MacMahon's Double Vision: Partition Diamonds Revisited

Plane partition diamonds were introduced by Andrews, Paule, and Riese (2001) as part of their study of MacMahon’s $\mathrm{\Omega }$-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.