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 nxn 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's in A and the number of 0'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'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's to the right of the 1 in the last row of an ASM, and the number of (n-1)'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.

28 pages; v2: published version

References in corpus (8)

• On the weighted enumeration of alternating sign matrices and descending plane partitions
• Multiply-refined enumeration of alternating sign matrices
• Punctured plane partitions and the q-deformed Knizhnik--Zamolodchikov and Hirota equations
• New enumeration formulas for alternating sign matrices and square ice partition functions
• The role of orthogonal polynomials in the six-vertex model and its combinatorial applications
• On refined enumerations of totally symmetric self-complementary plane partitions I
• Doubly-refined enumeration of Alternating Sign Matrices and determinants of 2-staircase Schur functions
• A formula for the doubly-refined enumeration of alternating sign matrices

Cited by in corpus (9)

• Diagonally and antidiagonally symmetric alternating sign matrices of odd order
• Multiply-refined enumeration of alternating sign matrices
• New enumeration formulas for alternating sign matrices and square ice partition functions
• Extreme diagonally and antidiagonally symmetric alternating sign matrices of odd order
• Arctic curves of the twenty-vertex model with domain wall boundaries
• Truncated determinants and the refined enumeration of Alternating Sign Matrices and Descending Plane Partitions
• Correlations in totally symmetric self-complementary plane partitions
• A bijection between permutation matrices and descending plane partitions without special parts, which respects the quadruplet of statistics considered by Behrend, Di Francesco and Zinn--Justin
• Weight-preserving bijections between integer partitions and a class of alternating sign trapezoids