In this work, we study several equivalence relations induced from the partitions of the sets of words of finite length. We have results on words over finite fields extending the work of Bacher (2002, Europ. J. Combinatorics, {\bf 23}, 141-147). Cardinalities of its equivalence classes and explicit relationships between two words are determined. Moreover, we deal with words of finite length over the ring $\mathbb{Z}/N\mathbb{Z}$ where $N$ is a positive integer. We have arithmetic results parallel to Bacher's.
Stankova and West showed that for any non-negative integer $s$ and any permutation $γ$ of $\{4,5,\dots,s+3\}$ there are as many permutations that avoid $231γ$ as there are that avoid $312γ$. We extend this result to the setting of words.
Episturmian morphisms constitute a powerful tool to study episturmian words. Indeed, any episturmian word can be infinitely decomposed over the set of pure episturmian morphisms. Thus, an episturmian word can be defined by one of its morphic decompositions or, equivalently, by a certain directive word. Here we characterize pairs of words directing a common episturmian word. We also propose a way to uniquely define any episturmian word through a normalization of its directive words. As a consequence of these results, we characterize episturmian words having a unique directive word.
Covering arrays for words of length $t$ over a $d$ letter alphabet are $k \times n$ arrays with entries from the alphabet so that for each choice of $t$ columns, each of the $d^t$ $t$-letter words appears at least once among the rows of the selected columns. We study two schemes in which all words are not considered to be different. In the first case words are equivalent if they induce the same partition of a $t$ element set. In the second case, words of the same weight are equivalent. In both cases we produce logarithmic upper bounds on the minimum size $k=k(n)$ of a covering array. Definitiv…
Two same length words are $d$-equivalent if they have same descent set and same underlying alphabet. In particular, two same length permutations are $d$-equivalent if they have same descent set. The popularity of a pattern in a set of words is the overall number of copies of the pattern within the words of the set. We show the far-from-trivial fact that two patterns are $d$-equivalent if and only if they are equipopular over any $d$-equivalence class, and this equipopularity does not follow obviously from a trivial equidistribution.
Two finite words $u$ and $v$ are $k$-binomially equivalent if, for each word $x$ of length at most $k$, $x$ appears the same number of times as a subsequence (i.e., as a scattered subword) of both $u$ and $v$. This notion generalizes abelian equivalence. In this paper, we study the equivalence classes induced by the $k$-binomial equivalence with a special focus on the cardinalities of the classes. We provide an algorithm generating the $2$-binomial equivalence class of a word. For $k \geq 2$ and alphabet of $3$ or more symbols, the language made of lexicographically least elements of every $k$…
In the first part of this survey, we present classical notions arising in combinatorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our presentation tries to set up a unified framework with respect to a given binary relation. In the second part, we mainly focus on abelian equivalence, $k$-abelian equivalence, combinatorial coefficients and associated relations, Parikh matrices and $M$-equivalence. In particular, some new refinements of abelian equivalence are introduced.
A K-theoretic analogue of RSK insertion and Knuth equivalence relations was first introduced in 2006 by Buch, Kresch, Shimozono, Tamvakis, and Yong. The resulting K-Knuth equivalence relations on words and increasing tableaux on [n] has prompted investigation into the equivalence classes of tableaux arising from these relations. Of particular interest are the tableaux that are unique in their class, which we refer to as unique rectification targets (URTs). In this paper we give several new families of URTs and a bound on the length of intermediate words connecting two K-Knuth equivalent words.…
Parikh matrices have been a powerful tool in arithmetizing words by numerical quantities. However, the dependence on the ordering of the alphabet is inherited by Parikh matrices. Strong M-equivalence is proposed as a canonical alternative to M-equivalence to get rid of this undesirable property. Some characterization of strong M-equivalence for a restricted class of words is obtained. Finally, the existential counterpart of strong M-equivalence is introduced as well.
