Multiply-refined enumeration of alternating sign matrices

Abstract

Four natural boundary statistics and two natural bulk statistics are considered for alternating sign matrices (ASMs). Specifically, these statistics are the positions of the 1's in the first and last rows and columns of an ASM, and the numbers of generalized inversions and -1's in an ASM. Previously-known and related results for the exact enumeration of ASMs with prescribed values of some of these statistics are discussed in detail. A quadratic relation which recursively determines the generating function associated with all six statistics is then obtained. This relation also leads to various new identities satisfied by generating functions associated with fewer than six of the statistics. The derivation of the relation involves combining the Desnanot-Jacobi determinant identity with the Izergin-Korepin formula for the partition function of the six-vertex model with domain-wall boundary conditions.

62 pages; v3 slightly updated relative to published version

References in corpus (34)

• The arctic curve of the domain-wall six-vertex model
• Condensation of Determinants
• Six-Vertex, Loop and Tiling models: Integrability and Combinatorics
• Domino tilings and the six-vertex model at its free fermion point
• On the weighted enumeration of alternating sign matrices and descending plane partitions
• Emptiness formation probability in the domain-wall six-vertex model
• Factorial Schur functions and the Yang-Baxter equation
• Polynomial solutions of qKZ equation and ground state of XXZ spin chain at Delta = -1/2
• Totally Symmetric Self-Complementary Plane Partitions and Quantum Knizhnik-Zamolodchikov equation: a conjecture
• Punctured plane partitions and the q-deformed Knizhnik--Zamolodchikov and Hirota equations
• Keys and alternating sign matrices
• Bethe roots and 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
• Izergin-Korepin determinant reloaded
• On the problem of calculation of correlation functions in the six-vertex model with domain wall boundary conditions
• The role of orthogonal polynomials in the six-vertex model and its combinatorial applications
• Three-coloring statistical model with domain wall boundary conditions. I. Functional equations
• 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
• On refined enumerations of totally symmetric self-complementary plane partitions I
• Boundary correlation functions of the six and nineteen vertex models with domain wall boundary conditions
• Factorial Schur functions via the six vertex model
• 3-enumerated alternating sign matrices
• The Razumov-Stroganov conjecture: Stochastic processes, loops and combinatorics
• On the symmetry of the partition function of some square ice models
• Doubly-refined enumeration of Alternating Sign Matrices and determinants of 2-staircase Schur functions
• 1/N phenomenon for some symmetry classes of the odd alternating sign matrices
• On refined enumerations of totally symmetric self-complementary plane partitions II
• Fully packed loop models on finite geometries
• A formula for the doubly-refined enumeration of alternating sign matrices
• Alternating paths of fully packed loops and inversion number
• Lambda Determinants
• Truncated determinants and the refined enumeration of Alternating Sign Matrices and Descending Plane Partitions

Cited by in corpus (10)

• Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory
• Diagonally and antidiagonally symmetric alternating sign matrices of odd order
• 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
• Quantum integrable combinatorics of Schur polynomials
• Factorization in the multirefined tangent method
• A pipe dream perspective on totally symmetric self-complementary plane partitions
• Alternating sign matrices and totally symmetric plane partitions
• Correlations in totally symmetric self-complementary plane partitions