A doubly-refined enumeration of alternating sign matrices and descending plane partitions

Abstract

It was shown recently by the authors that, for any n, there is equality between the distributions of certain triplets of statistics on n×n alternating sign matrices (ASMs) and descending plane partitions (DPPs) with each part at most n. The statistics for an ASM A are the number of generalized inversions in A, the number of −1ʼs in A and the number of 0ʼs to the left of the 1 in the first row of A, and the respective statistics for a DPP D are the number of nonspecial parts in D, the number of special parts in D and the number of nʼs in D. Here, the result is generalized to include a fourth statistic for each type of object, where this is the number of 0ʼs to the right of the 1 in the last row of an ASM, and the number of (n−1)ʼs plus the number of rows of length n−1 in a DPP. This generalization is proved using the known equality of the three-statistic generating functions, together with relations which express each four-statistic generating function in terms of its three-statistic counterpart. These relations are obtained by applying the Desnanot–Jacobi identity to determinantal expressions for the generating functions, where the determinants arise from standard methods involving the six-vertex model with domain-wall boundary conditions for ASMs, and nonintersecting lattice paths for DPPs.