Ferrers Graphs, D-Permutations, and Surjective Staircases

TL;DR

This paper introduces new hyperplane arrangements linked to Ferrers graphs, providing combinatorial interpretations of their characteristic polynomials through permutation enumeration and developing generalized surjective staircases with explicit generating functions.

Contribution

It presents a novel family of hyperplane arrangements, establishes their connection to Ferrers graphs, and generalizes surjective staircases with explicit enumeration formulas.

Findings

Intersection lattices are isomorphic to bond lattices of Ferrers graphs.

Characteristic polynomials are interpreted combinatorially via permutation enumeration.

Explicit generating functions are derived for certain families of arrangements.

Abstract

We introduce a new family of hyperplane arrangements inspired by the homogenized Linial arrangement (which was recently introduced by Hetyei), and show that the intersection lattices of these arrangements are isomorphic to the bond lattices of Ferrers graphs. Using recent work of Lazar and Wachs we are able to give combinatorial interpretations of the characteristic polynomials of these arrangements in terms of permutation enumeration. For certain infinite families of these hyperplane arrangements, we are able to give generating function formulas for their characteristic polynomials. To do so, we develop a generalization of Dumont's surjective staircases, and introduce a polynomial which enumerates these generalized surjective staircases according to several statistics. We prove a recurrence for these polynomials and show that in certain special cases this recurrence can be solved explicitly to yield a generating function. We also prove refined versions of several of these results using the theory of complex hyperplane arrangements.