Mills, Robbins, and Rumsey conjectured, and Zeilberger proved, thatthe number of alternating sign matrices of order $n$ equalsA(n):=(1!4!7!...(3n-2)!)/(n!(n+1)!...(2n-1)!).Mills, Robbins, and Rumsey also made the stronger conjecture thatthe number of such matriceswhose (unique) '1' of the first row is at the rth columnequalsA(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.
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