The new dinv is not so new

In [Duane, Garsia, Zabrocki 2013] the authors introduced a new dinv statistic, denoted ndinv, on the two part case of the shuffle conjecture (Haglund et al. 2005) in order to prove a compositional refinement. Though in [Hicks, Kim 2013] a non-recursive (but algorithmic) definition of ndinv has been given, this statistic still looks a bit unnatural. In this paper we "unveil the mystery" around the ndinv, by showing bijectively that the ndinv actually matches the usual dinv statistic in a special case of the generalized Delta conjecture in [Haglund, Remmel, Wilson 2018]. Moreover, we give also a non-compositional proof of the "ehh" case of the shuffle conjecture (after [Garsia, Xin, Zabrocki 2014]) by bijectively proving a relation with the two part case of the Delta conjecture

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

Abstract. In a recent paper, Duane, Garsia, and Zabrocki introduced a new statistic, “ndinv”, on a family of parking functions. The definition was guided by a recursion satisfied by the polynomial 〈∆hmCp1Cp2... Cp k 1, en〉, for ∆hm a Macdonald eigenoperator, Cp i a modified Hall-Littlewood operator and (p1, p2,..., pk) a composition of n. Using their new statistics, they are able to give a new interpretation for the polynomial 〈∇en, hjhn−j 〉 as a q,t numerator of parking functions by area and ndinv. We recall that in the shuffle conjecture, parking functions are q,t enumerated by area and diagonal inversion number (dinv). Since their definition is recursive, they pose the problem of obtaining a non recursive definition. We solved this problem by giving an explicit formula for ndinv similar to the classical definition of dinv. In this paper, we describe the work we did to construct this formula and to prove that the resulting ndinv is the same as the one recursively defined by Duane, Garsia, and Zabrocki. Résumé. Dans un travail récent Duane, Garsia et Zabrocki ont introduit une nouvelle statistique, “ndinv ” pour une famille de Fonctions Parking. Ce “ndinv ” découle d’une récurrence satisfaite par le polynôme 〈∆hmCp1Cp2 · · · Cpk 1, en〉, oú ∆h