one may construct steady words of any length by picking letters from arbitrary …
Let $s_n$ be the number of words consisting of the ternary alphabet consisting of the digits 0, 1, and 2 such that no subword (or factor) is a square (a word concatenated with itself, e.g., $11$, $1212$, or $102102$). From computational evidence, $s_n$ grows exponentially at a rate of about $1.317277^n$. While known upper bounds are already relatively close to the conjectured rate, effective lower bounds are much more difficult to obtain. In this paper, we construct a $54$-Brinkhuis $952$-triple, which leads to an improved lower bound on the number of $n$-letter ternary squarefree words: $952^…
In a recent paper, Harju posed three open problems concerning square-free arithmetic progressions in infinite words. In this note we solve two of them.
We consider partial words with a unique position starting a power. We show that over a $k$ letter alphabet, a partial word with a unique position starting a square can contain at most $k$ squares. This is in contrast to full words which can contain at most one power if a unique position starts a power. For certain higher powers we exhibit binary partial words containing three powers all of which start at the same position.
A {\em square} is a word of the form $uu$. In this paper we prove that for a given finite word $w$, the number of distinct square factors of $w$ is bounded by $|w|-|\Alphabet(w)|+1$, where $|w|$ denotes the length of $w$ and $|\Alphabet(w)|$ denotes the number of distinct letters in $w$. This result answers a conjecture of Fraenkel and Simpson stated in 1998.
In this paper, we study binary and ternary matrices that are used for CDMA applications that are injective on binary or ternary user vectors. In other words, in the absence of additive noise, the interference of overloaded CDMA can be removed completely. Some new algorithms are proposed for constructing such matrices. Also, using an information theoretic approach, we conjecture the extent to which such CDMA matrix codes exist. For overloaded case, we also show that some of the codes derived from our algorithms perform better than the binary Welch Bound Equality codes; the decoding is ML but of…
We show that every infinite word $ω$ on a finite subset of $\mathbb{Z}$ must contain arbitrarily large factors $B_1B_2$ which are "close" to being \textit{additive squares}. We also show that for all $k>1, \ ω$ must contain a factor $U_1U_2 ... U_k$ where $U_1,U_2,..., U_k$ all have the same \textit{average.}
We consider sets of factors that can be avoided in square-free words on two-generator free groups. The elements of the group are presented in terms of 0,1,2,3 such that 0 and 2 (resp.,1 and 3) are inverses of each other so that 02, 20, 13 and 31 do not occur in a reduced word. A Dean word is a reduced word that does not contain occurrences of $uu$ for any nonempty $u$. Dean showed in 1965 that there exist infinite square-free reduced words. We show that if $w$ is a Dean word of length at least 59 then there are at most six reduced words of length 3 avoided by $w$. We construct an infinite Dean…
A Tighter Upper Bound for the Number of Distinct Squares in Circular Words
Hangzhou International Innovation Institute of Beihang University（北京航空航天大学杭州国际创新学院）；LMIB-School of Mathematical Sciences, Beihang University（北京航空航天大学数学科学学院）
Carpi constructed an infinite word over a 4-letter alphabet that avoids squares in all subsequences indexed by arithmetic progressions of odd difference. We show a connection between Carpi's construction and the paperfolding words. We extend Carpi's result by constructing uncountably many words that avoid squares in arithmetic progressions of odd difference. We also construct infinite words avoiding overlaps and infinite words avoiding arbitrarily large squares in arithmetic progressions of odd difference. We use these words to construct labelings of the 2-dimensional integer lattice such that…
A word is square-free if it does not contain any square (a word of the form $XX$), and is extremal square-free if it cannot be extended to a new square-free word by inserting a single letter at any position. Grytczuk, Kordulewski, and Niewiadomski proved that there exist infinitely many ternary extremal square-free words. We establish that there are no extremal square-free words over any alphabet of size at least 17.
Let $u \shuffle v$ denote the set of all shuffles of the words $u$ and $v$. It is shown that for each integer $n \geq 3$ there exists a square-free ternary word $u$ of length $n$ such that $u\shuffle u$ contains a square-free word. This property is then shown to also hold for infinite words, i.e., there exists an infinite square-free word $u$ on three letters such that $u$ can be shuffled with itself to produce an infinite square-free word $w \in u \shuffle u$.
A position $p$ in a word $w$ is critical if the minimal local period at $p$ is equal to the global period of $w$. According to the Critical Factorisation Theorem all words of length at least two have a critical point. We study the number $η(w)$ of critical points of square-free ternary words $w$, i.e., words over a three letter alphabet. We show that the sufficiently long square-free words $w$ satisfy $η(w) \le |w|-5$ where $|w|$ denotes the length of $w$. Moreover, the bound $|w|-5$ is reached by infinitely many words. On the other hand, every square-free word $w$ has at least $|w|/4$ critica…
We are interested in the maximal number of distinct squares in a word. This problem was introduced by Fraenkel and Simpson, who presented a bound of 2n for a word of length n, and conjectured that the bound was less than n. Being that the problem is on repetitions, their solution relies on Fine and Wilf's Periodicity lemma. Ilie then refined their result and presented a bound of 2n-O(log n). Lam used an induction to get a bound of 95n/48. Deza, Franek and Thierry achieved a bound of 11n/6 through a combinatorial approach. Using the properties of the core of the interrupt, presented by Thierry,…
For any integer $k>2$, the infinite $k$-bonacci word $W^{(k)}$, on the infinite alphabet is defined as the fixed point of the morphism $φ_k:\mathbb{N}\rightarrow \mathbb{N}^2 \cup \mathbb{N}$, where \begin{equation*} φ_k(ki+j) = \left\{ \begin{array}{ll} (ki)(ki+j+1) & \text{if } j = 0,\cdots ,k-2, (ki+j+1)& \text{if } j =k-1. \end{array} \right. \end{equation*} The finite $k$-bonacci word $W^{(k)}_n$ is then defined as the prefix of $W^{(k)}$ whose length is the $(n+k)$-th $k$-bonacci number. We obtain the structure of all square factors occurring in $W^{(k)}$. Moreover, we prove that the cri…
A power is a word of the form $\underbrace{uu...u}_{k \; \text{times}}$, where $u$ is a word and $k$ is a positive integer and a square is a word of the form $uu$. Fraenkel and Simpson conjectured in 1998 that the number of distinct squares in a word is bounded by the length of the word. This conjecture was proven recently by Brlek and Li. Besides, there exists a stronger upper bound for binary words conjectured by Jonoska, Manea and Seki stating that for a word of length $n$ over the alphabet $\left\{a, b\right\}$, if we let $k$ be the least of the number of a's and the number of b's and $k \…
Joint Graduate School of Mathematics for Innovation, Kyushu University（九州大学数学创新联合研究生院）；Graduate School of Informatics and Engineering, University of Electro-Communications（电子通信大学信息与工程研究生院）；Department of Computer Science and Engineering, University of Yamanashi（山梨大学计算机科学与工程系）；Department of Informatics, Kyushu University（九州大学信息学系）
A word $\bar{w} = \bar{u}\bar{u}$ is a $long$ $square$ if $\bar{u}$ is of length at least 3; a word $\bar{w}$ is $long$-$square$-$free$ if $\bar{w}$ contains no sub-word that is a long square. We can use words to generate graph colorings; a graph coloring is called $long$-$repetition$-$free$ if the word formed by the coloring of each path in the graph is long-square-free. Our results show that every rooted tree of radius less than or equal to seven is long-repetition-free two-colorable. We also prove there exists a class of trees which are not long-repetition-free two-colorable.
