# Borders, Palindrome Prefixes, and Square Prefixes

## TL;DR

This paper establishes bijections between words with no even palindromic prefixes and unbordered words, providing asymptotic counts and extending results to words with no square prefixes, including a conjecture resolution.

## Contribution

It introduces explicit bijections linking palindromic prefix restrictions to unbordered words and solves a conjecture on words with no square prefix.

## Key findings

- Number of words with no even palindromic prefix equals unbordered words.
- Asymptotic enumeration for words with no palindromic prefixes.
- Resolution of a 2013 conjecture on words with no square prefix.

## Abstract

We show that the number of length-n words over a k-letter alphabet having no even palindromic prefix is the same as the number of length-n unbordered words, by constructing an explicit bijection between the two sets. A slightly different but analogous result holds for those words having no odd palindromic prefix. Using known results on borders, we get an asymptotic enumeration for the number of words having no even (resp., odd) palindromic prefix . We obtain an analogous result for words having no nontrivial palindromic prefix. Finally, we obtain similar results for words having no square prefix, thus proving a 2013 conjecture of Chaffin, Linderman, Sloane, and Wilks.