Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Words that almost commute   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 $h(n)$.

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The categories of this paper are: chem-ph
10
The categories of this paper are: q-bio
4
The categories of this paper are: cond-mat
26
The categories of this paper are: bayes-an
11
The categories of this paper are: hep-lat
32
The categories of this paper are: atom-ph
16
The categories of this paper are: stat
37
The categories of this paper are: supr-con
28
The categories of this paper are: alg-geom
36
The categories of this paper are: cmp-lg
14
The categories of this paper are: math-ph
19
The categories of this paper are: eess
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The categories of this paper are: q-fin
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The categories of this paper are: econ
21
The categories of this paper are: adap-org
25
The categories of this paper are: patt-sol