Properties of palindromes in finite words

TL;DR

This paper explores the properties and enumeration of palindromes within finite words, introducing methods to generate all palindromes of a given length and analyzing their complexity compared to subword complexity.

Contribution

It presents new methods for constructing palindromes from De Bruijn words and analyzes the shape of the palindrome complexity function, providing bounds and exact counts.

Findings

Palindrome complexity differs from subword complexity in shape.

Upper bounds established for average number of palindromes in words.

Exact formulas derived for palindromes of length 1 and 2 in words of length n.

Abstract

We present a method which displays all palindromes of a given length from De Bruijn words of a certain order, and also a recursive one which constructs all palindromes of length $n+1$ from the set of palindromes of length $n$. We show that the palindrome complexity function, which counts the number of palindromes of each length contained in a given word, has a different shape compared with the usual (subword) complexity function. We give upper bounds for the average number of palindromes contained in all words of length $n$, and obtain exact formulae for the number of palindromes of length 1 and 2 contained in all words of length $n$.