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

2025-10-15

AI Summary

This paper addresses the bijective labeling problem for regions of hyperplane arrangements associated with Ferrers diagrams. The core contribution is the introduction of four parity-restricted permutation models—namely, the original model and three variants—each establishing an explicit bijection with the arrangement’s regions. By constructing three rigorous bijections, the work unifies Hetyei’s and Lazar–Wachs’ enumerative results on two families of Genocchi numbers and, for the first time, reveals a natural combinatorial connection between down-up and up-down Genocchi numbers at the structural level. As a consequence, a Seidel-like triangular array is derived, yielding a novel unified combinatorial interpretation for both Genocchi number families. The methodology integrates bijective combinatorics, permutation statistics, hyperplane arrangement theory, and Ferrers diagram analysis. This work advances the deep interplay between restricted permutation models and hyperplane arrangement enumeration, significantly enriching the combinatorial understanding of Genocchi numbers.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Bijectively label hyperplane arrangement regions with parity-restricted permutations
• Establish equivalence between Hetyei and Lazar-Wachs enumerative results
• Introduce three new permutation variants with analogous parity restrictions

Innovation

• Bijective labeling with parity-restricted permutation ascents
• Introducing three variants of parity-restricted permutations
• Establishing bijections between distinct permutation models