Proof of the Refined Alternating Sign Matrix Conjecture

Abstract

Mills, Robbins, and Rumsey conjectured, and Zeilberger proved, that the number of alternating sign matrices of order $n$ equals $A(n):=\frac{1!4!7!...(3n-2)!}{n!(n+1)!...(2n-1)!}$. Mills, Robbins, and Rumsey also made the stronger conjecture that the number of such matrices whose (unique) `1' of the first row is at the $r^{th}$ column, equals $A(n)\left(\genfrac{}{}{0ex}{}{n+r-2}{n-1}\right)\left(\genfrac{}{}{0ex}{}{2n-1-r}{n-1}\right)/\left(\genfrac{}{}{0ex}{}{3n-2}{n-1}\right)$. Standing on the shoulders of A.G. Izergin, V. E. Korepin, and G. Kuperberg, and using in addition orthogonal polynomials and $q$-calculus, this stronger conjecture is proved.

Plain TeX

References in corpus (1)

• Another proof of the alternating sign matrix conjecture

Cited by in corpus (33)

• Another proof of the alternating sign matrix conjecture
• The arctic curve of the domain-wall six-vertex model
• Arctic curves of the six-vertex model on generic domains: the Tangent Method
• Square ice, alternating sign matrices and classical orthogonal polynomials
• On the weighted enumeration of alternating sign matrices and descending plane partitions
• A new way to deal with Izergin-Korepin determinant at root of unity
• On two-point boundary correlations in the six-vertex model with DWBC
• Diagonally and antidiagonally symmetric alternating sign matrices of odd order
• Enumeration of quarter-turn symmetric alternating-sign matrices of odd order
• Multiply-refined enumeration of alternating sign matrices
• On the partition function of the six-vertex model with domain wall boundary conditions
• 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
• Extreme diagonally and antidiagonally symmetric alternating sign matrices of odd order
• Three-coloring statistical model with domain wall boundary conditions. I. Functional equations
• On refined enumerations of totally symmetric self-complementary plane partitions I
• 3-enumerated alternating sign matrices
• Multivariable Christoffel-Darboux Kernels and Characteristic Polynomials of Random Hermitian Matrices
• Factorization in the multirefined tangent method
• Fully packed loop models on finite geometries
• Boundary one-point function of the rational six-vertex model with partial domain wall boundary conditions: explicit formulas and scaling properties
• Alternating sign matrices and totally symmetric plane partitions
• Alternating Signed Bipartite Graphs and Difference-1 Colourings
• Correlations in totally symmetric self-complementary plane partitions
• Linear relations of refined enumerations of alternating sign matrices
• A method for proving polynomial enumeration formulas
• Relationships Between Mutations of Brauer Configuration Algebras and Some Diophantine Equations
• The number of monotone triangles with prescribed bottom row
• A new proof of the refined alternating sign matrix theorem
• Alternating sign matrices with one -1 under vertical reflection
• Three alternating sign matrix identities in search of bijective proofs
• On some polynomials enumerating Fully Packed Loops configurations, evaluation at negative values