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Identifier

St000959: Permutations ⟶ ℤ

Values

[1] => 1

[1,2] => 1

[2,1] => 1

[1,2,3] => 1

[1,3,2] => 1

[2,1,3] => 1

[2,3,1] => 2

[3,1,2] => 2

[3,2,1] => 5

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

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

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

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

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

[1,4,3,2] => 5

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

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

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

[2,3,4,1] => 6

[2,4,1,3] => 6

[2,4,3,1] => 18

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

[3,1,4,2] => 6

[3,2,1,4] => 5

[3,2,4,1] => 18

[3,4,1,2] => 22

[3,4,2,1] => 62

[4,1,2,3] => 6

[4,1,3,2] => 18

[4,2,1,3] => 18

[4,2,3,1] => 73

[4,3,1,2] => 62

[4,3,2,1] => 210

[1,2,3,4,5] => 1

[1,2,3,5,4] => 1

[1,2,4,3,5] => 1

[1,2,4,5,3] => 2

[1,2,5,3,4] => 2

[1,2,5,4,3] => 5

[1,3,2,4,5] => 1

[1,3,2,5,4] => 2

[1,3,4,2,5] => 2

[1,3,4,5,2] => 6

[1,3,5,2,4] => 6

[1,3,5,4,2] => 18

[1,4,2,3,5] => 2

[1,4,2,5,3] => 6

[1,4,3,2,5] => 5

[1,4,3,5,2] => 18

[1,4,5,2,3] => 22

[1,4,5,3,2] => 62

[1,5,2,3,4] => 6

[1,5,2,4,3] => 18

[1,5,3,2,4] => 18

[1,5,3,4,2] => 73

[1,5,4,2,3] => 62

[1,5,4,3,2] => 210

[2,1,3,4,5] => 1

[2,1,3,5,4] => 2

[2,1,4,3,5] => 2

[2,1,4,5,3] => 6

[2,1,5,3,4] => 6

[2,1,5,4,3] => 18

[2,3,1,4,5] => 2

[2,3,1,5,4] => 6

[2,3,4,1,5] => 6

[2,3,4,5,1] => 24

[2,3,5,1,4] => 24

[2,3,5,4,1] => 86

[2,4,1,3,5] => 6

[2,4,1,5,3] => 24

[2,4,3,1,5] => 18

[2,4,3,5,1] => 86

[2,4,5,1,3] => 106

[2,4,5,3,1] => 352

[2,5,1,3,4] => 24

[2,5,1,4,3] => 86

[2,5,3,1,4] => 86

[2,5,3,4,1] => 418

[2,5,4,1,3] => 352

[2,5,4,3,1] => 1382

[3,1,2,4,5] => 2

[3,1,2,5,4] => 6

[3,1,4,2,5] => 6

[3,1,4,5,2] => 24

[3,1,5,2,4] => 24

[3,1,5,4,2] => 86

[3,2,1,4,5] => 5

[3,2,1,5,4] => 18

[3,2,4,1,5] => 18

[3,2,4,5,1] => 86

[3,2,5,1,4] => 86

[3,2,5,4,1] => 354

[3,4,1,2,5] => 22

[3,4,1,5,2] => 106

[3,4,2,1,5] => 62

[3,4,2,5,1] => 352

[3,4,5,1,2] => 508

[3,4,5,2,1] => 1634

[3,5,1,2,4] => 106

[3,5,1,4,2] => 462

>>> Load all 872 entries. <<<

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Description

The number of strong Bruhat factorizations of a permutation.
This is, the number of factorizations $\pi = t_1 \cdots t_\ell$ for transpositions $\{ t_i \mid 1 \leq i \leq \ell\}$ such that $t_1 \cdots t_i$ has more inversions than $t_1 \cdots t_{i-1}$ for all $1 \leq i \leq \ell$.

Code

@cached_function
def Transpositions(n):
    return Permutations(n).conjugacy_class([2]+[1]*(n-2))

@cached_function
def strong_Bruhat_factorizations(pi):
    inv = pi.number_of_inversions()
    if inv == 0:
        return 1
    else:
        facs = 0
        for t in Transpositions(len(pi)):
            tau = pi*t
            if tau.number_of_inversions() < inv:
                facs += strong_Bruhat_factorizations(tau)
        return facs

def statistic(pi):
    return strong_Bruhat_factorizations(pi)

Created

Aug 28, 2017 at 16:39 by Christian Stump

Updated

Aug 28, 2017 at 16:39 by Christian Stump