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 (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.

v2: published version

References in corpus (9)

• Six-Vertex, Loop and Tiling models: Integrability and Combinatorics
• Quantum Knizhnik-Zamolodchikov equation: reflecting boundary conditions and combinatorics
• Punctured plane partitions and the q-deformed Knizhnik--Zamolodchikov and Hirota equations
• The role of orthogonal polynomials in the six-vertex model and its combinatorial applications
• A Natural Bijection between Permutations and a Family of Descending Plane Partitions
• Open boundary Quantum Knizhnik-Zamolodchikov equation and the weighted enumeration of Plane Partitions with symmetries
• 3-enumerated alternating sign matrices
• The Razumov-Stroganov conjecture: Stochastic processes, loops and combinatorics
• Fully packed loop models on finite geometries

Cited by in corpus (25)

• Stochastic six-vertex model
• Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory
• 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
• A doubly-refined enumeration of alternating sign matrices and descending plane partitions
• Extreme diagonally and antidiagonally symmetric alternating sign matrices of odd order
• Arctic curves of the twenty-vertex model with domain wall boundaries
• The nineteen-vertex model and alternating sign matrices
• Quantum integrable combinatorics of Schur polynomials
• Symmetry classes of alternating sign matrices in the nineteen-vertex model
• Alternating sign trapezoids and a constant term approach
• Sum rules for the supersymmetric eight-vertex model
• A new determinant for the $Q$-enumeration of alternating sign matrices
• Linear recurrences for cylindrical networks
• A Positive integral property on the ground state of the two-boundary Temperley--Lieb Hamiltonian
• A Fourfold Refined Enumeration of Alternating Sign Trapezoids
• Truncated determinants and the refined enumeration of Alternating Sign Matrices and Descending Plane Partitions
• Inversions and the Gog-Magog problem
• A direct bijection between descending plane partitions with no special parts and permutation matrices
• A Catalan Subset of Descending Plane Partitions
• Weight-preserving bijections between integer partitions and a class of alternating sign trapezoids
• Alternating sign matrices with reflective symmetry and plane partitions: $n+3$ pairs of equivalent statistics and a Cauchy-type identity
• Centered polygon numbers, heptagons and nonagons, and the Robbins numbers
• 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