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

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.