Palindromic richness

Abstract

In this paper, we study combinatorial and structural properties of a new class of finite and infinite words that are 'rich' in palindromes in the utmost sense. A characteristic property of so-called rich words is that all complete returns to any palindromic factor are themselves palindromes. These words encompass the well-known episturmian words, originally introduced by the second author together with X. Droubay and G. Pirillo in 2001. Other examples of rich words have appeared in many different contexts. Here we present the first unified approach to the study of this intriguing family of words.Amongst our main results, we give an explicit description of the periodic rich infinite words and show that the recurrent balanced rich infinite words coincide with the balanced episturmian words. We also consider two wider classes of infinite words, namely weakly rich words and almost rich words (both strictly contain all rich words, but neither one is contained in the other). In particular, we classify all recurrent balanced weakly rich words. As a consequence, we show that any such word on at least three letters is necessarily episturmian; hence weakly rich words obey Fraenkel's conjecture. Likewise, we prove that a certain class of almost rich words obeys Fraenkel's conjecture by showing that the recurrent balanced ones are episturmian or contain at least two distinct letters with the same frequency.Lastly, we study the action of morphisms on (almost) rich words with particular interest in morphisms that preserve (almost) richness. Such morphisms belong to the class of Pmorphisms that was introduced by A. Hof, O. Knill, and B. Simon in 1995.

Counting distinct palindromes in a word in linear time

“…A word w with this maximal number |w| + 1 of distinct palindromes is called rich or full by various authors. Subsequently, the notion of palindromic richness has been extended to infinite words and several connected results have been obtained in both the finite and infinite contexts (see, e.g., [11, Section 6.2.2] for a recent survey, [12] for a unified study of structural and combinatorial properties of rich words and [5,16]). S. Labbé indicated to us that a word w can be rich in palindromes (in the sense that it contains many distinct palindromes) without being full of palindromes (in the sense that it cannot contain more).…”

Episturmian words: a survey

“…The theory of rich words has recently been further developed in a series of papers [61,29,42,30]. In independent work, Ambrož, Frougny, Masáková, and Pelantová [8] have considered the same class of words which they call full words, following the earlier work of Brlek, Hamel, Nivat, and Reutenauer in [26].…”

“…More recently, it was proved in [61] that any recurrent balanced rich infinite word is necessarily episturmian, and hence such words obey Fraenkel's conjecture (recall that rich words were defined Section 6.2.2).…”

On Words with the Zero Palindromic Defect

“…Let us recall that a language L ⊂ A * is recurrent if for any two words u, v ∈ L there exists w ∈ L such that u is a prefix of w and v is a suffix of w. Using results of Glen et al [16], Vesti in [34] formulated a sufficient condition which prevents two rich words u, v to be simultaneously factors of another rich word. His proposition uses the notion of longest palindromic suffix of a factor u, denoted lps(u) and longest palindromic prefix of a factor u, denoted lpp(u).…”

“…any prefix of u has a unioccurrent longest palindromic suffix ( [14]); 3. for any palindromic factor w of u, every complete return word of w is a palindrome ( [16]); 4. for any factor w of u, every factor of u that contains w only as its prefix and w only as its suffix is a palindrome (…”