Counting scattered palindromes in a finite word

Abstract

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 .