- FindStat

Identifier

Mp00108: Permutations —cycle type⟶ Integer partitions
St000182: Integer partitions ⟶ ℤ

Values

[1] => [1] => 1

[1,2] => [1,1] => 1

[2,1] => [2] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,3,4,5,2] => [4,1] => 30

[1,3,5,2,4] => [4,1] => 30

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

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

[1,4,2,5,3] => [4,1] => 30

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

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

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

[1,4,5,3,2] => [4,1] => 30

[1,5,2,3,4] => [4,1] => 30

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

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

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

[1,5,4,2,3] => [4,1] => 30

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

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

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

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

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

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

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

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

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

[2,3,4,1,5] => [4,1] => 30

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

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

[2,3,5,4,1] => [4,1] => 30

[2,4,1,3,5] => [4,1] => 30

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

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

[2,4,3,5,1] => [4,1] => 30

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

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

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

[2,5,1,4,3] => [4,1] => 30

[2,5,3,1,4] => [4,1] => 30

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

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

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

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

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

[3,1,4,2,5] => [4,1] => 30

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

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

[3,1,5,4,2] => [4,1] => 30

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

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

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

[3,2,4,5,1] => [4,1] => 30

[3,2,5,1,4] => [4,1] => 30

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

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

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

[3,4,2,1,5] => [4,1] => 30

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

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

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

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

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

>>> Load all 35020 entries. <<<

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$F_{1} = q$

$F_{2} = 2\ q$

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

$F_{4} = q + 3\ q^{3} + 12\ q^{6} + 8\ q^{8}$

$F_{5} = q + 10\ q^{10} + 15\ q^{15} + 40\ q^{20} + 24\ q^{24} + 30\ q^{30}$

$F_{6} = q + 30\ q^{15} + 80\ q^{40} + 45\ q^{45} + 180\ q^{90} + 240\ q^{120} + 144\ q^{144}$

$F_{7} = q + 21\ q^{21} + 70\ q^{70} + 210\ q^{105} + 420\ q^{210} + 280\ q^{280} + 840\ q^{420} + 1008\ q^{504} + 630\ q^{630} + 720\ q^{720} + 840\ q^{840}$

Description

The number of permutations whose cycle type is the given integer partition.
This number is given by

$\{ \pi \in \mathfrak{S}_n : \text{type}(\pi) = \lambda\} = \frac{n!}{\lambda_1 \cdots \lambda_k \mu_1(\lambda)! \cdots \mu_n(\lambda)!}$
where

$\mu_j(\lambda)$ denotes the number of parts of

$\lambda$ equal to

$j$ .
All permutations with the same cycle type form a wikipedia:Conjugacy class.

Map

cycle type

Description

The cycle type of a permutation as a partition.