Binary words avoiding xx^Rx and strongly unimodal sequences

TL;DR

This paper studies the growth rates of binary words avoiding specific patterns, showing that avoiding xx^Rx results in intermediate growth and establishing a connection to strongly unimodal sequences.

Contribution

It demonstrates that avoiding the pattern xx^Rx leads to intermediate growth in binary words and provides a simpler analysis compared to previous work on xxx^R.

Findings

Growth of binary words avoiding xx^Rx is intermediate between polynomial and exponential.

A bijection is established between such words and strongly unimodal sequences.

Analysis for xx^Rx avoidance is simpler than for xxx^R.

Abstract

In previous work, Currie and Rampersad showed that the growth of the number of binary words avoiding the pattern xxx^R was intermediate between polynomial and exponential. We now show that the same holds for the growth of the number of binary words avoiding the pattern xx^Rx. Curiously, the analysis for xx^Rx is much simpler than that for xxx^R. We derive our results by giving a bijection between the set of binary words avoiding xx^Rx and a class of sequences closely related to the class of "strongly unimodal sequences."