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Identifier

Mp00252: Permutations —restriction⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001900: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{2} = 2\ q$

$F_{3} = 6\ q$

$F_{4} = 12\ q + 12\ q^{2}$

$F_{5} = 30\ q + 60\ q^{2} + 30\ q^{3}$

$F_{6} = 144\ q + 180\ q^{2} + 330\ q^{4} + 66\ q^{5}$

$F_{7} = 840\ q + 1008\ q^{2} + 280\ q^{3} + 735\ q^{4} + 630\ q^{5} + 595\ q^{7} + 847\ q^{8} + 105\ q^{9}$

Description

The number of distinct irreducible representations contained in the higher Lie character for an integer partition.

Map

conjugate

Description

Return the conjugate partition of the partition.
The conjugate partition of the partition

$\lambda$ of

$n$ is the partition

$\lambda^*$ whose Ferrers diagram is obtained from the diagram of

$\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.

Map

cycle type

Description

The cycle type of a permutation as a partition.

Map

restriction

Description

The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.