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Enumeration of quarter-turn-symmetric alternating-sign matrices of odd order. (English) Zbl 1177.15035
Theor. Math. Phys. 149, No. 3, 1639-1650 (2006); translation from Teor. Mat. Fiz. 149, No. 3, 395-408 (2006).
Summary: Kuperberg showed that the partition function of the square-ice model related to 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 written similarly. In particular, this allows proving Robbins’s conjectures related to the enumeration of quarter-turn-symmetric alternating-sign matrices.

MSC:
15A99 Basic linear algebra
05A15 Exact enumeration problems, generating functions
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
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