Pattern Avoidability with Involution

TL;DR

This paper explores the conditions under which certain patterns with involutions can be avoided in infinite words, focusing on how alphabet size influences pattern avoidability.

Contribution

It introduces a formal framework for pattern avoidability with involutions and determines the minimal alphabet sizes needed for avoiding specific patterns.

Findings

Pattern a t(a) a is avoidable over three letters.

Pattern a a a is avoidable over two letters.

Avoidability depends on alphabet size and pattern structure.

Abstract

An infinte word w avoids a pattern p with the involution t if there is no substitution for the variables in p and no involution t such that the resulting word is a factor of w. We investigate the avoidance of patterns with respect to the size of the alphabet. For example, it is shown that the pattern a t(a) a can be avoided over three letters but not two letters, whereas it is well known that a a a is avoidable over two letters.