Alternating sign matrices and totally symmetric plane partitions

TL;DR

This paper introduces a new family of symmetric functions linked to totally symmetric plane partitions, establishing connections with alternating sign matrices and extending enumeration techniques for related combinatorial objects.

Contribution

It defines a new Schur positive symmetric function family, links it to alternating sign matrices and plane partitions, and provides a novel antisymmetrizer-to-determinant formula with a bijective proof.

Findings

New connection between ASMs and symmetric plane partitions

A determinant formula for antisymmetrizer-to-determinant transformation

Enumeration results for shifted plane partitions

Abstract

We introduce a new family $\mathcal{A}_{n,k}$ of Schur positive symmetric functions, which are defined as sums over totally symmetric plane partitions. In the first part, we show that, for $k=1$, this family is equal to a multivariate generating function involving $n+3$ variables of objects that extend alternating sign matrices (ASMs), which have recently been introduced by the authors. This establishes a new connection between ASMs and a class of plane partitions, thereby complementing the fact that ASMs are equinumerous with totally symmetric self-complementary plane partitions as well as with descending plane partitions. The proof is based on a new antisymmetrizer-to-determinant formula for which we also provide a bijective proof. In the second part, we relate three specialisation of $\mathcal{A}_{n,k}$ to a weighted enumeration of certain well-known classes of column strict shifted plane partitions that generalise descending plane partitions.