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*       Copyright (C) 2019 The FindStatCrew <info@findstat.org>             *
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Statistic identifier: St001825

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Collection: Decorated permutations

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Description: The size of the Grassmannian interval associated with a decorated permutation.

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References: [1]   Billey, S. C., Weaver, J. E. Criteria for smoothness of Positroid varieties via pattern avoidance, Johnson graphs, and spirographs [[arXiv:2207.06508]]

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Code:
def anti_exceedance_positions(x):
    """
    sage: x = DecoratedPermutation([5,4,1,2,7,6,9,-8,3])
    sage: anti_exceedance_positions(x)
    [3, 4, 9, 6]
    """
    pi = x.to_signed_permutation().permutation()
    pi_inv = pi.inverse()
    n = len(pi)
    anti = []
    for i in range(1, n+1):
        if i < pi_inv(i) or x[i-1] == i:
            anti.append(pi_inv(i))
    return anti

def to_grassmann_interval(x):
    """
    sage: x = DecoratedPermutation([5,4,1,2,7,6,9,-8,3])
    sage: to_grassmann_interval(x)
    ([1, 2, 6, 3, 5, 4, 7, 9, 8], [3, 4, 6, 9, 1, 2, 5, 7, 8])
    """
    n = len(x)
    I = anti_exceedance_positions(x)
    u1 = []
    u2 = []
    v1 = []
    v2 = []
    for i in range(1, n+1):
        if i in I:
            v1.append(i)
            u1.append(abs(x[i-1]))
        else:
            v2.append(i)
            u2.append(abs(x[i-1]))
    return Permutation(u1+u2), Permutation(v1+v2)

def interval_size(x):
    u, v = to_grassmann_interval(x)
    return sum(1 for pi in u.bruhat_greater() if pi.bruhat_lequal(v))


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Statistic values:

