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In mathematics, an alternating sign matrix is a square matrix of 0s, 1s, and −1s such that the sum of each row and column is 1 and the nonzero entries in each row and column alternate in sign. These matrices generalize permutation matrices and arise naturally when using Dodgson condensation to compute a determinant. They are also closely related to the six-vertex model with domain wall boundary conditions from statistical mechanics. They were first defined by William Mills, David Robbins, and Howard Rumsey in the former context.

Example

An example of an alternating sign matrix (that is not also a permutation matrix) is

Puzzle picture

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

Alternating sign matrix conjecture

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

$$\prod _{{k=0}}^{{n-1}}{\frac {(3k+1)!}{(n+k)!}}={\frac {1!4!7!\cdots (3n-2)!}{n!(n+1)!\cdots (2n-1)!}}.$$

The first few terms in this sequence for n = 0, 1, 2, 3, … are

1, 1, 2, 7, 42, 429, 7436, 218348, … (sequence A005130 in the OEIS).

This conjecture was first proved by Doron Zeilberger in 1992.[1] In 1995, Greg Kuperberg gave a short proof[2] based on the Yang–Baxter equation for the six-vertex model with domain-wall boundary conditions, that uses a determinant calculation due to Anatoli Izergin.[3] A third proof was given by Ilse Fischer using what is called the operator method.[4]

Razumov–Stroganov conjecture

In 2001, A. Razumov and Y. Stroganov conjectured a connection between O(1) loop model, fully packed loop model (FPL) and ASMs.[5] This conjecture was proved in 2010 by Cantini and Sportiello.[6]

References

Zeilberger, Doron, "Proof of the alternating sign matrix conjecture", The Electronic Journal of Combinatorics 3 (1996), R13.
Kuperberg, Greg, "Another proof of the alternating sign matrix conjecture", International Mathematics Research Notes (1996), 139-150.
"Determinant formula for the six-vertex model", A. G. Izergin et al. 1992 J. Phys. A: Math. Gen. 25 4315.
Fischer, Ilse (2005). "A new proof of the refined alternating sign matrix theorem". Journal of Combinatorial Theory, Series A. 114 (2): 253–264. arXiv:math/0507270. Bibcode:2005math......7270F. doi:10.1016/j.jcta.2006.04.004.
Razumov, A.V., Stroganov Yu.G., Spin chains and combinatorics, Journal of Physics A, 34 (2001), 3185-3190.

L. Cantini and A. Sportiello, Proof of the Razumov-Stroganov conjectureJournal of Combinatorial Theory, Series A, 118 (5), (2011) 1549–1574,

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.
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.
Propp, James, The many faces of alternating-sign matrices, Discrete Mathematics and Theoretical Computer Science, Special issue on Discrete Models: Combinatorics, Computation, and Geometry (July 2001).
Razumov, A. V., Stroganov Yu. G., Combinatorial nature of ground state vector of O(1) loop model, Theor. Math. Phys., 138 (2004), 333–337.
Razumov, A. V., Stroganov Yu. G., O(1) loop model with different boundary conditions and symmetry classes of alternating-sign matrices], Theor. Math. Phys., 142 (2005), 237–243, arXiv:cond-mat/0108103
Robbins, David P., The story of 1 , 2 , 7 , 42 , 429 , 7436 , … {\displaystyle 1,2,7,42,429,7436,\dots } {\displaystyle 1,2,7,42,429,7436,\dots }, The Mathematical Intelligencer, 13 (2), 12–19 (1991), doi:10.1007/BF03024081.
Zeilberger, Doron, Proof of the refined alternating sign matrix conjecture, New York Journal of Mathematics 2 (1996), 59–68.

Alternating sign matrix entry in MathWorld
Alternating sign matrices entry in the FindStat database