Let $\lambda$ and $\mu$ be partitions of some integer $n \geq 1$. Let $d$ be the number of parts in $\mu$ and let $\bar{\mu} \in \{1,\dotsc,d\}^n$ denote the string $1^{\mu_1} 2^{\mu_2} \dotsb d^{\mu_d}$ (i.e., $\mu_1$ ones, $\mu_2$ twos, etc.). Let $N_{\lambda\mu}$ denote the number of permutations of cycle type $\lambda$ that leave the string $\bar{\mu}$ invariant.

Does this quantity have a name or is it perhaps related to some other known combinatorial quantity? Is there any simple formula or algorithm for computing it?

**Example**

One can check that $N_{(2,1,1,1),(3,2)} = 4$ because in the string 11122 there are three ways of swapping ones and one way of swapping twos.

**Computing $N_{\lambda\mu}$**

More generally one can go about computing $N_{\lambda\mu}$ by packing the cycles $\lambda$ into the histogram $\mu$. We call $\{S_1, \dotsc, S_d\}$ a *packing of $\lambda$ into $\mu$* if $S_i$ are disjoint sets such that their union is $\{1, \dotsc, k\}$, where $k$ is the number of parts in $\lambda$, and $\mu_i = \sum_{j \in S_i} \lambda_j$ for all $i \in \{1, \dotsc, d\}$. Then it seems that
$$
\begin{align}
N_{\lambda\mu}
&= \sum_{\{S_1,\dots,S_d\}} \prod_{i=1}^d \binom{\mu_i}{\lambda_{S_i}} \prod_{j \in S_i} (\lambda_j-1)! \\
&= \sum_{\{S_1,\dots,S_d\}} \prod_{i=1}^d \frac{\mu_i!}{\prod_{j \in S_i} \lambda_j}
\end{align}
$$
where the sum is over all packings of $\lambda$ into $\mu$ and $\binom{\mu_i}{\lambda_{S_i}}$ denotes the multinomial coefficient with the parts of $\lambda$ indexed by $S_i$ at the bottom. Is there any simpler way of doing this?