New enumeration formulas for alternating sign matrices and square ice partition functions

Abstract

The refined enumeration of alternating sign matrices (ASMs) of given order having prescribed behavior near one or more of their boundary edges has been the subject of extensive study, starting with the Refined Alternating Sign Matrix Conjecture of Mills-Robbins-Rumsey, its proof by Zeilberger, and more recent work on doubly-refined and triply-refined enumeration by several authors. In this paper we extend the previously known results on this problem by deriving explicit enumeration formulas for the "top-left-bottom" (triply-refined) and "top-left-bottom-right" (quadruply-refined) enumerations. The latter case solves the problem of computing the full boundary correlation function for ASMs. The enumeration formulas are proved by deriving new representations, which are of independent interest, for the partition function of the square ice model with domain wall boundary conditions at the "combinatorial point" 2π/3.

35 pages

References in corpus (8)

• Six-Vertex, Loop and Tiling models: Integrability and Combinatorics
• On the weighted enumeration of alternating sign matrices and descending plane partitions
• Emptiness formation probability in the domain-wall six-vertex model
• Multiply-refined enumeration of alternating sign matrices
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Cited by in corpus (8)

• 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
• A doubly-refined enumeration of alternating sign matrices and descending plane partitions
• Quantum integrable combinatorics of Schur polynomials
• Factorization in the multirefined tangent method
• Inversions and the Gog-Magog problem
• Correlations in totally symmetric self-complementary plane partitions