On refined enumerations of some symmetry classes of ASMs

Abstract

Using determinant representations for partition functions of the corresponding square ice models and the method proposed recently by one of the authors, we investigate refined enumerations of vertically symmetric alternating-sign matrices, off-diagonally symmetric alternating-sign matrices and alternating-sign matrices with U-turn boundary. For all these cases the explicit formulas for refined enumerations are found. It particular, Kutin-Yuen conjecture is proved.