Macdonald cumulants, $G$-inversion polynomials and $G$-parking functions

TL;DR

This paper introduces a combinatorial formula for Macdonald cumulants that generalizes existing formulas, leading to new proofs of positivity properties and connections to parking functions and Tutte polynomials.

Contribution

It provides a new combinatorial formula for Macdonald cumulants, extending Haglund's formula, and applies it to prove positivity and factorization properties, as well as conjectures related to Macdonald polynomials.

Findings

Proves a combinatorial formula for Macdonald cumulants.

Establishes $q,t$-positivity of Macdonald cumulants in key bases.

Connects Macdonald cumulants to $G$-parking functions and Tutte polynomials.

Abstract

We prove a combinatorial formula for Macdonald cumulants which generalizes the celebrated formula of Haglund for Macdonald polynomials. We provide several applications of our formula. Firstly, it gives a new, constructive proof of a strong factorization property of Macdonald polynomials proven recently by the author of this paper. Moreover it proves that Macdonald cumulants are $q,t$--positive in the monomial and in the fundamental quasisymmetric bases. Furthermore, we use our formula to prove the recent higher-order Macdonald positivity conjecture for the coefficients of the Schur polynomials indexed by hooks. Our combinatorial formula relates Macdonald cumulants to the generating function of $G$-parking functions, or equivalently to a certain specialization of the Tutte polynomials.