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-----------------------------------------------------------------------------
Statistic identifier: St001773
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Collection: Signed permutations
-----------------------------------------------------------------------------
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"))
-----------------------------------------------------------------------------
Statistic 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
[-1,3,-2,-4] => 1
[-1,-3,2,4] => 2
[-1,-3,2,-4] => 1
[-1,-3,-2,4] => 2
[-1,-3,-2,-4] => 1
[1,3,4,2] => 3
[1,3,4,-2] => 3
[1,3,-4,2] => 3
[1,3,-4,-2] => 3
[1,-3,4,2] => 3
[1,-3,4,-2] => 3
[1,-3,-4,2] => 2
[1,-3,-4,-2] => 2
[-1,3,4,2] => 2
[-1,3,4,-2] => 2
[-1,3,-4,2] => 2
[-1,3,-4,-2] => 2
[-1,-3,4,2] => 2
[-1,-3,4,-2] => 2
[-1,-3,-4,2] => 1
[-1,-3,-4,-2] => 1
[1,4,2,3] => 3
[1,4,2,-3] => 3
[1,4,-2,3] => 3
[1,4,-2,-3] => 2
[1,-4,2,3] => 3
[1,-4,2,-3] => 3
[1,-4,-2,3] => 3
[1,-4,-2,-3] => 2
[-1,4,2,3] => 2
[-1,4,2,-3] => 2
[-1,4,-2,3] => 2
[-1,4,-2,-3] => 1
[-1,-4,2,3] => 2
[-1,-4,2,-3] => 2
[-1,-4,-2,3] => 2
[-1,-4,-2,-3] => 1
[1,4,3,2] => 2
[1,4,3,-2] => 2
[1,4,-3,2] => 4
[1,4,-3,-2] => 3
[1,-4,3,2] => 2
[1,-4,3,-2] => 2
[1,-4,-3,2] => 3
[1,-4,-3,-2] => 3
[-1,4,3,2] => 1
[-1,4,3,-2] => 1
[-1,4,-3,2] => 3
[-1,4,-3,-2] => 2
[-1,-4,3,2] => 1
[-1,-4,3,-2] => 1
[-1,-4,-3,2] => 2
[-1,-4,-3,-2] => 2
[2,1,3,4] => 3
[2,1,3,-4] => 2
[2,1,-3,4] => 2
[2,1,-3,-4] => 1
[2,-1,3,4] => 3
[2,-1,3,-4] => 2
[2,-1,-3,4] => 2
[2,-1,-3,-4] => 1
[-2,1,3,4] => 3
[-2,1,3,-4] => 2
[-2,1,-3,4] => 2
[-2,1,-3,-4] => 1
[-2,-1,3,4] => 3
[-2,-1,3,-4] => 2
[-2,-1,-3,4] => 2
[-2,-1,-3,-4] => 1
[2,1,4,3] => 2
[2,1,4,-3] => 2
[2,1,-4,3] => 2
[2,1,-4,-3] => 2
[2,-1,4,3] => 2
[2,-1,4,-3] => 2
[2,-1,-4,3] => 2
[2,-1,-4,-3] => 2
[-2,1,4,3] => 2
[-2,1,4,-3] => 2
[-2,1,-4,3] => 2
[-2,1,-4,-3] => 2
[-2,-1,4,3] => 2
[-2,-1,4,-3] => 2
[-2,-1,-4,3] => 2
[-2,-1,-4,-3] => 2
[2,3,1,4] => 3
[2,3,1,-4] => 2
[2,3,-1,4] => 3
[2,3,-1,-4] => 2
[2,-3,1,4] => 3
[2,-3,1,-4] => 2
[2,-3,-1,4] => 3
[2,-3,-1,-4] => 2
[-2,3,1,4] => 3
[-2,3,1,-4] => 2
[-2,3,-1,4] => 3
[-2,3,-1,-4] => 2
