Alternating sign matrix
mathematical model also called the Razumov–Stroganov conjecture
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
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}}.}
Alternating sign matrix theorem
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
!
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