Let $g$ be an element of the permutation group $S_n$ and let $\eta$ be the cyclic permutation $(1,2,\dots,n)$. Define $D(g)$ as the number of cycles in permutation $g$.
I am aware of the fact that the maximum of $D(g) + D(\eta^{-1} \circ g)$ is $n+1$, corresponding to the non-crossing partitions, $NC_n$. I am further aware of the fact that the number of such partitions with $D(g) = k$ is given by the Narayana number $N(n,k)$. I would like to understand further the structure of these permutations.
More precisely, define the set of $g$'s such that $g \in NC_n$ and $D(g) =k$ as $X_{n,k}$. For $h \in X_{n,k}$, we define $(m^{(h)}_1, m^{(h)}_2, \dots, m^{(h)}_n)$ as a list where $m^{(h)}_i$ is the number of cycles in $h$ of length $i$. We must have $\sum_i m^{(h)}_i = k$. How many elements of $X_{n,k}$ have $(m^{(h)}_1, m^{(h)}_2, \dots, m^{(h)}_n)$ and $(\tilde{m}^{(\eta^{-1}\circ h)}_1, \tilde{m}^{(\eta^{-1}\circ h)}_2, \dots, \tilde{m}^{(\eta^{-1}\circ h)}_n)$ where $m$'s and $\tilde{m}$'s are, in general, different. (Note that $\sum_i \tilde{m}_i^{(\eta^{-1}\circ h)} = n +1-k$).
