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arXiv · 2112.02074

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

Shane Chern
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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.

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Shane Chern. 2021-12-03. Parity considerations for drops in cycles on $\{1,2,\ldots,n\}$. https://arxiv.org/abs/2112.02074

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