[+]         => 1
[-]         => 1
[+,+]       => 1
[-,+]       => 1
[+,-]       => 1
[-,-]       => 1
[2,1]       => 2
[+,+,+]     => 1
[-,+,+]     => 1
[+,-,+]     => 1
[+,+,-]     => 1
[-,-,+]     => 1
[-,+,-]     => 1
[+,-,-]     => 1
[-,-,-]     => 1
[+,3,2]     => 2
[-,3,2]     => 2
[2,1,+]     => 2
[2,1,-]     => 2
[2,3,1]     => 4
[3,1,2]     => 4
[3,+,1]     => 2
[3,-,1]     => 2
[+,+,+,+]   => 1
[-,+,+,+]   => 1
[+,-,+,+]   => 1
[+,+,-,+]   => 1
[+,+,+,-]   => 1
[-,-,+,+]   => 1
[-,+,-,+]   => 1
[-,+,+,-]   => 1
[+,-,-,+]   => 1
[+,-,+,-]   => 1
[+,+,-,-]   => 1
[-,-,-,+]   => 1
[-,-,+,-]   => 1
[-,+,-,-]   => 1
[+,-,-,-]   => 1
[-,-,-,-]   => 1
[+,+,4,3]   => 2
[-,+,4,3]   => 2
[+,-,4,3]   => 2
[-,-,4,3]   => 2
[+,3,2,+]   => 2
[-,3,2,+]   => 2
[+,3,2,-]   => 2
[-,3,2,-]   => 2
[+,3,4,2]   => 4
[-,3,4,2]   => 4
[+,4,2,3]   => 4
[-,4,2,3]   => 4
[+,4,+,2]   => 2
[-,4,+,2]   => 2
[+,4,-,2]   => 2
[-,4,-,2]   => 2
[2,1,+,+]   => 2
[2,1,-,+]   => 2
[2,1,+,-]   => 2
[2,1,-,-]   => 2
[2,1,4,3]   => 4
[2,3,1,+]   => 4
[2,3,1,-]   => 4
[2,3,4,1]   => 8
[2,4,1,3]   => 10
[2,4,+,1]   => 4
[2,4,-,1]   => 4
[3,1,2,+]   => 4
[3,1,2,-]   => 4
[3,1,4,2]   => 8
[3,+,1,+]   => 2
[3,-,1,+]   => 2
[3,+,1,-]   => 2
[3,-,1,-]   => 2
[3,+,4,1]   => 4
[3,-,4,1]   => 4
[3,4,1,2]   => 14
[3,4,2,1]   => 8
[4,1,2,3]   => 8
[4,1,+,2]   => 4
[4,1,-,2]   => 4
[4,+,1,3]   => 4
[4,-,1,3]   => 4
[4,+,+,1]   => 2
[4,-,+,1]   => 2
[4,+,-,1]   => 2
[4,-,-,1]   => 2
[4,3,1,2]   => 8
[4,3,2,1]   => 4
[+,+,+,+,+] => 1
[-,+,+,+,+] => 1
[+,-,+,+,+] => 1
[+,+,-,+,+] => 1
[+,+,+,-,+] => 1
[+,+,+,+,-] => 1
[-,-,+,+,+] => 1
[-,+,-,+,+] => 1
[-,+,+,-,+] => 1
[-,+,+,+,-] => 1
[+,-,-,+,+] => 1
[+,-,+,-,+] => 1
[+,-,+,+,-] => 1
[+,+,-,-,+] => 1
[+,+,-,+,-] => 1
[+,+,+,-,-] => 1
[-,-,-,+,+] => 1
[-,-,+,-,+] => 1
[-,-,+,+,-] => 1
[-,+,-,-,+] => 1
[-,+,-,+,-] => 1
[-,+,+,-,-] => 1
[+,-,-,-,+] => 1
[+,-,-,+,-] => 1
[+,-,+,-,-] => 1
[+,+,-,-,-] => 1
[-,-,-,-,+] => 1
[-,-,-,+,-] => 1
[-,-,+,-,-] => 1
[-,+,-,-,-] => 1
[+,-,-,-,-] => 1
[-,-,-,-,-] => 1
[+,+,+,5,4] => 2
[-,+,+,5,4] => 2
[+,-,+,5,4] => 2
[+,+,-,5,4] => 2
[-,-,+,5,4] => 2
[-,+,-,5,4] => 2
[+,-,-,5,4] => 2
[-,-,-,5,4] => 2
[+,+,4,3,+] => 2
[-,+,4,3,+] => 2
[+,-,4,3,+] => 2
[+,+,4,3,-] => 2
[-,-,4,3,+] => 2
[-,+,4,3,-] => 2
[+,-,4,3,-] => 2
[-,-,4,3,-] => 2
[+,+,4,5,3] => 4
[-,+,4,5,3] => 4
[+,-,4,5,3] => 4
[-,-,4,5,3] => 4
[+,+,5,3,4] => 4
[-,+,5,3,4] => 4
[+,-,5,3,4] => 4
[-,-,5,3,4] => 4
[+,+,5,+,3] => 2
[-,+,5,+,3] => 2
[+,-,5,+,3] => 2
[+,+,5,-,3] => 2
[-,-,5,+,3] => 2
[-,+,5,-,3] => 2
[+,-,5,-,3] => 2
[-,-,5,-,3] => 2
[+,3,2,+,+] => 2
[-,3,2,+,+] => 2
[+,3,2,-,+] => 2
[+,3,2,+,-] => 2
[-,3,2,-,+] => 2
