Generalized Energy Statistics and Kostka--Macdonald Polynomials

TL;DR

This paper interprets the t=1 specialization of modified Macdonald polynomials as generating functions of energy statistics on paths from Box-Ball Systems, introducing a one-parameter generalization with conjectured identical distributions.

Contribution

It provides a new combinatorial interpretation of Macdonald polynomials using energy statistics on BBS-paths and introduces a novel one-parameter generalization with conjectured properties.

Findings

Energy statistics on BBS-paths relate to Macdonald polynomial specializations

Introduces a one-parameter generalization of energy statistics

Conjectures that the generalized statistics share the same distribution

Abstract

We give an interpretation of the t=1 specialization of the modified Macdonald polynomial as a generating function of the energy statistics defined on the set of paths arising in the context of Box-Ball Systems (BBS-paths for short). We also introduce one parameter generalizations of the energy statistics on the set of BBS-paths which all, conjecturally, have the same distribution.