Further applications of a power series method for pattern avoidance

TL;DR

This paper explores advanced algebraic techniques to analyze pattern avoidance in words, demonstrating that certain complex patterns are avoidable over small alphabets, thus extending previous theoretical bounds.

Contribution

It improves bounds on pattern avoidability, showing patterns with many variables and sufficient length are avoidable on binary alphabets, expanding prior results.

Findings

Patterns with k variables of length at least 4^k are avoidable on binary alphabet

The algebraic Golod technique confirms exponential growth of avoiding words

Extends previous bounds on pattern avoidability in combinatorics on words

Abstract

In combinatorics on words, a word w over an alphabet Sigma is said to avoid a pattern p over an alphabet Delta if there is no factor x of w and no non-erasing morphism h from Delta^* to Sigma^* such that h(p) = x. Bell and Goh have recently applied an algebraic technique due to Golod to show that for a certain wide class of patterns p there are exponentially many words of length n over a 4-letter alphabet that avoid p. We consider some further consequences of their work. In particular, we show that any pattern with k variables of length at least 4^k is avoidable on the binary alphabet. This improves an earlier bound due to Cassaigne and Roth.