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

TL;DR

This work develops parity-pattern restrictions that connect region counts of hyperplane arrangements to Genocchi numbers, introducing four permutation models and four labelings of the Ferrers-graph arrangement $\mathcal{K}_{2n}$. It constructs three new bijections and a rich Seidel-like triangle tying median Genocchi numbers $h_n$ and Genocchi numbers $g_n$ to permutation statistics and to various known models (D-permutations, collapsed permutations, and Dumont-type permutations). The authors provide a recursive bijection for the first labeling and three complementary bijections that link the new models to established combinatorial structures, yielding a unified combinatorial framework for the genus of Genocchi-related sequences. They also highlight structural recurrences and potential divisibility phenomena, suggesting directions for further Seidel-triangle explorations and extensions (e.g., FFLY26).

Abstract

Hetyei introduced in 2019 the homogenized Linial arrangement and showed that its regions are counted by the median Genocchi numbers. In the course of devising a different proof of Hetyei's result, Lazar and Wachs considered another hyperplane arrangement that is associated with certain bipartite graph called Ferrers graph. We bijectively label the regions of this latter arrangement with permutations whose ascents are subject to a parity restriction. This labeling not only establishes the equivalence between two enumerative results due to Hetyei and Lazar-Wachs, repectively, but also motivates us to derive and investigate a Seidel-like triangle that interweaves Genocchi numbers of both kinds. Applying similar ideas, we introduce three more variants of permutations with analogous parity restrictions. We provide labelings for regions of the aforementioned arrangement using these three sets of restricted permutations as well. Furthermore, bijections from our first permutation model to two previously known permutation models are established.

Figures (9)

• Figure 1: The equivalence between Theorems \ref{['thm:hetyei']} and \ref{['thm:Lazar-Wachs']}
• Figure 2: The Hasse diagram of a region $R\in\mathcal{R}(\mathcal{K}_{18})$
• Figure 3: The recursively defined bijection $\Phi_k$ from $\mathfrak{G}^{\mathrm{I}}_{2n,2k}$ to $\mathcal{D}^{\mathrm{III}}_{2n,2k}$
• Figure 4: The composed bijection $\Theta=\vartheta^{-1}\circ\theta: \mathcal{CO}_{2n+2}\to\mathfrak{G}^{\mathrm{I}}_{2n}$
• Figure :

Theorems & Definitions (45)

• Theorem 1.1: LW22
• Conjecture 1.2: LW22
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 2.1: het19
• Proposition 2.2
• Definition 2.3
• Definition 3.1