The number of colourings of a polygon such that the multiplicities of a colour are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.

The rowmotion of a signed permutation with respect to the sorting order.
The sorting order on signed permutations (with respect to the Coxeter element $-n, 1, 2,\dots, n-1$) is defined in [1].

The signed permutation with all signs positive.

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.