In this paper we investigate local to global phenomena for a new family of complexity functions of infinite words indexed by $k \in \Ni \cup \{+\infty\}$ where $\Ni$ denotes the set of positive integers. Two finite words $u$ and $v$ in $A^*$ are said to be $k$-Abelian equivalent if for all $x \in A^*$ of length less than or equal to $k$, the number of occurrences of $x$ in $u$ is equal to the number of occurrences of $x$ in $v$. This defines a family of equivalence relations $\sim_k$ on $A^*$, bridging the gap between the usual notion of Abelian equivalence (when $k = 1$) and equality (when $k…
Parikh matrices have been extensively investigated due to their usefulness in studying subword occurrences in words. Due to the dependency of Parikh matrices on the ordering of the alphabet, strong M-equivalence was proposed as an order-independent alternative to M-equivalence in studying words possessing the same Parikh matrix. This paper introduces and studies the notions of strong 2.t and strong 3.t transformations in determining when two ternary words are strongly M-equivalent. The irreducibility of strong 2.t transformations are then scrutinized, exemplified by a structural characterizati…
In this paper we introduce and study a family of complexity functions of infinite words indexed by $k \in \ints ^+ \cup {+\infty}.$ Let $k \in \ints ^+ \cup {+\infty}$ and $A$ be a finite non-empty set. Two finite words $u$ and $v$ in $A^*$ are said to be $k$-Abelian equivalent if for all $x\in A^*$ of length less than or equal to $k,$ the number of occurrences of $x$ in $u$ is equal to the number of occurrences of $x$ in $v.$ This defines a family of equivalence relations $\thicksim_k$ on $A^*,$ bridging the gap between the usual notion of Abelian equivalence (when $k=1$) and equality (when $…
Vincular or dashed patterns resemble classical patterns except that some of the letters within an occurrence are required to be adjacent. We prove several infinite families of Wilf-equivalences for k-ary words involving vincular patterns containing a single dash, which explain the majority of the equivalences witnessed for such patterns of length four. When combined with previous results, numerical evidence, and some arguments in specific cases, we obtain the complete Wilf-classification for all vincular patterns of length four containing a single dash. In some cases, our proof shows further t…
Two paths are equivalent modulo a given string $τ$, whenever they have the same length and the positions of the occurrences of $τ$ are the same in both paths. This equivalence relation was introduced for Dyck paths in \cite{BP}, where the number of equivalence classes was evaluated for any string of length 2. In this paper, we evaluate the number of equivalence classes in the set of ballot paths for any string of length 2 and 3, as well as in the set of Dyck paths for any string of length 3.
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
We investigate the number of sets of words that can be formed from a finite alphabet, counted by the total length of the words in the set. An explicit expression for the counting sequence is derived from the generating function, and asymptotics for large alphabet respectively large total word length are discussed. Moreover, we derive a Gaussian limit law for the number of words in a random finite language.
We introduce an equivalence relation, called cobordism, for words and study cobordism invariants of words inspired by methods of low-dimensional topology.
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 study the palindromic length of factors of infinite words fixed by morphisms of the so-called class $\mathcal{P}$ introduced by Hof, Knill and Simon. We show that it grows at most logarithmically with the length of the factor. For the Fibonacci word and the Thue-Morse word we provide estimates on the constants of the growth. We also construct an infinite word rich in palindromes for which the palindromic length grows as $\sqrt{n}$.
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）
The Rearrangement Conjecture states that if two words over $\mathbb{P}$ are Wilf-equivalent in the factor order on $\mathbb{P}^\ast$ then they are rearrangements of each other. We introduce the notion of strong Wilf-equivalence and prove that if two words over $\mathbb{P}$ are strongly Wilf-equivalent then they are rearrangements of each other. We further conjecture that Wilf-equivalence implies strong Wilf-equivalence.
Two words are $k$-binomially equivalent whenever they share the same subwords, i.e., subsequences, of length at most $k$ with the same multiplicities. This is a refinement of both abelian equivalence and the Simon congruence. The $k$-binomial complexity of an infinite word $\mathbf{x}$ maps the integer $n$ to the number of classes in the quotient, by this $k$-binomial equivalence relation, of the set of factors of length $n$ occurring in $\mathbf{x}$. This complexity measure has not been investigated very much. In this paper, we characterize the $k$-binomial complexity of the Thue--Morse word.…
We survey known results and open problems in abelian combinatorics on words. Abelian combinatorics on words is the extension to the commutative setting of the classical theory of combinatorics on words. The extension is based on \emph{abelian equivalence}, which is the equivalence relation defined in the set of words by having the same Parikh vector, that is, the same number of occurrences of each letter of the alphabet. In the past few years, there was a lot of research on abelian analogues of classical definitions and properties in combinatorics on words. This survey aims to gather these res…
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.
Two words $u$ and $v$ are $k$-abelian equivalent if, for each word $x$ of length at most $k$, $x$ occurs equally many times as a factor in both $u$ and $v$. The notion of $k$-abelian equivalence is an intermediate notion between the abelian equivalence and the equality of words. In this paper, we study the equivalence classes induced by the $k$-abelian equivalence, mainly focusing on the cardinalities of the classes. In particular, we are interested in the number of singleton $k$-abelian classes, i.e., classes containing only one element. We find a connection between the singleton classes and …
The method we have applied in "A. Bernini, L. Ferrari, R. Pinzani, Enumerating permutations avoiding three Babson-Steingrimsson patterns, Ann. Comb. 9 (2005), 137--162" to count pattern avoiding permutations is adapted to words. As an application, we enumerate several classes of words simultaneously avoiding two generalized patterns of length 3.
