Related papers: Some variations on Lyndon words
We extend the left-to-right Lyndon factorisation of a word to the left Lyndon tree construction of a Lyndon word. It yields an algorithm to sort the prefixes of a Lyndon word according to the infinite ordering defined by Dolce et al.…
A generalized lexicographic order on words is a lexicographic order where the total order of the alphabet depends on the position of the comparison. A generalized Lyndon word is a finite word which is strictly smallest among its class of…
A word $w$ over an alphabet $\Sigma$ is a Lyndon word if there exists an order defined on $\Sigma$ for which $w$ is lexicographically smaller than all of its conjugates (other than itself). We introduce and study \emph{universal Lyndon…
$V$-order is a global order on strings related to Unique Maximal Factorization Families (UMFFs), which are themselves generalizations of Lyndon words. $V$-order has recently been proposed as an alternative to lexicographical order in the…
Let $\A$ be a finite non-empty set and $\preceq $ a total order on $\A^\nats$ verifying the following lexicographic like condition: For each $n\in \nats$ and $u, v\in \A^n,$ if $u^\omega \prec v^\omega$ then $ux\prec vy$ for all $x, y \in…
Lyndon words have been largely investigated and showned to be a useful tool to prove interesting combinatorial properties of words. In this paper we state new properties of both Lyndon and inverse Lyndon factorizations of a word $w$, with…
An infinite word is an infinite Lyndon word if it is smaller, with respect to the lexicographic order, than all its proper suffixes, or equivalently if it has infinitely many finite Lyndon words as prefixes. A characterization of binary…
A generalized lexicographical order on infinite words is defined by choosing for each position a total order on the alphabet. This allows to define generalized Lyndon words. Every word in the free monoid can be factorized in a unique way as…
In this paper, we extend the notion of Lyndon word to transfinite words. We prove two main results. We first show that, given a transfinite word, there exists a unique factorization in Lyndon words that are densely non-increasing, a…
The notion of inverse Lyndon word is related to the classical notion of Lyndon word. More precisely, inverse Lyndon words are all and only the nonempty prefixes of the powers of the anti-Lyndon words, where an anti-Lyndon word with respect…
We say that a family $\mathcal{W}$ of strings over $\Sigma^+$ forms a Unique Maximal Factorization Family (UMFF) if and only if every $w \in \mathcal{W}$ has a unique maximal factorization. Further, an UMFF $\mathcal{W}$ is called a…
Given two finite words $u$ and $v$ of equal length, define the \emph{overlap gap between $u$ and $v$}, denoted $og(u,v)$, as the least integer $m$ for which there exist words $x$ and $x'$ of length $m$ such that $xu=vx'$ or $ux=x'v$.…
The Lyndon array stores, at each position of a word, the length of the longest maximal Lyndon subword starting at that position, and plays an important role in combinatorics on words, for example in the construction of fundamental data…
The Chen-Fox-Lyndon theorem states that every finite word over a fixed alphabet can be uniquely factorized as a lexicographically nonincreasing sequence of Lyndon words. This theorem can be used to define the family of Lyndon words in a…
The prefix palindromic length $PPL_u(n)$ of an infinite word $u$ is the minimal number of concatenated palindromes needed to express the prefix of length $n$ of $u$. In a 2013 paper with Puzynina and Zamboni we stated the conjecture that…
An infinite permutation $\alpha$ is a linear ordering of $\mathbb N$. We study properties of infinite permutations analogous to those of infinite words, and show some resemblances and some differences between permutations and words. In this…
Motivated by applications to string processing, we introduce variants of the Lyndon factorization called inverse Lyndon factorizations. Their factors, named inverse Lyndon words, are in a class that strictly contains anti-Lyndon words, that…
The theorem of factorisation forests shows the existence of nested factorisations -- a la Ramsey -- for finite words. This theorem has important applications in semigroup theory, and beyond. The purpose of this paper is to illustrate the…
The notion of Lyndon word and Lyndon factorization has shown to have unexpected applications in theory as well in developing novel algorithms on words. A counterpart to these notions are those of inverse Lyndon word and inverse Lyndon…
We define a new class of languages of $\omega$-words, strictly extending $\omega$-regular languages. One way to present this new class is by a type of regular expressions. The new expressions are an extension of $\omega$-regular expressions…