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Identifier

St000134: Alternating sign matrices ⟶ ℤ

Values

[[1]] => 1

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

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

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

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

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

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

[[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]] => 3

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

[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]] => 8

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

[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]] => 8

[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]] => 8

[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]] => 8

[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]] => 8

[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]] => 8

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

[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]] => 8

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

[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]] => 8

[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]] => 8

[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]] => 8

[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]] => 8

[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]] => 8

[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]] => 8

[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]] => 8

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

[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]] => 8

[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]] => 8

[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]] => 8

[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]] => 8

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

[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]] => 8

[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]] => 8

[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]] => 8

[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]] => 8

[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]] => 8

[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]] => 8

[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]] => 8

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

[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]] => 8

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

[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]] => 8

[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]] => 8

[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]] => 8

[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]] => 8

[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]] => 8

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

[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]] => 8

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

[[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]] => 5

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 2

[[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]] => 5

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 5

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 4

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 5

[[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]] => 2

[[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]] => 10

[[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]] => 10

[[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]] => 10

[[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]] => 10

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Description

The size of the orbit of an alternating sign matrix under gyration.
Wieland defined gyration in [2] and proved that the action rotates the link pattern of the associated fully-packed loop. It was also proved in [1] that gyration is in equivariant bijection with rowmotion on the alternating sign matrix poset.

References

[1] Striker, J., Williams, N. Promotion and rowmotion MathSciNet:2950491
[2] Wieland, B. A large dihedral symmetry of the set of alternating sign matrices MathSciNet:1773294

Code

def statistic(asm):
    fixed = asm
    asm = asm.gyration()
    i = 1
    while not (asm == fixed):
        asm = asm.gyration()
        i = i + 1
    return i

Created

Jun 20, 2013 at 00:19 by Jessica Striker

Updated

Oct 17, 2015 at 14:02 by Christian Stump