Let $\lambda=(\lambda\_1,\lambda\_2)$, $\mu=(\mu\_1,\mu\_2)$ be two compositions of $n$. Just to remind that $\lambda,\mu$ are not necessarily partitions. Denote $S\_{\lambda}$ and $S\_{\mu}$ the Young subgroups of $S\_n$. Say, $S\_{\lambda}=S\_{\{1,2,\ldots,\lambda\_1\}}\times S\_{\{\lambda\_1+1,\lambda\_1+2,\ldots,\lambda\_1+\lambda\_2\}}$. For each $\sigma$ in $S\_n$, the $(S\_{\lambda},S\_{\mu})$-double coset of $\sigma$ is the set 
$$S\_{\lambda}\sigma S\_{\mu}=\{s\_{\lambda}\sigma s\_{\mu}|s\_{\lambda}\in S\_{\lambda},s\_{\mu}\in S\_{\mu}\}.$$


The set of all double cosets is denoted $S\_{\lambda}\setminus S\_n/S\_{\mu}$.
 The cardinality of the set $S\_{\lambda}\setminus S\_n/S\_{\mu}$ is $\min\{\lambda\_1,\lambda\_2,\mu\_1,\mu\_2\}+1$. I want to ask if there exists an easy way to find out these representative elements of double cosets?



Is there a way to find a representative set for double cosets of groups?
$S\_n$ is a Coxeter group (w.r.t. the neighbour-transpositions) and the Young subgroups are its parabolic subgroup. Therefore there is a canonical way to describe coset and double-coset representatives: There is a unique element of minimal length in each of these (double)cosets.


This also provides algorithmic ways to efficiently iterating through the set of (double)cosets by enumerating the representatives.


