  Mills, Robbins, and Rumsey conjectured, and Zeilberger proved, that the
number of alternating sign matrices of order $n$ equals $A(n):={{1!4!7! ...
(3n-2)!} \over {n!(n+1)! ... (2n-1)!}}$. 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 $r^{th}$ column, equals $A(n) {{n+r-2} \choose
{n-1}}{{2n-1-r} \choose {n-1}}/ {{3n-2} \choose {n-1}}$. 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.
