Homogenized Graphical Shi Arrangements and Deformed Dumont Permutations

Abstract

We introduce the homogenized graphical Shi arrangement associated with a simple undirected graph $G$, which serves as a broad generalization of classical deformations of the braid arrangement, including the Shi and homogenized Linial arrangements studied by Lazar and Wachs. We demonstrate that the intersection lattices of certain homogenized graphical Shi arrangements are isomorphic to the bond lattices of naturally associated graphs. For a distinguished family of graphs, by employing non-broken-circuit (NBC) techniques, we obtain explicit combinatorial interpretations of the arrangement's Möbius function and characteristic polynomial. We achieve this by introducing generalisations of previously studied combinatorial objects: $\mathcal{R}$-deformed increasing-decreasing ($\mathcal{R}$-DID) forests and $\mathcal{R}$-D-permutations, establishing bijections between them to interpret the coefficients of the characteristic polynomial in terms of $\mathcal{R}$-D-permutations with prescribed number of cycles. Furthermore, we explore refinements of $\mathcal{R}$-D-permutations by their starting letters. Finally, we also resolve an open conjecture posed by Deutsch, Kitaev, and Remmel concerning the equidistribution of specific parity-constrained descent and ascent statistics.