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…
We show that for every $n \geq 1$ and over any finite alphabet, there is a word whose circular factors of length $n$ have a one-to-one correspondence with the set of primitive words. In particular, we prove that such a word can be obtained by a greedy algorithm, or by concatenating all Lyndon words of length $n$ in increasing lexicographic order. We also look into connections between de Bruijn graphs of primitive words and Lyndon graphs. Finally, we also show that the shortest word that contains every $p$-power of length $pn$ over a $k$-letter alphabet has length between $pk^n$ and roughly $(p…
A run in a word is a periodic factor whose length is at least twice its period and which cannot be extended to the left or right (by a letter) to a factor with greater period. In recent years a great deal of work has been done on estimating the maximum number of runs that can occur in a word of length $n$. A number of associated problems have also been investigated. In this paper we consider a new variation on the theme. We say that the total run length (TRL) of a word is the sum of the lengths of the runs in the word and that $τ(n)$ is the maximum TRL over all words of length $n$. We show tha…
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.
Kitaev, Liese, Remmel, and Sagan recently defined generalized factor order on words comprised of letters from a partially ordered set $(P, \leq_P)$ by setting $u \leq_P w$ if there is a subword $v$ of $w$ of the same length as $u$ such that the $i$-th character of $v$ is greater than or equal to the $i$-th character of $u$ for all $i$. This subword $v$ is called an embedding of $u$ into $w$. For the case where $P$ is the positive integers with the usual ordering, they defined the weight of a word $w = w_1\ldots w_n$ to be $\text{wt}(w) = x^{\sum_{i=1}^n w_i} t^{n}$, and the corresponding weigh…
We prove several Wilf-equivalences for vincular patterns of length 4, some of which generalize to infinite families of vincular patterns. We also present functional equations for the generating functions for the number of permutations of length n avoiding a single pattern for the patterns 124-3, 134-2, 231-4, 241-3, 132-4, and 142-3. This nearly completes the Wilf-classification of vincular patterns of length 4. As a corollary, these results imply Wilf-equivalences for certain barred patterns of length 5 with a single bar.
We say that a word $w$ of length $kn$ is a $k$-\textit{antipower} if it can be written in the form $w_1 \cdots w_k$, where each $w_i$ is a distinct word of length $n$. We analyze prefixes of the Thue-Morse word $\textbf{t}$ and lengths of antipowers occurring in them. Define $Γ(k)$ to be the largest odd $n$ such that the prefix of $\textbf{t}$ of length $kn$ is not a $k$-antipower, and $γ(k)$ to be the smallest odd $n$ such that the corresponding prefix is a $k$-antipower. We provide strong bounds on the asymptotic values of $γ(k)$ and $Γ(k)-γ(k)$. Our bounds on $γ(k)$ affirmatively answer one…
The aim of the paper is to enumerate all closed knight paths of length n over a square board of size n+1. The closed knight paths of length 4, 6 and 8 are classified up to equivalence. We determine that there are exactly 3 equivalence classes of closed knight paths of length 4, exactly 25 equivalence classes of closed knight paths of length 6 and exactly 478 equivalence classes of closed knight paths of length 8.
We define an analog of David Little's algorithm for reduced words in type B, and investigate its main properties. In particular, we show that our algorithm preserves the recording tableau of Kraśkiewicz insertion, and that it provides a bijective realization of the Type B transition equations in Schubert calculus. Many other aspects of type A theory carry over to this new setting. Our primary tool is a shifted version of the dual equivalence graphs defined by Assaf and further developed by Roberts. We provide an axiomatic characterization of shifted dual equivalence graphs, and use them to pro…
A word is "crucial" with respect to a given set of "prohibited words" (or simply "prohibitions") if it avoids the prohibitions but it cannot be extended to the right by any letter of its alphabet without creating a prohibition. A "minimal crucial word" is a crucial word of the shortest length. A word W contains an "abelian k-th power" if W has a factor of the form X_1X_2...X_k where X_i is a permutation of X_1 for 2<= i <= k. When k=2 or 3, one deals with "abelian squares" and "abelian cubes", respectively. In 2004 (arXiv:math/0205217), Evdokimov and Kitaev showed that a minimal crucial word o…
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…
Catalan words are particular growth-restricted words over the set of non-negative integers, and they represent still another combinatorial class counted by the Catalan numbers. We study the distribution of descents on the sets of Catalan words avoiding a pattern of length at most three: for each such a pattern $p$ we provide a bivariate generating function where the coefficient of $x^ny^k$ in its series expansion is the number of length $n$ Catalan words with $k$ descents and avoiding $p$. As a byproduct, we enumerate the set of Catalan words avoiding $p$, and we provide the popularity of desc…
The notion of palindromic length of a finite word, as well as an infinite word, was first introduced by Frid, Puzynina and Zamboni\cite{FRID2013737}. They conjectured that if the palindromic length of an infinite word is bounded, then this sequence is eventually periodic. In this article, we give a formal expression of the palindromic length of Thue-Morse sequence and find all sequences which have the same palindromic length as Thue-Morse's.
