Combinatorial formulas for symmetric Macdonald polynomials by superizations

TL;DR

This paper introduces new, more concise combinatorial formulas for symmetric Macdonald polynomials using novel statistics and superization techniques, unifying and extending previous results in the field.

Contribution

It provides the fewest-term combinatorial formulas for Macdonald polynomials and recovers several existing formulas through superization methods.

Findings

New formulas with fewer terms for symmetric Macdonald polynomials

Recovery of three existing formulas via superization

Introduction of a new statistic on super fillings

Abstract

In this paper, we derive new combinatorial formulas for symmetric Macdonald polynomials $P_{\lambda}(X;q,t)$ and integral Macdonald polynomials $J_{\lambda}(X;q,t)$, in terms of several new statistics and the major index for a partition $\lambda$. Compared to previous formulas, these new ones contain the fewest terms. Moreover, three existing formulas for symmetric Macdonald polynomials established by Corteel--Mandelshtam--Williams (2022), Corteel--Haglund--Mandelshtam--Mason--Williams (2022) and Mandelshtam (2025) are recovered. Our proof relies on a new statistic on super fillings, employing the superization formulas of Haglund--Haimain--Loehr (2005) and Ayyer--Mandelshtam--Martin (2023), together with our recent approach to modified Macdonald polynomials.