Refined Enumerations of Some Symmetry Classes of Alternating-Sign Matrices

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.