Alternating sign matrix

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In mathematics, an alternating sign matrix is a square matrix of 0s, 1s, and −1s such that the non-zero entries of each row begin and end with 1 and alternate in sign, and the non-zero entries of each column have the same property. These matrices arise naturally when using Dodgson condensation to compute a determinant. They are also closely related to the square ice model from statistical mechanics. They were first defined by William Mills, David Robbins, and Howard Rumsey in the former context.

For example, the permutation matrices are alternating sign matrices, as is

$\begin{bmatrix} 0&0&1&0\\ 1&0&0&0\\ 0&1&-1&1\\ 0&0&1&0 \end{bmatrix}$

The alternating sign matrix conjecture states that the number of $n\times n$ alternating sign matrices is

$\frac{1! 4! 7! \cdots (3n-2)!}{n! (n+1)! \cdots (2n-1!)}$

This conjecture was first proved by Doron Zeilberger in 1992. In 1995, Greg Kuperberg gave a short proof that uses the Yang-Baxter equation, and a determinant formula due to Anatoli Izergin and Vladimir Korepin, applied to the square ice interpretation.

[edit] References and further reading

Bressoud, David M., Proofs and Confirmations, MAA Spectrum, Mathematical Associations of America, Washington, D.C., 1999.
Bressoud, David M. and Propp, James, How the alternating sign matrix conjecture was solved, Notices of the American Mathematical Society, 46 (1999), 637-646.
Kuperberg, Greg, Another proof of the alternating sign matrix conjecture, International Mathematics Research Notes (1996), 139-150.
Mills, William H., Robbins, David P., and Rumsey, Howard, Jr., Proof of the Macdonald conjecture, Inventiones Mathematicae, 66 (1982), 73-87.
Mills, William H., Robbins, David P., and Rumsey, Howard, Jr., Alternating sign matrices and descending plane partitions, Journal of Combinatorial Theory, Series A, 34 (1983), 340-359.
Robbins, David P., The story of $1, 2, 7, 42, 429, 7436, \cdots$ , The Mathematical Intelligencer, 13 (1991), 12-19.
Zeilberger, Doron, Proof of the alternating sign matrix conjecture, Electronic Journal of Combinatorics 3 (1996), R13.
Zeilberger, Doron, Proof of the refined alternating sign matrix conjecture, New York Journal of Mathematics 2 (1996), 59-68.