A simpler formula for the number of diagonal inversions of an (m,n)-Parking Function and a returning Fermionic formula

TL;DR

This paper simplifies the dinv statistic for parking functions in rectangular lattices, proves its non-negativity, and derives a fermionic formula for the count of such functions in specific lattice shapes.

Contribution

It introduces a simplified, non-negative dinv statistic for parking functions and establishes a fermionic formula in rectangular lattice cases, extending classical results.

Findings

Simplified the dinv statistic for parking functions.

Proved the non-negativity of the dinv statistic.

Derived a fermionic formula for parking functions in n x (n+1) lattices.

Abstract

Recent results have placed the classical shuffle conjecture of Haglund et al. in a broader context of an infinite family of conjectures about parking functions in any rectangular lattice. The combinatorial side of the new conjectures has been defined using a complicated generalization of the dinv statistic which is composed of three parts and which is not obviously non-negative. Here we simplify the definition of dinv, prove that it is always non-negative, and give a geometric description of the statistic in the style of the classical case. We go on to show that in the n x (n+1) lattice, parking functions satisfy a fermionic formula that is similar to the one given in the classical case by Haglund and Loehr.