On the number of episturmian palindromes

Discrete Mathematics, 2010

The palindrome complexity function pal w of a word w attaches to each n ∈ N the number of palindromes (factors equal to their mirror images) of length n contained in w. The number of all the nonempty palindromes in a finite word is called the total palindrome complexity of that word. We present exact bounds for the total palindrome complexity and construct words which have any palindrome complexity between these bounds, for binary alphabets as well as for alphabets with the cardinal greater than 2. Denoting by M q (n) the average number of palindromes in all words of length n over an alphabet with q letters, we present an upper bound for M q (n) and prove that the limit of M q (n)/n is 0. A more elaborate estimation leads to M q (n) = O( √ n).

2006

We present a method which displays all palindromes of a given length from De Bruijn words of a certain order, and also a recursive one which constructs all palindromes of length n + 1 from the set of palindromes of length n. We show that the palindrome complexity function, which counts the number of palindromes of each length contained in a given word, has a different shape compared with the usual (subword) complexity function. We give upper bounds for the average number of palindromes contained in all words of length n, and obtain exact formulae for the number of palindromes of length 1 and 2 contained in all words of length n.

Cornell University - arXiv, 2021

We investigate the scattered palindromic subwords in a finite word. We start by characterizing the words with the least number of scattered palindromic subwords. Then, we give an upper bound for the total number of palindromic subwords in a word of length n in terms of Fibonacci number Fn by proving that at most Fn new scattered palindromic subwords can be created on the concatenation of a letter to a word of length n − 1. We propose a conjecture on the maximum number of scattered palindromic subwords in a word of length n with q distinct letters. We support the conjecture by showing its validity for words where q ≥ n 2 .

Theoretical Informatics and Applications, 2009

We study the palindromic complexity of infinite words $u_\\beta$, the fixed points of the substitution over a binary alphabet, $\\phi(0)=0^a1$, $\\phi(1)=0^b1$, with $a-1\\geq b\\geq 1$, which are canonically associated with quadratic non-simple Parry numbers $\\beta$.

European Journal of Combinatorics, 2018

We extend classical Ostrowski numeration systems, closely related to Sturmian words, by allowing a wider range of coefficients, so that possible representations of a number n better reflect the structure of the associated characteristic Sturmian word. In particular, this extended numeration system helps to catch occurrences of palindromes in a characteristic Sturmian word and thus to prove for Sturmian words the following conjecture stated in 2013 by Puzynina, Zamboni and the author: If a word is not periodic, then for every Q > 0 it has a prefix which cannot be decomposed to a concatenation of at most Q palindromes.

Journal of Combinatorial Theory, Series A, 2015

We regard a finite word u = u 1 u 2 · · · u n up to word isomorphism as an equivalence relation on {1, 2, . . . , n} where i is equivalent to j if and only if u i = u j . Some finite words (in particular all binary words) are generated by palindromic relations of the form k ∼ j +i−k for some choice of 1 ≤ i ≤ j ≤ n and k ∈ {i, i + 1, . . . , j}. That is to say, some finite words u are uniquely determined up to word isomorphism by the position and length of some of its palindromic factors. In this paper we study the function µ(u) defined as the least number of palindromic relations required to generate u. We show that if x is an infinite word such that µ(u) ≤ 2 for each factor u of x, then x is ultimately periodic. On the other hand we establish the existence of non-ultimately periodic words for which µ(u) ≤ 3 for each factor u of x, and obtain a complete classification of such words on a binary alphabet (which includes the well known class of Sturmian words). In contrast, for the Thue-Morse word, we show that the function µ is unbounded.

Theoretical Computer Science, 2013

Trapezoidal words are words having at most n+1 distinct factors of length n for every n ≥ 0. They therefore encompass finite Sturmian words. We give combinatorial characterizations of trapezoidal words and exhibit a formula for their enumeration. We then separate trapezoidal words into two disjoint classes: open and closed. A trapezoidal word is closed if it has a factor that occurs only as a prefix and as a suffix; otherwise it is open. We investigate open and closed trapezoidal words, in relation with their special factors. We prove that Sturmian palindromes are closed trapezoidal words and that a closed trapezoidal word is a Sturmian palindrome if and only if its longest repeated prefix is a palindrome. We also define a new class of words, semicentral words, and show that they are characterized by the property that they can be written as uxyu, for a central word u and two different letters x, y. Finally, we investigate the prefixes of the Fibonacci word with respect to the property of being open or closed trapezoidal words, and show that the sequence of open and closed prefixes of the Fibonacci word follows the Fibonacci sequence.

Ars Combinatoria, 2003

A composition of a positive integer n consists of an ordered sequence of positive integers whose sum is n. A palindromic composition is one for which the sequence is the same from left to right as from right to left. This paper shows various ways of generating all palindromic compositions, counts the number of times each integer appears as a summand among all the palindromic compositions of n, and describes several patterns among the numbers generated in the process of enumeration.

Theoretical Informatics and Applications, 2008

In this paper we solve some open problems related to (pseudo)palindrome closure operators and to the infinite words generated by their iteration, that is, standard episturmian and pseudostandard words. We show that if ϑ is an involutory antimorphism of A * , then both ϑ-palindromic closures of any factor of a ϑ-standard word are also factors of some ϑ-standard word. We also introduce the class of pseudostandard words with "seed", obtained by iterated pseudopalindrome closure starting from a nonempty word. We prove that pseudostandard words with seed are morphic images of standard episturmian words. Moreover, for any given pseudostandard word s with seed, there exists an integer N such that for any n ≥ N , s has at most one right (resp. left) special factor of length n. * The work for this paper has been supported by the Italian Ministry of Education under Project COFIN 2005 -Automi e Linguaggi Formali: aspetti matematici e applicativi.

International Journal of Apllied Mathematics

We study the classical complexity of k to k insertion words of a letter in Sturmian words. Then, we determine the Abelian complexity and palindromic complexity of these words. Finally, we show that the k to k insertion of a letter x in Sturmian words preserves the palindromic richness of Sturmian words if and only if k = 1.