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For a finite subset $A$ of $\mathbb{Z}_{>0}$, Lazar and Wachs (2019)conjectured that the number of cycles on $A$ with only even-odd drops is equalto the number of D-cycles on $A$. In this note, we introduce cycles on amultiset with only even-odd drops and prove bijectively a multiset version oftheir conjecture. As a consequence, the number of cycles on $[2n]$ with onlyeven-odd drops equals the Genocchi number $g_n$. With Laguerre histories as anintermediate structure, we also construct a bijection between a class ofpermutations of length $2n-1$ known to be counted by $g_n$ invented by Dumontand the cycles on $[2n]$ with only even-odd drops.

Cycles on a multiset with only even-odd drops