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Identifier

St001528: Permutations ⟶ ℤ

Values

[1] => 1

[1,2] => 1

[2,1] => 1

[1,2,3] => 1

[1,3,2] => 3

[2,1,3] => 3

[2,3,1] => 2

[3,1,2] => 2

[3,2,1] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,4,5,3] => 20

[1,2,5,3,4] => 20

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

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

[1,3,2,5,4] => 45

[1,3,4,2,5] => 20

[1,3,4,5,2] => 45

[1,3,5,2,4] => 45

[1,3,5,4,2] => 20

[1,4,2,3,5] => 20

[1,4,2,5,3] => 45

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

[1,4,3,5,2] => 20

[1,4,5,2,3] => 45

[1,4,5,3,2] => 45

[1,5,2,3,4] => 45

[1,5,2,4,3] => 20

[1,5,3,2,4] => 20

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

[1,5,4,2,3] => 45

[1,5,4,3,2] => 45

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

[2,1,3,5,4] => 45

[2,1,4,3,5] => 45

[2,1,4,5,3] => 44

[2,1,5,3,4] => 44

[2,1,5,4,3] => 45

[2,3,1,4,5] => 20

[2,3,1,5,4] => 44

[2,3,4,1,5] => 45

[2,3,4,5,1] => 44

[2,3,5,1,4] => 44

[2,3,5,4,1] => 45

[2,4,1,3,5] => 45

[2,4,1,5,3] => 44

[2,4,3,1,5] => 20

[2,4,3,5,1] => 45

[2,4,5,1,3] => 44

[2,4,5,3,1] => 44

[2,5,1,3,4] => 44

[2,5,1,4,3] => 45

[2,5,3,1,4] => 45

[2,5,3,4,1] => 20

[2,5,4,1,3] => 44

[2,5,4,3,1] => 44

[3,1,2,4,5] => 20

[3,1,2,5,4] => 44

[3,1,4,2,5] => 45

[3,1,4,5,2] => 44

[3,1,5,2,4] => 44

[3,1,5,4,2] => 45

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

[3,2,1,5,4] => 45

[3,2,4,1,5] => 20

[3,2,4,5,1] => 45

[3,2,5,1,4] => 45

[3,2,5,4,1] => 20

[3,4,1,2,5] => 45

[3,4,1,5,2] => 44

[3,4,2,1,5] => 45

[3,4,2,5,1] => 44

[3,4,5,1,2] => 44

[3,4,5,2,1] => 44

[3,5,1,2,4] => 44

[3,5,1,4,2] => 45

>>> Load all 1200 entries. <<<

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$F_{1} = q$

$F_{2} = 2\ q$

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

$F_{4} = q + 6\ q^{6} + 8\ q^{8} + 9\ q^{9}$

$F_{5} = q + 10\ q^{10} + 20\ q^{20} + 44\ q^{44} + 45\ q^{45}$

$F_{6} = q + 15\ q^{15} + 40\ q^{40} + 135\ q^{135} + 264\ q^{264} + 265\ q^{265}$

Description

The number of permutations such that the product with the permutation has the same number of fixed points.
More formally, given a permutation

$\pi$ , this is the number of permutations

$\sigma$ such that

$\pi$ and

$\pi\sigma$ have the same number of fixed points.
Note that the number of permutations

$\sigma$ such that

$\pi$ and

$\pi\sigma$ have the same cycle type is the size of the conjugacy class of

$\pi$ , St000690The size of the conjugacy class of a permutation..

References

[1] balli Number of Permutations? MathOverflow:144899

Code

def statistic(pi):
    m = pi.number_of_fixed_points()
    return sum(1 for tau in Permutations(pi.size()) if (pi*tau).number_of_fixed_points() == m)

Created

Apr 09, 2020 at 18:37 by Martin Rubey

Updated

Apr 09, 2020 at 18:37 by Martin Rubey