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Identifier

St001773: Signed permutations ⟶ ℤ

Values

[1] => 1

[-1] => 0

[1,2] => 2

[1,-2] => 1

[-1,2] => 1

[-1,-2] => 0

[2,1] => 1

[2,-1] => 1

[-2,1] => 1

[-2,-1] => 1

[1,2,3] => 3

[1,2,-3] => 2

[1,-2,3] => 2

[1,-2,-3] => 1

[-1,2,3] => 2

[-1,2,-3] => 1

[-1,-2,3] => 1

[-1,-2,-3] => 0

[1,3,2] => 2

[1,3,-2] => 2

[1,-3,2] => 2

[1,-3,-2] => 2

[-1,3,2] => 1

[-1,3,-2] => 1

[-1,-3,2] => 1

[-1,-3,-2] => 1

[2,1,3] => 2

[2,1,-3] => 1

[2,-1,3] => 2

[2,-1,-3] => 1

[-2,1,3] => 2

[-2,1,-3] => 1

[-2,-1,3] => 2

[-2,-1,-3] => 1

[2,3,1] => 2

[2,3,-1] => 2

[2,-3,1] => 2

[2,-3,-1] => 2

[-2,3,1] => 2

[-2,3,-1] => 2

[-2,-3,1] => 1

[-2,-3,-1] => 1

[3,1,2] => 2

[3,1,-2] => 2

[3,-1,2] => 2

[3,-1,-2] => 1

[-3,1,2] => 2

[-3,1,-2] => 2

[-3,-1,2] => 2

[-3,-1,-2] => 1

[3,2,1] => 1

[3,2,-1] => 1

[3,-2,1] => 3

[3,-2,-1] => 2

[-3,2,1] => 1

[-3,2,-1] => 1

[-3,-2,1] => 2

[-3,-2,-1] => 2

[1,2,3,4] => 4

[1,2,3,-4] => 3

[1,2,-3,4] => 3

[1,2,-3,-4] => 2

[1,-2,3,4] => 3

[1,-2,3,-4] => 2

[1,-2,-3,4] => 2

[1,-2,-3,-4] => 1

[-1,2,3,4] => 3

[-1,2,3,-4] => 2

[-1,2,-3,4] => 2

[-1,2,-3,-4] => 1

[-1,-2,3,4] => 2

[-1,-2,3,-4] => 1

[-1,-2,-3,4] => 1

[-1,-2,-3,-4] => 0

[1,2,4,3] => 3

[1,2,4,-3] => 3

[1,2,-4,3] => 3

[1,2,-4,-3] => 3

[1,-2,4,3] => 2

[1,-2,4,-3] => 2

[1,-2,-4,3] => 2

[1,-2,-4,-3] => 2

[-1,2,4,3] => 2

[-1,2,4,-3] => 2

[-1,2,-4,3] => 2

[-1,2,-4,-3] => 2

[-1,-2,4,3] => 1

[-1,-2,4,-3] => 1

[-1,-2,-4,3] => 1

[-1,-2,-4,-3] => 1

[1,3,2,4] => 3

[1,3,2,-4] => 2

[1,3,-2,4] => 3

[1,3,-2,-4] => 2

[1,-3,2,4] => 3

[1,-3,2,-4] => 2

[1,-3,-2,4] => 3

[1,-3,-2,-4] => 2

[-1,3,2,4] => 2

[-1,3,2,-4] => 1

[-1,3,-2,4] => 2

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Description

The number of minimal elements in Bruhat order not less than the signed permutation.
The minimal elements in question are biGrassmannian, that is both the element and its inverse have at most one descent.
This is the size of the essential set of the signed permutation, see [1].

References

[1] Reiner, V., Woo, A., Yong, A. Presenting the cohomology of a Schubert variety arXiv:0809.2981

Code

def statistic(pi):
    n = len(list(pi))
    B = SignedPermutations(n).bruhat_poset()
    return sum(1 for w in B.order_ideal_complement_generators([pi], "up"))

Created

Feb 13, 2022 at 17:49 by Martin Rubey

Updated

Feb 13, 2022 at 17:49 by Martin Rubey