A new `dinv' arising from the two part case of the Shuffle Conjecture

TL;DR

This paper introduces a new dinv statistic, ndinv, and proves a conjecture that replacing certain symmetric functions with modified Hall-Littlewood functions enumerates parking functions with specific diagonal path restrictions.

Contribution

The paper proves a conjecture linking symmetric functions and parking functions by deriving a recursion and defining a new dinv statistic, ndinv, for refined enumeration.

Findings

Derived a recursion for the symmetric function expression.

Constructed a new dinv statistic, ndinv, matching the polynomial enumeration.

Confirmed the conjecture relating Hall-Littlewood functions to parking functions with diagonal restrictions.

Abstract

In a recent paper J. Haglund showed that a certain symmetric function expresion enumerates by t^{area} q^{dinv} of the parking functions whose diagonal word is in the shuffle of 12...j and j+1...j+n with k of the cars j+1,...,j+n in the main diagonal including car j+n in the cell (1,1). In view of some recent conjectures of Haglund-Morse-Zabrocki it is natural to conjecture that replacing E_{n,k} by the modified Hall-Littlewood functions would yield a polynomial that enumerates the same collection of parking functions but now restricted by the requirement that the Dyck path supporting cars j+1,...,j+n hits the diagonal according to the composition p=(p_1,p_2,...,p_k). We prove here this conjecture by deriving a recursion for the symmetric function expression then using this recursion to construct a new dinv statistic we will denote ndinv and show that this polynomial enumerates the latter parking functions by t^{area} q^{ndinv}.