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Refined enumerations of some symmetry classes of alternating-sign matrices. (English) Zbl 1178.05011
Theor. Math. Phys. 141, No. 3, 1609-1630 (2004); translation from Teor. Mat. Fiz. 141, No. 3, 323-347 (2004).
Summary: Using determinant representations for partition functions of the corresponding variants of square-ice models and the method recently proposed by one of us, we investigate refined enumerations of vertically symmetric alternating-sign matrices, off-diagonally symmetric alternating-sign matrices, and alternating-sign matrices with a U-turn boundary. For all these cases, we find explicit formulas for refined enumerations. In particular, we prove the Kutin-Yuen conjecture.

MSC:
05A15 Exact enumeration problems, generating functions
15B99 Special matrices
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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