Alternating sign matrix

[ 1 0 0 0 1 0 0 0 1 ] [ 1 0 0 0 0 1 0 1 0 ] [ 0 1 0 1 0 0 0 0 1 ] [ 0 1 0 1 − 1 1 0 1 0 ] [ 0 1 0 0 0 1 1 0 0 ] [ 0 0 1 1 0 0 0 1 0 ] [ 0 0 1 0 1 0 1 0 0 ] {\displaystyle {\begin{matrix}{\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}\qquad {\begin{bmatrix}1&0&0\\0&0&1\\0&1&0\end{bmatrix}}\\{\begin{bmatrix}0&1&0\\1&0&0\\0&0&1\end{bmatrix}}\qquad {\begin{bmatrix}0&1&0\\1&-1&1\\0&1&0\end{bmatrix}}\qquad {\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}}\\{\begin{bmatrix}0&0&1\\1&0&0\\0&1&0\end{bmatrix}}\qquad {\begin{bmatrix}0&0&1\\0&1&0\\1&0&0\end{bmatrix}}\end{matrix}}}

The seven alternating sign matrices of size 3

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.^[1] 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.

The permutation matrices are precisely the alternating sign matrices that don't contain −1.

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

[ 0 0 1 0 1 0 0 0 0 1 − 1 1 0 0 1 0 ] . {\displaystyle {\begin{bmatrix}0&0&1&0\\1&0&0&0\\0&1&-1&1\\0&0&1&0\end{bmatrix}}.}

The alternating sign matrix theorem states that the number of n × n {\displaystyle n\times n} alternating sign matrices is

∏ k = 0 n − 1 ( 3 k + 1 ) ! ( n + k ) ! = 1 ! 4 ! 7 ! ⋯ ( 3 n − 2 ) ! n ! ( n + 1 ) ! ⋯ ( 2 n − 1 ) ! . {\displaystyle \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 theorem was first proved by Doron Zeilberger in 1992.^[2] In 1995, Greg Kuperberg gave a short proof^[3] 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.^[4] In 2005, a third proof was given by Ilse Fischer using what is called the operator method.^[5]

In 2001, A. Razumov and Y. Stroganov conjectured a connection between O(1) loop model, fully packed loop model and alternating sign matrices.^[6] This conjecture was proved in 2010 by Cantini and Sportiello.^[7]

• Bressoud, David M., Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, MAA Spectrum, Mathematical Associations of America, Washington, D.C., 1999.ISBN 978-0521666466
• 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 } , 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