A new determinant for the Q-enumeration of alternating sign matrices

Abstract

Fischer provided a new type of binomial determinant for the number of alternating sign matrices involving the third root of unity. In this paper we prove that her formula, when replacing the third root of unity by an indeterminate q, actually gives the (q−1+2+q)-enumeration of alternating sign matrices. By evaluating a generalisation of this determinant we are able to reprove a conjecture of Mills, Robbins and Rumsey stating that the Q-enumeration is a product of two polynomials in Q. Further we provide a closed product formula for the generalised determinant in the 0-, 1-, 2- and 3-enumeration case, leading to new proofs of the 1-, 2- and 3-enumeration of alternating sign matrices, and a factorisation in the 4-enumeration case. Finally we relate the 1-enumeration case of our generalised determinant to the determinant evaluations of Ciucu, Eisenkölbl, Krattenthaler and Zare, which count weighted cyclically symmetric lozenge tilings of a hexagon with a triangular hole and are a generalisation of a famous result by Andrews. As a result, we obtain alternative proofs of their determinantal evaluations using the Desnanot-Jacobi identity (Dodgson condensation).