[-2,-3,1,4] => 2
[-2,-3,1,-4] => 1
[-2,-3,-1,4] => 2
[-2,-3,-1,-4] => 1
[2,3,4,1] => 3
[2,3,4,-1] => 3
[2,3,-4,1] => 3
[2,3,-4,-1] => 3
[2,-3,4,1] => 3
[2,-3,4,-1] => 3
[2,-3,-4,1] => 2
[2,-3,-4,-1] => 2
[-2,3,4,1] => 3
[-2,3,4,-1] => 3
[-2,3,-4,1] => 3
[-2,3,-4,-1] => 3
[-2,-3,4,1] => 2
[-2,-3,4,-1] => 2
[-2,-3,-4,1] => 1
[-2,-3,-4,-1] => 1
[2,4,1,3] => 3
[2,4,1,-3] => 3
[2,4,-1,3] => 3
[2,4,-1,-3] => 2
[2,-4,1,3] => 3
[2,-4,1,-3] => 3
[2,-4,-1,3] => 3
[2,-4,-1,-3] => 2
[-2,4,1,3] => 2
[-2,4,1,-3] => 2
[-2,4,-1,3] => 3
[-2,4,-1,-3] => 2
[-2,-4,1,3] => 2
[-2,-4,1,-3] => 2
[-2,-4,-1,3] => 3
[-2,-4,-1,-3] => 2
[2,4,3,1] => 2
[2,4,3,-1] => 2
[2,4,-3,1] => 4
[2,4,-3,-1] => 3
[2,-4,3,1] => 2
[2,-4,3,-1] => 2
[2,-4,-3,1] => 3
[2,-4,-3,-1] => 3
[-2,4,3,1] => 2
[-2,4,3,-1] => 2
[-2,4,-3,1] => 3
[-2,4,-3,-1] => 2
[-2,-4,3,1] => 2
[-2,-4,3,-1] => 2
[-2,-4,-3,1] => 2
[-2,-4,-3,-1] => 2
[3,1,2,4] => 3
[3,1,2,-4] => 2
[3,1,-2,4] => 3
[3,1,-2,-4] => 2
[3,-1,2,4] => 3
[3,-1,2,-4] => 2
[3,-1,-2,4] => 2
[3,-1,-2,-4] => 1
[-3,1,2,4] => 3
[-3,1,2,-4] => 2
[-3,1,-2,4] => 3
[-3,1,-2,-4] => 2
[-3,-1,2,4] => 3
[-3,-1,2,-4] => 2
[-3,-1,-2,4] => 2
[-3,-1,-2,-4] => 1
[3,1,4,2] => 3
[3,1,4,-2] => 3
[3,1,-4,2] => 3
[3,1,-4,-2] => 3
[3,-1,4,2] => 2
[3,-1,4,-2] => 2
[3,-1,-4,2] => 2
[3,-1,-4,-2] => 2
[-3,1,4,2] => 3
[-3,1,4,-2] => 3
[-3,1,-4,2] => 2
[-3,1,-4,-2] => 2
[-3,-1,4,2] => 3
[-3,-1,4,-2] => 3
[-3,-1,-4,2] => 2
[-3,-1,-4,-2] => 2
[3,2,1,4] => 2
[3,2,1,-4] => 1
[3,2,-1,4] => 2
[3,2,-1,-4] => 1
[3,-2,1,4] => 4
[3,-2,1,-4] => 3
[3,-2,-1,4] => 3
[3,-2,-1,-4] => 2
[-3,2,1,4] => 2
[-3,2,1,-4] => 1
[-3,2,-1,4] => 3
[-3,2,-1,-4] => 2
[-3,-2,1,4] => 3
[-3,-2,1,-4] => 2
[-3,-2,-1,4] => 3
[-3,-2,-1,-4] => 2
[3,2,4,1] => 2
[3,2,4,-1] => 2
[3,2,-4,1] => 2
[3,2,-4,-1] => 2
[3,-2,4,1] => 3
[3,-2,4,-1] => 3
[3,-2,-4,1] => 3
[3,-2,-4,-1] => 3
[-3,2,4,1] => 3
[-3,2,4,-1] => 3
[-3,2,-4,1] => 2
[-3,2,-4,-1] => 2
[-3,-2,4,1] => 3
[-3,-2,4,-1] => 3
[-3,-2,-4,1] => 2
[-3,-2,-4,-1] => 2
[3,4,1,2] => 3
[3,4,1,-2] => 3
[3,4,-1,2] => 3
[3,4,-1,-2] => 2
[3,-4,1,2] => 3
[3,-4,1,-2] => 3
[3,-4,-1,2] => 3
[3,-4,-1,-2] => 2
[-3,4,1,2] => 3
[-3,4,1,-2] => 3
[-3,4,-1,2] => 4
[-3,4,-1,-2] => 2