Frid, Puzynina and Zamboni (2013) defined the palindromic length of a finite word $w$ as the minimal number of palindromes whose concatenation is equal to $w$. For an infinite word $u$ we study $PL_{u}$, that is, the function that assigns to each positive integer $n$, the maximal palindromic length of factors of length $n$ in $u$. Recently, Frid (2018) proved that $\limsup_{n\to\infty} PL_{u}(n)=+\infty$ for any Sturmian word $u$. We show that there is a constant $K>0$ such that $PL_{u}(n)\leq K\ln n$ for every Sturmian word $u$, and that for each non-decreasing function $f$ with property $\li…
The prefix palindromic length $p_{\mathbf{u}}(n)$ of an infinite word $\mathbf{u}$ is the minimal number of concatenated palindromes needed to express the prefix of length $n$ of $\mathbf{u}$. This function is surprisingly difficult to study; in particular, the conjecture that $p_{\mathbf{u}}(n)$ can be bounded only if $\mathbf{u}$ is ultimately periodic is open since 2013. A more recent conjecture concerns the prefix palindromic length of the period doubling word: it seems that it is not $2$-regular, and if it is true, this would give a rare if not unique example of a non-regular function of …
We show that the number of prefix normal binary words of length $n$ is $2^{n-Θ((\log n)^2)}$. We also show that the maximum number of binary words of length $n$ with a given fixed prefix normal form is $2^{n-O(\sqrt{n\log n})}$.
A word $w=w_1w_2\cdots w_n$ is alternating if either $w_1<w_2>w_3<w_4>\cdots$ (when the word is up-down) or $w_1>w_2<w_3>w_4<\cdots$ (when the word is down-up). The study of alternating words avoiding classical permutation patterns was initiated by the authors in~\cite{GKZ}, where, in particular, it was shown that 123-avoiding up-down words of even length are counted by the Narayana numbers. However, not much was understood on the structure of 123-avoiding up-down words. In this paper, we fill in this gap by introducing the notion of a cut-pair that allows us to subdivide the set of words in q…
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,…
A word $w=w_1w_2\cdots w_n$ is alternating if either $w_1<w_2>w_3<w_4>\cdots$ (when the word is up-down) or $w_1>w_2<w_3>w_4<\cdots$ (when the word is down-up). In this paper, we initiate the study of (pattern-avoiding) alternating words. We enumerate up-down (equivalently, down-up) words via finding a bijection with order ideals of a certain poset. Further, we show that the number of 123-avoiding up-down words of even length is given by the Narayana numbers, which is also the case, shown by us bijectively, with 132-avoiding up-down words of even length. We also give formulas for enumerating a…
In their paper \cite{DokosDwyer:Permutat12}, Dokos et al. conjecture that the major index statistic is equidistributed among 1423-avoiding, 2413-avoiding, and 2314-avoiding permutations. In this paper we confirm this conjecture by constructing two major index preserving bijections, $Θ:S_n(1423)\to S_n(2413)$ and $Ω:S_n(2314)\to S_n(2413)$. In fact, we show that $Θ$ (respectively, $Ω$) preserves numerous other statistics including the descent set, right-to-left maximum (respectively, left-to-right minimum), and a new statistic we call top-steps (respectively, bottom-steps).
Given two random finite sequences from $[k]^n$ such that a prefix of the first sequence is a suffix of the second, we examine the length of their longest common subsequence. If $\ell$ is the length of the overlap, we prove that the expected length of an LCS is approximately $\max(\ell, \mathbb{E}[L_n])$, where $L_n$ is the length of an LCS between two independent random sequences. We also obtain tail bounds on this quantity.
