The number of graphs with given frequency partition.
The frequency partition of a graph on $n$ vertices is the partition obtained from its degree sequence by recording and sorting the frequencies of the numbers that occur.
For example, the complete graph on $n$ vertices has frequency partition $(n)$. The path on $n$ vertices has frequency partition $(n-2,2)$, because its degree sequence is $(2,\dots,2,1,1)$. The star graph on $n$ vertices has frequency partition is $(n-1, 1)$, because its degree sequence is $(n-1,1,\dots,1)$.
There are two graphs having frequency partition $(2,1)$: the path and an edge together with an isolated vertex.

Return the signed permutation with all signs reversed.

The partition corresponding to the even cycles.
A cycle of length $\ell$ of a signed permutation $\pi$ can be written in two line notation as
$$\begin{array}{cccc} a_1 & a_2 & \dots & a_\ell \\ \pi(a_1) & \pi(a_2) & \dots & \pi(a_\ell) \end{array}$$
where $a_i > 0$ for all $i$, $a_{i+1} = |\pi(a_i)|$ for $i < \ell$ and $a_1 = |\pi(a_\ell)|$.
The cycle is even, if the number of negative elements in the second row is even.
This map records the integer partition given by the lengths of the odd cycles.
The integer partition of even cycles together with the integer partition of the odd cycles determines the conjugacy class of the signed permutation.

The signed permutation with all signs positive.