[-3,-4,1,2] => 2
[-3,-4,1,-2] => 2
[-3,-4,-1,2] => 2
[-3,-4,-1,-2] => 1
[3,4,2,1] => 2
[3,4,2,-1] => 2
[3,4,-2,1] => 4
[3,4,-2,-1] => 3
[3,-4,2,1] => 2
[3,-4,2,-1] => 2
[3,-4,-2,1] => 4
[3,-4,-2,-1] => 3
[-3,4,2,1] => 2
[-3,4,2,-1] => 2
[-3,4,-2,1] => 4
[-3,4,-2,-1] => 3
[-3,-4,2,1] => 1
[-3,-4,2,-1] => 1
[-3,-4,-2,1] => 2
[-3,-4,-2,-1] => 2
[4,1,2,3] => 3
[4,1,2,-3] => 3
[4,1,-2,3] => 3
[4,1,-2,-3] => 2
[4,-1,2,3] => 3
[4,-1,2,-3] => 3
[4,-1,-2,3] => 2
[4,-1,-2,-3] => 1
[-4,1,2,3] => 3
[-4,1,2,-3] => 3
[-4,1,-2,3] => 3
[-4,1,-2,-3] => 2
[-4,-1,2,3] => 3
[-4,-1,2,-3] => 3
[-4,-1,-2,3] => 2
[-4,-1,-2,-3] => 1
[4,1,3,2] => 2
[4,1,3,-2] => 2
[4,1,-3,2] => 4
[4,1,-3,-2] => 3
[4,-1,3,2] => 2
[4,-1,3,-2] => 2
[4,-1,-3,2] => 3
[4,-1,-3,-2] => 2
[-4,1,3,2] => 2
[-4,1,3,-2] => 2
[-4,1,-3,2] => 3
[-4,1,-3,-2] => 3
[-4,-1,3,2] => 2
[-4,-1,3,-2] => 2
[-4,-1,-3,2] => 2
[-4,-1,-3,-2] => 2
[4,2,1,3] => 2
[4,2,1,-3] => 2
[4,2,-1,3] => 3
[4,2,-1,-3] => 2
[4,-2,1,3] => 3
[4,-2,1,-3] => 3
[4,-2,-1,3] => 3
[4,-2,-1,-3] => 2
[-4,2,1,3] => 2
[-4,2,1,-3] => 2
[-4,2,-1,3] => 3
[-4,2,-1,-3] => 2
[-4,-2,1,3] => 3
[-4,-2,1,-3] => 3
[-4,-2,-1,3] => 3
[-4,-2,-1,-3] => 2
[4,2,3,1] => 2
[4,2,3,-1] => 2
[4,2,-3,1] => 4
[4,2,-3,-1] => 3
[4,-2,3,1] => 3
[4,-2,3,-1] => 3
[4,-2,-3,1] => 3
[4,-2,-3,-1] => 2
[-4,2,3,1] => 2
[-4,2,3,-1] => 2
[-4,2,-3,1] => 3
[-4,2,-3,-1] => 3
[-4,-2,3,1] => 3
[-4,-2,3,-1] => 3
[-4,-2,-3,1] => 2
[-4,-2,-3,-1] => 2
[4,3,1,2] => 2
[4,3,1,-2] => 2
[4,3,-1,2] => 2
[4,3,-1,-2] => 1
[4,-3,1,2] => 4
[4,-3,1,-2] => 4
[4,-3,-1,2] => 4
[4,-3,-1,-2] => 2
[-4,3,1,2] => 2
[-4,3,1,-2] => 2
[-4,3,-1,2] => 2
[-4,3,-1,-2] => 1
[-4,-3,1,2] => 3
[-4,-3,1,-2] => 3
[-4,-3,-1,2] => 3
[-4,-3,-1,-2] => 2
[4,3,2,1] => 1
[4,3,2,-1] => 1
[4,3,-2,1] => 3
[4,3,-2,-1] => 2
[4,-3,2,1] => 3
[4,-3,2,-1] => 3
[4,-3,-2,1] => 4
[4,-3,-2,-1] => 3
[-4,3,2,1] => 1
[-4,3,2,-1] => 1
[-4,3,-2,1] => 3
[-4,3,-2,-1] => 2
[-4,-3,2,1] => 2
[-4,-3,2,-1] => 2
[-4,-3,-2,1] => 3
[-4,-3,-2,-1] => 3
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Created: Feb 13, 2022 at 17:49 by Martin Rubey
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Last Updated: Feb 13, 2022 at 17:49 by Martin Rubey