Let $n \ge 1$ be an integer. Denote by $\Pi^n $ the group of permutations of $\left\{1,\cdots,n\right\}$ and by $\Pi^{(2)}_{2n}$ the subset of all permutations of $\{1,\cdots, 2n \}$ that are composed of cycles of length two only. For the later there are $(2n-1)!!$ such permutations. Now, in that space of those permutations we define a certain mapping as follows:
\begin{equation} \Pi^{(2)}_{2 n} \ni \sigma \longrightarrow (\sigma^1, \sigma^2) \in \Pi^n \times \Pi^n \end{equation}
where $\sigma^{\xi+1} : (\lfloor (\sigma_{j,1}-\xi)/2 \rfloor + 1) \rightarrow (\lfloor (\sigma_{j,2}-\xi)/2 \rfloor + 1)$ for $j=1,\cdots,n$ and $\xi=0,1$. Here $\sigma_{j} := (\sigma_{j,1},\sigma_{j,2})$ is the $j$th cycle of the permutation $\sigma$. What we really mean by the notation is that for a given $\xi$ we consider a set of pairs $\left\{ (\lfloor (\sigma_{j,1}-\xi)/2 \rfloor + 1), (\lfloor (\sigma_{j,2}-\xi)/2 \rfloor + 1)\right\}_{j=1}^n $ and we rewrite it as ${\mathcal S}:=\left\{ j, \sigma^{\xi+1}_j \right\}_{j=1}^n $ reversing each pair if needed. This is done in a recursive way as follows. For each $j=1,\cdots,n$ out of the original sequence of pairs we select all the pairs that contain the number $j$ and swap them so that that number sits at the first position. There can be at most two such pairs. Out of those two pairs, which have been swapped if necessary, we accept that one where its second component is not yet included in the set ${\mathcal S}$ that we found so far.We also impose a constraint replacing the number $n+1$ in that sequence of pairs by one.
To give some motivation; this problem appears in a natural way when considering the spectral moments of the sample correlation matrix in a Wishart random ensemble with $N$ time series of length $T$ each.
To illustrate the problem I took $n=3$ and I generated all possible permutations $\sigma$ along with their images $\sigma^{1,2}$. The index of the permutation, the permutation itself $\sigma$, the permutations $\sigma^1$ and $\sigma^2$ are shown in the first second, third and fourth column respectively. The fifth column shows the contribution to the spectral moment which reads $N^{\#(\sigma^1)-1} T^{\#(\sigma^2)-n} \prod\limits_{\xi=1}^{\#(\sigma^1)} M_{c^{\sigma^1}_\xi}$ where $\#(\sigma^{1,2})$ are the numbers of cycles of the permutations $\sigma^{1,2}$ and $(c^{\sigma^1}_\xi)_{\xi=1}^{\#(\sigma^1)}$ are the cycle lengths of the permutation $\sigma^1$. Finally $M_p := (\sum\limits_{j=1}^N \lambda_j^p)/W$ for $p=1,2,\cdots$ are the spectral moments of the underlying correlation matrix and $\left(\lambda_j\right)_{j=1}^N$ are the eigenvalues of that matrix.
Now, can we find all possible permutations $\sigma$ such that the triple ${\mathcal T}:=\left\{ \#(\sigma^1),\#(\sigma^2),(c^{\sigma^1}_\xi)_{\xi=1}^{\#(\sigma^1)}\right\}$ is fixed? For example if we take ${\mathcal T}=\left\{1,1,(3)\right\}$ then there are four permutations $\sigma$ that give rise to that triple namely the permutation $5,8,9,11$.