# Alternating sign matrix

${\displaystyle {\begin{matrix}{\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}\qquad {\begin{bmatrix}1&0&0\\0&0&1\\0&1&0\end{bmatrix}}\\{\begin{bmatrix}0&1&0\\1&0&0\\0&0&1\end{bmatrix}}\qquad {\begin{bmatrix}0&1&0\\1&-1&1\\0&1&0\end{bmatrix}}\qquad {\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}}\\{\begin{bmatrix}0&0&1\\1&0&0\\0&1&0\end{bmatrix}}\qquad {\begin{bmatrix}0&0&1\\0&1&0\\1&0&0\end{bmatrix}}\end{matrix}}}$
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.[citation needed] 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.

## Examples

A permutation matrix is an alternating sign matrix, and an alternating sign matrix is a permutation matrix if and only if no entry equals −1.

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

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

## Alternating sign matrix theorem

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

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

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.[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] In 2005, a third proof was given by Ilse Fischer using what is called the operator method.[4]

## Razumov–Stroganov problem

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

1. ^ Zeilberger, Doron, "Proof of the alternating sign matrix conjecture", Electronic Journal of Combinatorics 3 (1996), R13.
2. ^ Kuperberg, Greg, "Another proof of the alternating sign matrix conjecture", International Mathematics Research Notes (1996), 139-150.
3. ^ "Determinant formula for the six-vertex model", A. G. Izergin et al. 1992 J. Phys. A: Math. Gen. 25 4315.
4. ^ 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.
5. ^ Razumov, A.V., Stroganov Yu.G., Spin chains and combinatorics, Journal of Physics A, 34 (2001), 3185-3190.
6. ^ L. Cantini and A. Sportiello, Proof of the Razumov-Stroganov conjectureJournal of Combinatorial Theory, Series A, 118 (5), (2011) 1549–1574,