Growth rate of binary words avoiding $xxx^R$

TL;DR

This paper investigates the growth rate of binary words avoiding a specific pattern, establishing bounds that grow faster than polynomial but slower than exponential, characterized by functions of the form $Cn^{ ext{lg} n + c}$.

Contribution

It provides the first bounds on the number of such words, showing they grow asymptotically like functions of the form $Cn^{ ext{lg} n + c}$, advancing understanding of pattern-avoiding words.

Findings

Bounds on the number of words are asymptotically equivalent to $Cn^{ ext{lg} n + c}$ functions.

Growth rate is neither polynomial nor exponential, but intermediate.

Establishes a new understanding of pattern avoidance in binary words.

Abstract

Consider the set of those binary words with no non-empty factors of the form $xxx^R$. Du, Mousavi, Schaeffer, and Shallit asked whether this set of words grows polynomially or exponentially with length. In this paper, we demonstrate the existence of upper and lower bounds on the number of such words of length $n$, where each of these bounds is asymptotically equivalent to a (different) function of the form $Cn^{\lg n+c}$, where $C$, $c$ are constants.