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Identifier

St000889: Alternating sign matrices ⟶ ℤ

Values

[[1]] => 1

[[1,0],[0,1]] => 1

[[0,1],[1,0]] => 1

[[1,0,0],[0,1,0],[0,0,1]] => 1

[[0,1,0],[1,0,0],[0,0,1]] => 1

[[1,0,0],[0,0,1],[0,1,0]] => 1

[[0,1,0],[1,-1,1],[0,1,0]] => 1

[[0,0,1],[1,0,0],[0,1,0]] => 2

[[0,1,0],[0,0,1],[1,0,0]] => 2

[[0,0,1],[0,1,0],[1,0,0]] => 1

[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]] => 1

[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]] => 1

[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]] => 1

[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]] => 1

[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]] => 2

[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]] => 2

[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]] => 1

[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]] => 1

[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]] => 1

[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]] => 1

[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]] => 1

[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]] => 2

[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]] => 2

[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]] => 1

[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]] => 2

[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]] => 2

[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]] => 4

[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]] => 2

[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]] => 4

[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]] => 2

[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]] => 2

[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]] => 2

[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]] => 2

[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]] => 4

[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]] => 4

[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]] => 2

[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]] => 1

[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]] => 1

[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]] => 2

[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]] => 2

[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]] => 2

[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]] => 1

[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]] => 3

[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]] => 3

[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]] => 3

[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]] => 2

[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]] => 2

[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]] => 2

[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]] => 3

[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]] => 3

[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]] => 3

[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]] => 1

[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 1

[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 1

[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 1

[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 1

[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 2

[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 2

[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 1

[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]] => 1

[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]] => 1

[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => 1

[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => 1

[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => 2

[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => 2

[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => 1

[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => 2

[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => 2

[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => 4

[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => 2

[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => 4

[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => 2

[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => 2

[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => 2

[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => 2

[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => 4

[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => 4

[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => 2

[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => 1

[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => 1

[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => 2

[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => 2

[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => 2

[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => 1

[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => 3

[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]] => 3

[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]] => 3

[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]] => 2

[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]] => 2

[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]] => 2

[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]] => 3

[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]] => 3

[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]] => 3

[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]] => 1

[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]] => 1

[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]] => 1

[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => 1

[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => 1

[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => 2

[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => 2

[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => 1

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Description

The number of alternating sign matrices with the same antidiagonal sums.
The X-ray of an alternating sign matrix $A$ is the vector of the sums of the antidiagonals of $A$, read from top to bottom.
This statistic records the number of alternating sign matrices having the same X-ray as the given matrix. The analogous concept for permutations is called the degeneracy of the X-ray of the permutation in [1], see St000886The number of permutations with the same antidiagonal sums..
The number of alternating sign matrices determined by their X-ray is the Catalan number, see [2].

References

[1] Bebeacua, C., Mansour, T., Postnikov, A., Severini, S. On the X-rays of permutations arXiv:math/0506334
[2] Rubey, M. The set of alternating sign matrices which are determined by their X-ray is a member of the Catalan family arXiv:1707.06802

Code

def ASM_X_ray(A):
    P = A.to_matrix()
    n = P.nrows()
    return tuple(sum(P[k-1-j][j] for j in range(max(0, k-n), min(k,n)))
                 for k in range(1,2*n))

@cached_function
def ASM_X_rays(n):
    return sorted(ASM_X_ray(A) for A in AlternatingSignMatrices(n))

def statistic(A):
    return ASM_X_rays(A.to_matrix().nrows()).count(ASM_X_ray(A))

Created

Jul 14, 2017 at 11:43 by Martin Rubey

Updated

Dec 25, 2017 at 17:25 by Martin Rubey