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

Abstract

We prove a conjecture of Mills, Robbins and Rumsey [Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (3) (1983) 340–359] that, for any n, k, m and p, the number of n × n alternating sign matrices (ASMs) for which the 1 of the first row is in column k + 1 and there are exactly m −1ʼs 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 n × n 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 Lindström–Gessel–Viennot theorem, together with a bijection between DPPs and certain sets of nonintersecting lattice paths.