[-,3,2,+,-] => 2
[+,3,2,-,-] => 2
[-,3,2,-,-] => 2
[+,3,2,5,4] => 4
[-,3,2,5,4] => 4
[+,3,4,2,+] => 4
[-,3,4,2,+] => 4
[+,3,4,2,-] => 4
[-,3,4,2,-] => 4
[+,3,4,5,2] => 8
[-,3,4,5,2] => 8
[+,3,5,2,4] => 10
[-,3,5,2,4] => 10
[+,3,5,+,2] => 4
[-,3,5,+,2] => 4
[+,3,5,-,2] => 4
[-,3,5,-,2] => 4
[+,4,2,3,+] => 4
[-,4,2,3,+] => 4
[+,4,2,3,-] => 4
[-,4,2,3,-] => 4
[+,4,2,5,3] => 8
[-,4,2,5,3] => 8
[+,4,+,2,+] => 2
[-,4,+,2,+] => 2
[+,4,-,2,+] => 2
[+,4,+,2,-] => 2
[-,4,-,2,+] => 2
[-,4,+,2,-] => 2
[+,4,-,2,-] => 2
[-,4,-,2,-] => 2
[+,4,+,5,2] => 4
[-,4,+,5,2] => 4
[+,4,-,5,2] => 4
[-,4,-,5,2] => 4
[+,4,5,2,3] => 14
[-,4,5,2,3] => 14
[+,4,5,3,2] => 8
[-,4,5,3,2] => 8
[+,5,2,3,4] => 8
[-,5,2,3,4] => 8
[+,5,2,+,3] => 4
[-,5,2,+,3] => 4
[+,5,2,-,3] => 4
[-,5,2,-,3] => 4
[+,5,+,2,4] => 4
[-,5,+,2,4] => 4
[+,5,-,2,4] => 4
[-,5,-,2,4] => 4
[+,5,+,+,2] => 2
[-,5,+,+,2] => 2
[+,5,-,+,2] => 2
[+,5,+,-,2] => 2
[-,5,-,+,2] => 2
[-,5,+,-,2] => 2
[+,5,-,-,2] => 2
[-,5,-,-,2] => 2
[+,5,4,2,3] => 8
[-,5,4,2,3] => 8
[+,5,4,3,2] => 4
[-,5,4,3,2] => 4
[2,1,+,+,+] => 2
[2,1,-,+,+] => 2
[2,1,+,-,+] => 2
[2,1,+,+,-] => 2
[2,1,-,-,+] => 2
[2,1,-,+,-] => 2
[2,1,+,-,-] => 2
[2,1,-,-,-] => 2
[2,1,+,5,4] => 4
[2,1,-,5,4] => 4
[2,1,4,3,+] => 4
[2,1,4,3,-] => 4
[2,1,4,5,3] => 8
[2,1,5,3,4] => 8
[2,1,5,+,3] => 4
[2,1,5,-,3] => 4
[2,3,1,+,+] => 4
[2,3,1,-,+] => 4
[2,3,1,+,-] => 4
[2,3,1,-,-] => 4
[2,3,1,5,4] => 8
[2,3,4,1,+] => 8
[2,3,4,1,-] => 8
[2,3,4,5,1] => 16
[2,3,5,1,4] => 22
[2,3,5,+,1] => 8
[2,3,5,-,1] => 8
[2,4,1,3,+] => 10
[2,4,1,3,-] => 10
[2,4,1,5,3] => 20
[2,4,+,1,+] => 4
[2,4,-,1,+] => 4
[2,4,+,1,-] => 4
[2,4,-,1,-] => 4
[2,4,+,5,1] => 8
[2,4,-,5,1] => 8
[2,4,5,1,3] => 34
[2,4,5,3,1] => 16
[2,5,1,3,4] => 22
[2,5,1,+,3] => 10
[2,5,1,-,3] => 10
[2,5,+,1,4] => 10
[2,5,-,1,4] => 10
[2,5,+,+,1] => 4
[2,5,-,+,1] => 4
[2,5,+,-,1] => 4
[2,5,-,-,1] => 4
[2,5,4,1,3] => 20
[2,5,4,3,1] => 8
[3,1,2,+,+] => 4
[3,1,2,-,+] => 4
[3,1,2,+,-] => 4
[3,1,2,-,-] => 4
[3,1,2,5,4] => 8
[3,1,4,2,+] => 8
[3,1,4,2,-] => 8
[3,1,4,5,2] => 16
[3,1,5,2,4] => 20
[3,1,5,+,2] => 8
[3,1,5,-,2] => 8
[3,+,1,+,+] => 2
[3,-,1,+,+] => 2
[3,+,1,-,+] => 2
[3,+,1,+,-] => 2
[3,-,1,-,+] => 2
[3,-,1,+,-] => 2
[3,+,1,-,-] => 2
[3,-,1,-,-] => 2
[3,+,1,5,4] => 4
[3,-,1,5,4] => 4
[3,+,4,1,+] => 4
[3,-,4,1,+] => 4
[3,+,4,1,-] => 4
[3,-,4,1,-] => 4
[3,+,4,5,1] => 8
[3,-,4,5,1] => 8
[3,+,5,1,4] => 10
[3,-,5,1,4] => 10
[3,+,5,+,1] => 4
[3,-,5,+,1] => 4
[3,+,5,-,1] => 4
[3,-,5,-,1] => 4

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Created: Jul 23, 2022 at 18:16 by Martin Rubey

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Last Updated: Jul 23, 2022 at 18:16 by Martin Rubey