Fredholm determinant formula for frozen-corner ASM enumeration

Establish that, for all integers n ≥ 1 and 1 ≤ s ≤ n, the frozen-corner enumeration B_{n,s}—the number of n×n alternating sign matrices (ASMs) with an s×s square of zero entries in a corner—satisfies B_{n,s} = A_n det_{1≤i,j≤s}(I − M), where A_m denotes the total number of m×m ASMs and M is the s×s matrix with entries M_{ij} = (A_{n−s+j} / A_{n−s+j−1}) · (1/(2πi)^2) ∮_{C_0} ∮_{C_0} [f_i^+(z) f_j^−(w) / (1 − z − w)] dz dw for i,j = 1,…,s, with C_0 a small closed contour around the origin, g_N(z) = 2F1(−N+1, N; 2N; 1 − z), and f_i^{±}(z) = [1 ± (−1)^i z] · (1 − z)^{i−1} · z^i · g_{n−s+i}(z).

Background

The paper studies a refinement of ASM enumeration where one enforces an s×s block of zeros (a frozen square) in a corner. The authors denote this count by B_{n,s} and develop integral and determinant representations to access asymptotics.

They conjecture a Fredholm-type determinant formula for B_{n,s}, introducing an explicit s×s matrix M with entries defined via a double contour integral built from hypergeometric generating functions g_N and auxiliary functions f_i^{±}. They provide extensive numerical checks but no proof, explicitly presenting the statement as a conjecture.