MacMahon's Double Vision: Partition Diamonds Revisited

TL;DR

This paper revisits the concept of partition diamonds, extending their analysis to d-fold cases using Stanley's P-partitions and deriving a closed-form generating function involving Euler–Mahonian polynomials.

Contribution

It introduces a new approach to d-fold partition diamonds via P-partitions and provides a closed-form generating function generalizing previous recursive formulas.

Findings

Derived a closed-form generating function for d-fold partition diamonds.

Connected partition diamonds with Euler–Mahonian polynomials and permutation statistics.

Extended the theory of partition diamonds to higher dimensions.

Abstract

Plane partition diamonds were introduced by Andrews, Paule, and Riese (2001) as part of their study of MacMahon's $\Omega$-operator in search for integer partition identities. More recently, Dockery, Jameson, Sellers, and Wilson (2024) extended this concept to $d$-fold partition diamonds and found their generating function in a recursive form. We approach $d$-fold partition diamonds via Stanley's (1972) theory of $P$-partitions and give a closed formula for a bivariate generalization of the Dockery--Jameson--Sellers--Wilson generating function; its main ingredient is the Euler--Mahonian polynomial encoding descent statistics of permutations.