Palindromes In Sturmian Strings

TL;DR

This paper characterizes the structure and count of maximal palindromes in Sturmian words, revealing a precise formula for their number based on morphic mappings and word structure.

Contribution

It provides a novel characterization of palindromes in Sturmian words and establishes an exact count formula for maximal palindromes in finite Sturmian words.

Findings

Number of maximal palindromes in a Sturmian word is 2|X|-2|Y|.

Set of maximal palindromes has cardinality based on morphic mappings.

Characterization of palindromes via morphisms in Sturmian words.

Abstract

Let p be a maximal palindrome in a Sturmian word s=ul_1pl_2v so that p is a palindrome and l_1pl_2 is not for letters l_1 and l_2. Let {\alpha}(p,p') be a morphism mapping letters a and b respectively to a^pb and a^p'b, |p-p'|=1. In this paper, we characterize the palindromes in a Sturmian word and show that the number of maximal palindromes in a Sturmian word X= {\alpha}(p,p')(Y) for finite Y and thus X is 2|X|-2|Y|. We show that the set of maximal palindromes in a finite Sturmian word X has the cardinality {\Sigma} i=1..n max(pi,p'i) where X is characterized by subsequent mappings of i=1..n.