An explicit formula for ndinv, a new statistic for two-shuffle parking functions

In a recent paper, Duane, Garsia, and Zabrocki introduced a new statistic, “ndinv”, on a family of parking functions. The definition was guided by their study of a recursion on 〈ΔhmCp1Cp2…Cpk1,en〉 for Δhm a Macdonald eigenoperator, Cpi a modified Hall–Littlewood operator, and (p1,p2,…,pk) a composition of n. Using their newly introduced statistic, one can give a new interpretation for 〈∇en,hjhn−j〉 as a sum of parking functions q,t counted by area and ndinv. This is a departure from the traditional sum, as stated by the shuffle conjecture, which q, t counts area and diagonal inversion number (dinv). Since their definition is necessarily recursive, they pose the problem of obtaining a non-recursive definition. In this paper, we solve this problem by giving an explicit formula for ndinv similar to the classical definition of dinv and prove it is equivalent to the ndinv of Duane, Garsia, and Zabrocki.