A connection between palindromic and factor complexity using return words

TL;DR

This paper establishes a fundamental link between palindromic and factor complexities in infinite words, showing their equivalence under certain conditions related to return words and reversal-closed factors.

Contribution

It proves the equivalence between palindromic return properties and a specific relation between palindromic and factor complexities in infinite words.

Findings

Complete returns to palindromes are palindromes if and only if a specific complexity relation holds.

The complexity relation P(n)+P(n+1)=C(n+1)-C(n)+2 characterizes the structure of words with symmetric factor sets.

The results connect palindromic and factor complexity through return words in reversal-closed infinite words.

Abstract

In this paper we prove that for any infinite word W whose set of factors is closed under reversal, the following conditions are equivalent: (I) all complete returns to palindromes are palindromes; (II) P(n) + P(n+1) = C(n+1) - C(n) + 2 for all n, where P (resp. C) denotes the palindromic complexity (resp. factor complexity) function of W, which counts the number of distinct palindromic factors (resp. factors) of each length in W.