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Identifier
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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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