- FindStat

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Identifier

St000065: Alternating sign matrices ⟶ ℤ

Values

[[1]] => 0

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

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

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

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

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

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

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

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

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

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

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

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

[[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]] => 0

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

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

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

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

[[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]] => 2

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

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

[[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]] => 0

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

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

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

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

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

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

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

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

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

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

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

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

[[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]] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

[[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]] => 0

[[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]] => 0

[[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]] => 0

[[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]] => 0

[[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]] => 0

[[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]] => 0

[[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]] => 0

[[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]] => 0

[[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]] => 2

[[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]] => 1

[[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]] => 1

[[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]] => 0

[[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]] => 1

[[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]] => 1

[[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]] => 0

[[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]] => 0

[[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]] => 1

[[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]] => 0

[[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]] => 0

[[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]] => 1

[[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]] => 0

[[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]] => 1

[[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]] => 1

[[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]] => 0

[[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]] => 1

[[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]] => 0

[[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]] => 1

[[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]] => 2

[[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]] => 1

[[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]] => 0

[[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]] => 0

[[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]] => 0

[[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]] => 0

[[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]] => 0

[[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]] => 1

[[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]] => 0

[[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]] => 0

[[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]] => 0

[[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]] => 0

[[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]] => 0

[[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]] => 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]] => 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]] => 0

[[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]] => 0

[[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]] => 0

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 6 + q$

$F_{4} = 24 + 16\ q + 2\ q^{2}$

$F_{5} = 120 + 200\ q + 94\ q^{2} + 14\ q^{3} + q^{4}$

Description

The number of entries equal to -1 in an alternating sign matrix.
The number of nonzero entries, St000890The number of nonzero entries in an alternating sign matrix. is twice this number plus the dimension of the matrix.

References

[1] Triangle of the number of alternating sign matrices according to the number of -1's OEIS:A180512

Code

def statistic(ASM):
    return ASM.number_negative_ones()

Created

Jun 11, 2013 at 05:15 by Jessica Striker

Updated

Jan 13, 2022 at 11:39 by Martin Rubey