Cycles on a multiset with only even-odd drops

For a finite subset A of 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 gn. 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 gn invented by Dumont and the cycles on [2n] with only even-odd drops. (c) 2021 Elsevier B.V. All rights reserved.