- FindStat

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Identifier

St000187: 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]] => 0

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

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

[[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]] => 0

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

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

[[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]] => 0

[[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]] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[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]] => 0

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

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

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

[[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]] => 0

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

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

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

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

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

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

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

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

[[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]] => 0

[[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]] => 1

[[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]] => 1

[[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]] => 0

[[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]] => 0

[[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]] => 0

[[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]] => 0

[[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]] => 1

[[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]] => 0

[[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]] => 0

[[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]] => -1

[[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]] => -1

[[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]] => 0

[[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]] => 1

[[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]] => 1

[[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]] => 0

[[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]] => -1

[[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]] => 0

[[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]] => 0

[[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]] => 0

[[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]] => 0

[[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]] => 1

[[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]] => 0

[[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]] => 0

[[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]] => 1

[[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]] => -1

[[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]] => -1

[[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]] => 1

[[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]] => 1

[[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]] => 0

[[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]] => -1

[[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]] => -1

[[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]] => 0

[[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]] => -1

[[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]] => -1

[[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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$F_{1} = q$

$F_{2} = q^{-1} + q$

$F_{3} = 3\ q^{-1} + 1 + 3\ q$

$F_{4} = 12\ q^{-1} + 16 + 14\ q$

$F_{5} = 86\ q^{-1} + 253 + 90\ q$

Description

The determinant of an alternating sign matrix.

References

[1] Amdeberhan, T. Determinantal questions on Alternate Sign Matrices MathOverflow:342370

Code

def statistic(a):
    return a.to_matrix().determinant()

Created

May 07, 2014 at 21:50 by Dakota Ihli

Updated

Sep 24, 2019 at 20:33 by Martin Rubey