Words that almost commute

TL;DR

This paper investigates words that are conjugates with a Hamming distance of 2, providing formulas to count such words and analyzing their growth, revealing no simple bound on their number as length increases.

Contribution

The paper introduces an efficient formula for counting length-$n$ words with conjugates differing by Hamming distance 2, advancing understanding of word conjugacy properties.

Findings

Derived formulas for counting specific conjugate words

Analyzed growth behavior of the count function $h(n)$

Established that $h(n)$ does not have a simple growth bound

Abstract

The \emph{Hamming distance} $\text{ham}(u,v)$ between two equal-length words $u$, $v$ is the number of positions where $u$ and $v$ differ. The words $u$ and $v$ are said to be \emph{conjugates} if there exist non-empty words $x,y$ such that $u=xy$ and $v=yx$. The smallest value $\text{ham}(xy,yx)$ can take on is $0$, when $x$ and $y$ commute. But, interestingly, the next smallest value $\text{ham}(xy,yx)$ can take on is $2$ and not $1$. In this paper, we consider conjugates $u=xy$ and $v=yx$ where $\text{ham}(xy,yx)=2$. More specifically, we provide an efficient formula to count the number $h(n)$ of length-$n$ words $u=xy$ over a $k$-letter alphabet that have a conjugate $v=yx$ such that $\text{ham}(xy,yx)=2$. We also provide efficient formulae for other quantities closely related to $h(n)$. Finally, we show that there is no one easily-expressible good bound on the growth of $h(n)$.