On k-abelian palindromes

Julien Cassaigne (1) , Juhani Karhumaki (2) , Svetlana Puzynina (3, 4)

1 I2M - Institut de Mathématiques de Marseille
2 Department of Mathematics and Statistics, University of Turku
3 Sobolev Institute of Mathematics
4 SPbU - St Petersburg State University

A word is called a palindrome if it is equal to its reversal. In the paper we consider a k-abelian modification of this notion. Two words are called k-abelian equivalent if they contain the same number of occurrences of each factor of length at most k. We say that a word is a k-abelian palindrome if it is k-abelian equivalent to its reversal. A question we deal with is the following: how many distinct palindromes can a word contain? It is well known that a word of length n can contain at most n + 1 distinct palindromes as its factors; such words are called rich. On the other hand, there exist infinite words containing only finitely many distinct palindromes as their factors; such words are called poor. We show that in the k-abelian case there exist infinite words containing finitely many distinct k-abelian palindromic factors. For rich words we show that there exist finite words of length ncontaining Theta(n(2)) distinct k-abelian palindromes as their factors.

Mots clés: infinite words; palindromes; rich words; k-abelian equivalence

Domaines: Combinatoire [math.CO]

Julien Cassaigne, Juhani Karhumaki, Svetlana Puzynina. On k-abelian palindromes. Information and Computation, 2018, 260, pp.89-98. ⟨10.1016/j.ic.2018.04.001⟩. ⟨hal-02060198⟩