combinat::alternatingSignMatrices – alternating sign matrices
combinat::alternatingSignMatrices represents the combinatorial class of alternating sign matrices.
Categories
Cat::GradedCombinatorialClass , Cat::FacadeDomain (matrix)
Details:
An alternating sign matrix of size math is a math square matrix of math's, math's, and math's such that the non-zero entries of each row begin and end with math's and alternate in sign, and the non-zero entries of each column have the same property. Here is a typical alternating sign matrix of size math:
[画像:math]
These matrices form a natural generalization of permutations, and appear in many different contexts like statistical mechanics.
Method count is inherited from Cat::GradedCombinatorialClass .
Method generator is inherited from Cat::GradedCombinatorialClass .
Method list is inherited from Cat::GradedCombinatorialClass .
We start by counting the number of alternating sign matrices; this uses the classical (and hard to prove!) formula math:
combinat::alternatingSignMatrices::count(n) $ n = 0..10
math
From this, we see that the first alternating sign matrix that is not a permutation matrix is of size math. Here it is:
combinat::alternatingSignMatrices::list(3)
[画像:math]
Background:
See the Wikipedia article on alternating sign matrices: http://en.wikipedia.org/wiki/Alternating_sign_matrix .
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