Proof of the Refined Alternating Sign Matrix Conjecture

Mills, Robbins, and Rumsey conjectured, and Zeilberger proved, that the number of alternating sign matrices of order n equals A(n):=1!4!7! ... (3n-2)! n!(n+1)! ... (2n-1)!. Mills, Robbins, and Rumsey also made the stronger conjecture that the number of such matrices whose (unique) `1' of the first row is at the rᵗʰ column, equals A(n) n+r-2 n-12n-1-r n-1/ 3n-2 n-1. Standing on the shoulders of A.G. Izergin, V. E. Korepin, and G. Kuperberg, and using in addition orthogonal polynomials and q-calculus, this stronger conjecture is proved.