Article · Forum of Mathematics Sigma · January 1, 2023
MacMahon’s statistics on higher-dimensional partitions
Alimzhan Amanov, Damir Yeliussizov
Abstract
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 stable 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 $\mathbb {Z}^d$ .