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Identifier

St000539: Permutations ⟶ ℤ

Values

[1,2] => 0

[2,1] => 1

[1,2,3] => 0

[1,3,2] => 1

[2,1,3] => 1

[2,3,1] => 1

[3,1,2] => 1

[3,2,1] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,3,4,5] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1200 entries. <<<

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$F_{2} = 1 + q$

$F_{3} = 1 + 4\ q + q^{2}$

$F_{4} = 1 + 8\ q + 6\ q^{2} + 8\ q^{3} + q^{4}$

$F_{5} = 1 + 12\ q + 23\ q^{2} + 48\ q^{3} + 23\ q^{4} + 12\ q^{5} + q^{6}$

$F_{6} = 1 + 16\ q + 59\ q^{2} + 137\ q^{3} + 147\ q^{4} + 147\ q^{5} + 137\ q^{6} + 59\ q^{7} + 16\ q^{8} + q^{9}$

Description

The number of odd inversions of a permutation.
An inversion

$i < j$ of a permutation is odd if

$i \not\equiv j\ (\operatorname{mod} 2)$ . See St000538The number of even inversions of a permutation. for even inversions.

References

[1] Klopsch, B., Voll, C. Igusa-type functions associated to finite formed spaces and their functional equations MathSciNet:2500892 arXiv:math/0603565
[2] Brenti, F., Carnevale, A. Odd length for even hyperoctahedral groups and signed generating functions arXiv:1606.01751
[3] Brenti, F., Carnevale, A. Odd length in Weyl groups arXiv:1709.03320

Code

def statistic(pi):
    return sum(1 for i,j in pi.inversions() if is_odd(j-i))

Created

Jun 22, 2016 at 12:32 by Christian Stump

Updated

Nov 06, 2017 at 21:57 by Martin Rubey