Frozen-corner enumeration of Alternating Sign Matrices

An Alternating Sign Matrix (ASM) is a square matrix with entries in $\{0,1,-1\}$, and such that: $i)$ in each row and columns, nonzero entries alternate in sign; $ii)$ for any given row or column, entries sum up to 1. We define the frozen-square enumeration as the enumeration of $n\times n$ ASMs under the refinement of having, located in a corner, an $s\times s$ square of entries that are all zeroes. We state a conjectural formula for such enumeration, in terms of a Fredholm type determinant of some $s\times s$ matrix whose entries are given explicitely. We provide numerical support in favour of our conjecture. We also illustrate the relevance of the conjectured formula in connection with the limit shape observed in large ASMs, its fluctuations, and the Tracy--Widom distribution.