Alternating sign matrix

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.

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

\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 $$n\times n$$ 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, ….

This theorem was first proved by Doron Zeilberger in 1992. In 1995, Greg Kuperberg gave a short proof 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. In 2005, a third proof was given by Ilse Fischer using what is called the operator method.

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. This conjecture was proved in 2010 by Cantini and Sportiello.