The poset perspective on alternating sign matrices

TL;DR

This paper explores the structure of alternating sign matrices through poset theory, revealing connections with various combinatorial objects and deriving new formulas for tournament generating functions.

Contribution

It introduces a poset-based framework linking ASMs to multiple combinatorial objects and proves a new expansion formula for tournament generating functions.

Findings

Bijection between order ideals of certain posets and ASMs, TSSCPPs, and other objects

New expansion formula for tournament generating function involving TSSCPPs

Unified poset perspective enhances understanding of combinatorial structures

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 put ASMs into a larger context by studying the order ideals of subposets of a certain poset, proving that they are in bijection with a variety of interesting combinatorial objects, including ASMs, totally symmetric self--complementary plane partitions (TSSCPPs), Catalan objects, tournaments, semistandard Young tableaux, and totally symmetric plane partitions. We use this perspective to prove an expansion of the tournament generating function as a sum over TSSCPPs which is analogous to a known formula involving ASMs.