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

作者：Behrend Roger E^*; Di Francesco Philippe; Zinn Justin Paul

来源：Journal of Combinatorial Theory - Series A, 2013, 120(2): 409-432.

DOI：10.1016/j.jcta.2012.09.004

• Alternating sign matrices
• Descending plane partitions
• Six-vertex model with domain-wall boundary conditions
• Nonintersecting lattice paths
• Desnanot-Jacobi identity

It was shown recently by the authors that, for any n, there is equality between the distributions of certain triplets of statistics on n x 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 -1%26apos;s in A and the number of 0%26apos;s 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 n%26apos;s in D. Here, the result is generalized to include a fourth statistic for each type of object, where this is the number of 0%26apos;s to the right of the 1 in the last row of an ASM, and the number of (n - 1)%26apos;s 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.

• 出版日期2013-2
• 单位中国地震局