On k-Abelian Palindromic Rich and Poor Words

Abstract

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. It is easy to see that there are no abelian poor words, and there exist words containing Θ(n ²) distinct abelian palindromes. We analyze these notions with respect to k-abelian equivalence. We show that in the k-abelian case there exist poor words containing finitely many distinct k-abelian palindromic factors, and there exist rich words containing Θ(n ²) distinct k-abelian palindromes as their factors. Therefore, for poor words the situation resembles normal words, while for rich words it is similar to the abelian case.