Proof of the alternating sign matrix conjecture

The number of $n \times n$ matrices whose entries are either $-1$, $0$, or $1$, whose row- and column- sums are all $1$, and such that in every row and every column the non-zero entries alternate in sign, is proved to be $$[1!4! \dots (3n-2)!] \over [n!(n+1)! \dots (2n-1)!],$$ as conjectured by Mills, Robbins, and Rumsey.

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