A unifying poset perspective on alternating sign matrices, plane partitions, Catalan objects, tournaments, and tableaux

TL;DR

This paper introduces a unifying poset framework that connects various combinatorial objects like ASMs, plane partitions, and tableaux, providing new bijections, counting formulas, and generating function expansions.

Contribution

It establishes a tetrahedral poset model that links multiple combinatorial objects, offering new bijections, product formulas, and generating function expansions.

Findings

Bijections between order ideals and combinatorial objects

Product formulas for counting order ideals

Expansion of tournament generating function over TSSCPPs

Abstract

Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1 whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We present a unifying perspective on ASMs and other combinatorial objects by studying a certain tetrahedral poset and its subposets. We prove the order ideals of these subposets are in bijection with a variety of interesting combinatorial objects, including ASMs, totally symmetric self-complementary plane partitions (TSSCPPs), staircase shaped semistandard Young tableaux, Catalan objects, tournaments, and totally symmetric plane partitions. We prove product formulas counting these order ideals and give the rank generating function of some of the corresponding lattices of order ideals. We also prove an expansion of the tournament generating function as a sum over TSSCPPs. This result is analogous to a result of Robbins and Rumsey expanding the tournament generating function as a sum over alternating sign matrices.