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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.
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Example
An example of an alternating sign matrix (that is not also a permutation matrix) is
Alternating sign matrix conjecture
The alternating sign matrix conjecture states that the number of
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. 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, which solves recurrence relations due to Vladimir Korepin.
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. This conjecture was proved in 2010 by Cantini and Sportiello.