Lazar–Wachs cycle-distribution conjecture for D- and E-permutations

Let $n\geq 1$. A permutation $\sigma\in\mathfrak{S}_{2n}$ is a D-permutation if $i\leq\sigma(i)$ whenever $i$ is odd and $i\geq\sigma(i)$ whenever $i$ is even. It is an E-permutation if $i>\sigma(i)$ implies that $i$ is even and $\sigma(i)$ is odd. Let $[2n]=\{1,\ldots,2n\}$, and let $\mathfrak{D}_{2n}$ and $\mathcal{E}_{2n}$ denote the sets of D-permutations and E-permutations on $[2n]$, respectively; write $\mathfrak{D}\mathcal{C}_{2n}$ and $\mathcal{E}\mathcal{C}_{2n}$ for the corresponding sets of D-cycles and E-cycles. For a permutation, a cycle means a cycle in its disjoint-cycle decomposition.

Lazar and Wachs' cycle-distribution conjecture. The number of D-permutations on $[2n]$ with $k$ cycles equals the number of E-permutations on $[2n]$ with $k$ cycles for all $k$. Consequently,

$$G_{2n}=|\mathcal{E}\mathcal{C}_{2n}|.$$

Here $G_{2n}$ denotes the Genocchi number indexed by $2n$; the equality predicts that it counts E-cycles just as the established result counts D-cycles. The conjecture is attributed to Lazar and Wachs and is presented in the paper as an open conjecture.