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Identifier

St001379: Permutations ⟶ ℤ

Values

[1] => 0

[1,2] => 0

[2,1] => 2

[1,2,3] => 0

[1,3,2] => 3

[2,1,3] => 2

[2,3,1] => 4

[3,1,2] => 3

[3,2,1] => 6

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

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

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

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

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

[1,4,3,2] => 8

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

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

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

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

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

[2,4,3,1] => 9

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

[3,1,4,2] => 7

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

[3,2,4,1] => 8

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

[3,4,2,1] => 10

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

[4,1,3,2] => 8

[4,2,1,3] => 7

[4,2,3,1] => 9

[4,3,1,2] => 8

[4,3,2,1] => 12

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

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

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

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

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

[1,2,5,4,3] => 10

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

[1,3,2,5,4] => 8

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

[1,3,4,5,2] => 7

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

[1,3,5,4,2] => 11

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

[1,4,2,5,3] => 9

[1,4,3,2,5] => 8

[1,4,3,5,2] => 10

[1,4,5,2,3] => 7

[1,4,5,3,2] => 12

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

[1,5,2,4,3] => 10

[1,5,3,2,4] => 9

[1,5,3,4,2] => 11

[1,5,4,2,3] => 10

[1,5,4,3,2] => 15

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

[2,1,3,5,4] => 7

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

[2,1,4,5,3] => 8

[2,1,5,3,4] => 7

[2,1,5,4,3] => 12

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

[2,3,1,5,4] => 9

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

[2,3,4,5,1] => 8

[2,3,5,1,4] => 7

[2,3,5,4,1] => 12

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

[2,4,1,5,3] => 10

[2,4,3,1,5] => 9

[2,4,3,5,1] => 11

[2,4,5,1,3] => 8

[2,4,5,3,1] => 13

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

[2,5,1,4,3] => 11

[2,5,3,1,4] => 10

[2,5,3,4,1] => 12

[2,5,4,1,3] => 11

[2,5,4,3,1] => 16

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

[3,1,2,5,4] => 8

[3,1,4,2,5] => 7

[3,1,4,5,2] => 9

[3,1,5,2,4] => 8

[3,1,5,4,2] => 13

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

[3,2,1,5,4] => 11

[3,2,4,1,5] => 8

[3,2,4,5,1] => 10

[3,2,5,1,4] => 9

[3,2,5,4,1] => 14

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

[3,4,1,5,2] => 11

[3,4,2,1,5] => 10

[3,4,2,5,1] => 12

[3,4,5,1,2] => 9

[3,4,5,2,1] => 14

[3,5,1,2,4] => 7

[3,5,1,4,2] => 12

>>> Load all 1200 entries. <<<

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$F_{1} = 1$

$F_{2} = 1 + q^{2}$

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

$F_{4} = 1 + q^{2} + 2\ q^{3} + 4\ q^{4} + 2\ q^{5} + 4\ q^{6} + 2\ q^{7} + 4\ q^{8} + 2\ q^{9} + q^{10} + q^{12}$

$F_{5} = 1 + q^{2} + 2\ q^{3} + 4\ q^{4} + 6\ q^{5} + 7\ q^{6} + 8\ q^{7} + 12\ q^{8} + 12\ q^{9} + 14\ q^{10} + 12\ q^{11} + 12\ q^{12} + 8\ q^{13} + 7\ q^{14} + 6\ q^{15} + 4\ q^{16} + 2\ q^{17} + q^{18} + q^{20}$

$F_{6} = 1 + q^{2} + 2\ q^{3} + 4\ q^{4} + 6\ q^{5} + 12\ q^{6} + 12\ q^{7} + 21\ q^{8} + 26\ q^{9} + 37\ q^{10} + 40\ q^{11} + 55\ q^{12} + 52\ q^{13} + 61\ q^{14} + 60\ q^{15} + 61\ q^{16} + 52\ q^{17} + 55\ q^{18} + 40\ q^{19} + 37\ q^{20} + 26\ q^{21} + 21\ q^{22} + 12\ q^{23} + 12\ q^{24} + 6\ q^{25} + 4\ q^{26} + 2\ q^{27} + q^{28} + q^{30}$

Description

The number of inversions plus the major index of a permutation.
This is, the sum of St000004The major index of a permutation. and St000018The number of inversions of a permutation..

References

[1] Wachs, M. L. An involution for signed Eulerian numbers MathSciNet:1158780

Code

def statistic(pi):
    return pi.major_index() + pi.number_of_inversions()

Created

Mar 30, 2019 at 08:54 by Christian Stump

Updated

Mar 30, 2019 at 08:54 by Christian Stump