Enumeration of quarter-turn symmetric alternating-sign matrices of odd order

Abstract

It was shown by Kuperberg that the partition function of the square-ice model related to the quarter-turn symmetric alternating-sign matrices of even order is the product of two similar factors. We propose a square-ice model whose states are in bijection with the quarter-turn symmetric alternating-sign matrices of odd order, and show that the partition function of this model can be also written in a similar way. This allows to prove, in particular, the conjectures by Robbins related to the enumeration of the quarter-turn symmetric alternating-sign matrices.

11 pages, 13 figures; minor corrections

References in corpus (7)

• Alternating sign matrices and domino tilings
• Another proof of the alternating sign matrix conjecture
• Proof of the Refined Alternating Sign Matrix Conjecture
• Symmetry Classes of Alternating Sign Matrices
• A new way to deal with Izergin-Korepin determinant at root of unity
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Cited by in corpus (8)

• On the weighted enumeration of alternating sign matrices and descending plane partitions
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• 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
• Extreme diagonally and antidiagonally symmetric alternating sign matrices of odd order
• Vertex Models and Spin Chains in Formulas and Pictures
• On the symmetry of the partition function of some square ice models