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).

Fischer为涉及统一的第三个根的交替符号矩阵的数量提供了一种新的二项式行列式。在本文中，我们证明了她的公式，当用不确定的q代替单位的第三个根时，实际上给出了$（q^{-\mathrm{1个}}+2+q）$-交替符号矩阵的枚举。通过评估这个决定，我们能够数落米尔斯的一个猜想的推广，罗宾斯和林士说明该Q -enumeration是两个多项式的产品Q。此外，我们提供了0、1、2和3枚举情况下广义行列式的封闭乘积公式，从而得到了交替符号矩阵的1、2和3枚举的新证明以及因式分解在4枚举的情况下。最后，我们将广义行列式的1枚举情况与Ciucu，Eisenkölbl，Krattenthaler和Zare的行列式评估联系起来，后者对具有三角形孔的六边形的加权循环对称菱形拼贴进行计数，并且是对Andrews著名结果的推广。结果，我们使用Desnanot-Jacobi身份（道奇森凝聚）获得了其行列式评估的替代证据。