MacMahon's statistics on higher-dimensional partitions

TL;DR

This paper explores combinatorial properties of higher-dimensional partitions, introduces a new statistic with a generating function matching MacMahon's conjecture, and establishes connections with probabilistic models.

Contribution

It presents a natural bijection for higher-dimensional partitions, introduces the corner-hook volume statistic, and develops higher-dimensional analogues of dual Grothendieck polynomials.

Findings

The corner-hook volume's generating function confirms MacMahon's conjecture.

Derived multivariable formulas generalize known plane partition formulas.

Established probabilistic links with last passage percolation models.

Abstract

We study some combinatorial properties of higher-dimensional partitions which generalize plane partitions. We present a natural bijection between $d$-dimensional partitions and $d$-dimensional arrays of nonnegative integers. This bijection has a number of important applications. We introduce a statistic on $d$-dimensional partitions, called the corner-hook volume, whose generating function has the formula of MacMahon's conjecture. We obtain multivariable formulas whose specializations give analogues of various formulas known for plane partitions. We also introduce higher-dimensional analogues of dual Grothendieck polynomials which are quasisymmetric functions and whose specializations enumerate higher-dimensional partitions of a given shape. Finally, we show probabilistic connections with a directed last passage percolation model in $\mathbb{Z}^d$.