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.

翻译：Hetyei于2019年引入了齐次Linial构型，并证明其区域数由中值热那契数给出。在构思Hetyei结果的不同证明过程中，Lazar与Wachs考察了另一个与特定二分图（称为Ferrers图）相关联的超平面构型。我们通过双射方式，用满足上升点奇偶性限制的排列来标记后一构型的区域。这一标记不仅建立了Hetyei与Lazar-Wachs各自枚举结果的等价性，还促使我们推导并研究一个交织两类热那契数的类Seidel三角。运用类似思想，我们引入另外三种具有类似奇偶性限制的排列变体，并同样使用这三类受限排列集为上述构型的区域提供标记。此外，我们建立了从首个排列模型到两个已知排列模型的双射。