The poset perspective on alternating sign matrices

Jessica Striker

doi:10.46298/dmtcs.2707

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. Les matrices à signe alternant (ASMs) sont des matrices carrées dont les coefficients sont 0,1 ou -1, telles que dans chaque ligne et chaque colonne la somme des entrées vaut 1 et les entrées non nulles ont des signes qui alternent. Nous incluons les ASMs dans un cadre plus vaste, en étudiant les idéaux des sous-posets d'un certain poset, dont nous prouvons qu'ils sont en bijection avec de nombreux objets combinatoires intéressants, tels que les ASMs, les partitions planes totalement symétriques autocomplémentaires (TSSCPPs), des objets comptés par les nombres de Catalan, les tournois, les tableaux semistandards, ou les partitions planes totalement symétriques. Nous utilisons ce point de vue pour démontrer un développement de la série génératrice des tournois en une somme portant sur les TSSCPPs, analogue à une formule déjà connue faisant appara\^ıtre les ASMs.

Highlights

Alternating sign matrices (ASMs) are defined as square matrices with entries 0, 1, or −1 whose rows and columns sum to 1 and alternate in sign, but have proved quite difficult to understand
In this paper we present a new perspective which sheds light on ASMs and Totally symmetric self–complementary plane partitions (TSSCPPs) and brings us closer to constructing a explicit ASM–TSSCPP bijection
The Hasse diagram of our new poset looks like a tetrahedron with one direction of edges missing: Inserting those extra edges yields a tetrahedral poset, denoted Tn, whose lattice of order ideals we find to

Summary

Introduction

Alternating sign matrices (ASMs) are defined as square matrices with entries 0, 1, or −1 whose rows and columns sum to 1 and alternate in sign, but have proved quite difficult to understand (and even count). Symmetric self–complementary plane partitions (TSSCPPs) are plane partitions, each equal to its complement and invariant under all permutations of the coordinate axes. TSSCPPs inside a 2n × 2n × 2n box are equinumerous with n × n ASMs, but no explicit bijection between these two sets of objects is known. In this paper we present a new perspective which sheds light on ASMs and TSSCPPs and brings us closer to constructing a explicit ASM–TSSCPP bijection

The tetrahedral poset

Combinatorial objects as subposet order ideals

Bijections with tournaments