
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.
Example.
An example of an alternating sign matrix (that is not also a permutation matrix) is
Alternating sign matrix theorem.
The "alternating sign matrix theorem" states that the number of formula_2 alternating sign matrices is
The first few terms in this sequence for "n" = 0, 1, 2, 3, … are
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. 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.