Geodesic
Refined Enumerations of Some Symmetry Classes of Alternating-Sign Matrices
Teoretičeskaâ i matematičeskaâ fizika, Tome 141 (2004) no. 3, pp. 323-347

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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.
Keywords: alternating-sign matrices, enumerations, square-ice model. 
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A. V. Razumov; Yu. G. Stroganov. Refined Enumerations of Some Symmetry Classes of Alternating-Sign Matrices. Teoretičeskaâ i matematičeskaâ fizika, Tome 141 (2004) no. 3, pp. 323-347. http://geodesic.mathdoc.fr/item/TMF_2004_141_3_a0/