A simpler formula for the number of diagonal inversions of an (m,n)(m,n)-parking function and a returning fermionic formula

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 that is composed of three parts and that is not obviously non-negative. Here we simplify the definition of dinv, prove that it is always non-negative, and give a graphical description of the statistic in the style of the classical case. We go on to show that in the (n−1)×n(n−1)×n lattice, parking functions satisfy a fermionic formula that is similar to the one given in the classical case by Haglund and Loehr.