On the weighted enumeration of alternating sign matrices and descending plane partitions

TL;DR

This paper proves a conjecture linking the enumeration of certain alternating sign matrices and descending plane partitions through determinant identities, using advanced combinatorial and statistical mechanics techniques.

Contribution

It establishes a new equality between the counts of specific ASMs and DPPs by expressing their generating functions as determinants and demonstrating their equivalence.

Findings

Confirmed the conjecture relating ASMs and DPPs counts

Derived determinant formulas for generating functions of ASMs and DPPs

Used bijections and elementary transformations to prove determinant equality

Abstract

We prove a conjecture of Mills, Robbins and Rumsey [Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (1983), 340-359] that, for any n, k, m and p, the number of nxn alternating sign matrices (ASMs) for which the 1 of the first row is in column k+1 and there are exactly m -1's and m+p inversions is equal to the number of descending plane partitions (DPPs) for which each part is at most n and there are exactly k parts equal to n, m special parts and p nonspecial parts. The proof involves expressing the associated generating functions for ASMs and DPPs with fixed n as determinants of nxn matrices, and using elementary transformations to show that these determinants are equal. The determinants themselves are obtained by standard methods: for ASMs this involves using the Izergin-Korepin formula for the partition function of the six-vertex model with domain-wall boundary conditions, together with a bijection between ASMs and configurations of this model, and for DPPs it involves using the Lindstrom-Gessel-Viennot theorem, together with a bijection between DPPs and certain sets of nonintersecting lattice paths.