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arXiv · 2009.00592

MacMahon's statistics on higher-dimensional partitions

Alimzhan Amanov·Damir Yeliussizov
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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 $\mathbb{Z}^d$.

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Alimzhan Amanov, Damir Yeliussizov. 2020-09-01. MacMahon's statistics on higher-dimensional partitions. https://arxiv.org/abs/2009.00592

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