Related papers: Avoiding conjugacy classes on the 5-letter alphabe…

In this paper, we study the relation between periodicity of two-dimensional words and their abelian pattern complexity. A pattern $\cal{P}$ in $\mathbb{Z}^n$ is the set of all translations of some finite subset $F$ of $\mathbb{Z}^n$. An…

A graph $G = (V, E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that, for any two distinct vertices $x, y \in V$, $xy \in E$ if and only if $x$ and $y$ alternate in $w$. Two letters $x$ and $y$ are said to…

A finite word $w$ is called \emph{rich} if it contains $\vert w\vert+1$ distinct palindromic factors including the empty word. For every finite rich word $w$ there are distinct nonempty palindromes $w_1, w_2,\dots,w_p$ such that…

We count permutations avoiding a nonconsecutive instance of a two- or three-letter pattern, that is, the pattern may occur but only as consecutive entries in the permutation. Two-letter patterns give rise to the Fibonacci numbers. The…

Let $\mathcal{C}_n$ denote the set of words $w=w_1\cdots w_n$ on the alphabet of positive integers satisfying $w_{i+1}\leq w_i+1$ for $1 \leq i \leq n-1$ with $w_1=1$. The members of $\mathcal{C}_n$ are known as Catalan words and are…

A word $w$ is called rich if it contains $| w|+1$ palindromic factors, including the empty word. We say that a rich word $w$ can be extended in at least two ways if there are two distinct letters $x,y$ such that $wx,wy$ are rich. Let $R$…

We study the conjugacy classes of the classical affine groups. We derive generating functions for the number of classes analogous to formulas of Wall and the authors for the classical groups. We use these to get good upper bounds for the…

This paper gives a definitive solution to the problem of describing conjugacy classes in arbitrary Coxeter groups in terms of cyclic shifts. Let $(W,S)$ be a Coxeter system. A cyclic shift of an element $w\in W$ is a conjugate of $w$ of the…

A graph $G = (V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy$ is an edge in $E$. Word-representable graphs are the subject of a long research…

We consider questions related to the structure of infinite words (over an integer alphabet) with bounded additive complexity, i.e., words with the property that the number of distinct sums exhibited by factors of the same length is bounded…

Given a set $F$ of words, one associates to each word $w$ in $F$ an undirected graph, called its extension graph, and which describes the possible extensions of $w$ on the left and on the right. We investigate the family of sets of words…

Let $H$ be an extension of a finite group $Q$ by a finite group $G$. Inspired by the results of duality theorems for \'etale gerbes on orbifolds, we describe the number of conjugacy classes of $H$ that maps to the same conjugacy class of…

We present an infinite family of formulas with reversal whose avoidability index is bounded between 4 and 5, and we show that several members of the family have avoidability index 5. This family is particularly interesting due to its size…

We introduce two classes of morphisms over the alphabet $A=\{0,1\}$ whose fixed points contain infinitely many antipalindromic factors. An antipalindrome is a finite word invariant under the action of the antimorphism…

We show that, if $w_1, \ldots , w_6$ are words which are not an identity of any (non-abelian) finite simple group, then $w_1(G)w_2(G) \cdots w_6(G) = G$ for all (non-abelian) finite simple groups $G$. In particular, for every word $w$,…

It is well known that Universal Cycles of $k$-letter words on an $n$-letter alphabet exist for all $k$ and $n$. In this paper, we prove that Universal Cycles exist for restricted classes of words, including: non-bijections, equitable words…

Any finite word $w$ of length $n$ contains at most $n+1$ distinct palindromic factors. If the bound $n+1$ is reached, the word $w$ is called rich. The number of rich words of length $n$ over an alphabet of cardinality $q$ is denoted…

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,…

In the study of infinite words, various notions of balancedness provide quantitative measures for how regularly letters or factors occur, and they find applications in several areas of mathematics and theoretical computer science. In this…

We show that for any permutation $w$ that avoids a certain set of 13 patterns of lengths 5 and 6, the Schubert polynomial $\mathfrak S_w$ can be expressed as the determinant of a matrix of elementary symmetric polynomials in a manner…