Cycles on a multiset with only even-odd drops

TL;DR

This paper proves a bijective correspondence between certain cycles with even-odd drops on multisets and D-cycles, establishing a combinatorial link to Genocchi numbers and involving Laguerre histories and Dumont's permutations.

Contribution

It introduces cycles on multisets with even-odd drops and proves a multiset version of Lazar and Wachs' conjecture using bijections.

Findings

Number of cycles with even-odd drops on [2n] equals the Genocchi number g_n.

Constructs a bijection between Dumont's permutations and cycles with even-odd drops.

Establishes a combinatorial interpretation of Genocchi numbers via cycles and Laguerre histories.

Abstract

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 equal to the number of D-cycles on $A$. In this note, we introduce cycles on a multiset with only even-odd drops and prove bijectively a multiset version of their conjecture. As a consequence, the number of cycles on $[2n]$ with only even-odd drops equals the Genocchi number $g_n$. With Laguerre histories as an intermediate structure, we also construct a bijection between a class of permutations of length $2n-1$ known to be counted by $g_n$ invented by Dumont and the cycles on $[2n]$ with only even-odd drops.