Cycles on a multiset with only even-odd drops

2021/08/09 by Lin, Zhicong, Yan, Sherry H. F. · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2108.03790

Abstract

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 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.

Cited by

• Parity patterns meet Genocchi numbers, I: four labelings and three bijections

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