# Stone, Harold S.

## Shuffle-Exchange Conjecture ★★★

Author(s): Beneš; Folklore; Stone

Given integers $k,n\ge2$, let $d(k,n)$ be the smallest integer $d\ge2$ such that the symmetric group $\frak S$ on the set of all words of length $n$ over a $k$-letter alphabet can be generated as $\frak S = (\sigma \frak G)^d$, where $\sigma\in \frak S$ is the \emph{shuffle permutation} defined by $\sigma(x_1 x_2 \dots x_{n}) = x_2 \dots x_{n} x_1$, and $\frak G$ is the \emph{exchange group} consisting of all permutations in $\frak S$ preserving the first $n-1$ letters in the words.

\begin{problem}[SE] Find $d(k,n)$. \end{problem}

\begin{conjecture}[SE] $d(k,n)=2n-1$. \end{conjecture}

Keywords:

## Shuffle-Exchange Conjecture (graph-theoretic form) ★★★

Author(s): Beneš; Folklore; Stone

Given integers $k,n \ge 2$, the \emph{2-stage Shuffle-Exchange graph/network}, denoted $\text{SE}(k,n)$, is the simple $k$-regular bipartite graph with the ordered pair $(U,V)$ of linearly labeled parts $U:=\{u_0,\dots,u_{t-1}\}$ and $V:=\{v_0,\dots,v_{t-1}\}$, where $t:=k^{n-1}$, such that vertices $u_i$ and $v_j$ are adjacent if and only if $(j - ki) \text{ mod } t < k$ (see Fig.1).

Given integers $k,n,r \ge 2$, the \emph{$r$-stage Shuffle-Exchange graph/network}, denoted $(\text{SE}(k,n))^{r-1}$, is the proper (i.e., respecting all the orders) concatenation of $r-1$ identical copies of $\text{SE}(k,n)$ (see Fig.1).

Let $r(k,n)$ be the smallest integer $r\ge 2$ such that the graph $(\text{SE}(k,n))^{r-1}$ is rearrangeable.

\begin{problem} Find $r(k,n)$. \end{problem}

\begin{conjecture} $r(k,n)=2n-1$. \end{conjecture}

Keywords: