# PhD Seminar • Algorithms and Complexity • Words that Almost Commute and Anti-Commute

Thursday, March 10, 2022 — 12:00 PM EST

## Please note: This PhD seminar will be given online.

Daniel Gabric, PhD candidate
David R. Cheriton School of Computer Science

Supervisor: Professor Jeffrey Shallit

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 $1$. 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 $h(n)$. Additionally, we give some results on the asymptotic behaviour of $h(n)$. Finally, we show some preliminary results on conjugates $u=xy$ and $v=yx$ that differ in all positions (i.e., $\text{ham}(xy,yx)=|xy|$).

Location
Online PhD seminar
200 University Avenue West

Waterloo, ON N2L 3G1
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