Parity considerations for drops in cycles on $\{1,2,\ldots,n\}$

TL;DR

This paper investigates specific cycle structures with parity restrictions on drops, deriving generating functions that confirm a conjecture relating to Genocchi numbers and uncover new enumerative identities.

Contribution

It introduces two bivariate generating functions for cycles with parity-restricted drops, confirming a conjecture and establishing a new connection to Genocchi medians.

Findings

One generating function confirms Lazar and Wachs' conjecture.

The other generating function relates cycles to Genocchi medians.

The paper provides explicit enumerative identities for parity-restricted cycles.

Abstract

In 2019, A. Lazar and M. L. Wachs conjectured that the number of cycles on $[2n]$ with only even-odd drops equals the $n$-th Genocchi number. In this paper, we restrict our attention to a subset of cycles on $[n]$ that in all drops in the cycle, the latter entry is odd. We deduce two bivariate generating functions for such a subset of cycles with an extra variable introduced to count the number of odd-odd and even-odd drops, respectively. One of the generating function identities confirms Lazar and Wachs' conjecture, while the other identity implies that the number of cycles on $[2n-1]$ with only odd-odd drops equals the $(n-2)$-th Genocchi median.