The number of characters of the symmetric group whose value on the partition is zero.
The maximal value for any given size is recorded in [2].

The signed permutation with all signs positive.

The Foata-Han bijection for signed permutations.
This map sends the flag major index St001433The flag major index of a signed permutation. to the flag inversion number St001428The number of B-inversions of a signed permutation..

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.