- FindStat

Identifier

Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000481: Integer partitions ⟶ ℤ

Values

[1] => [1] => 0

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

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

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

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

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

[4] => [1,1,1,1] => 1

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

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

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

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

[5] => [1,1,1,1,1] => 1

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

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

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

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

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

[1,1,1,1,1] => [5] => 0

[6] => [1,1,1,1,1,1] => 1

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

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

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

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

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

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

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

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

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

[1,1,1,1,1,1] => [6] => 0

[7] => [1,1,1,1,1,1,1] => 1

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

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

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

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

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

[4,1,1,1] => [4,1,1,1] => 1

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

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

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

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

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

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

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

[1,1,1,1,1,1,1] => [7] => 0

[8] => [1,1,1,1,1,1,1,1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,1,1,1,1,1] => [8] => 0

[9] => [1,1,1,1,1,1,1,1,1] => 1

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

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

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

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

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

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

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

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

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

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

[5,1,1,1,1] => [5,1,1,1,1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,1,1,1,1,1,1] => [9] => 0

[10] => [1,1,1,1,1,1,1,1,1,1] => 1

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

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

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

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

>>> Load all 272 entries. <<<

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$F_{1} = 1$

$F_{2} = 1 + q$

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

$F_{4} = 1 + 4\ q$

$F_{5} = 1 + 6\ q$

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

$F_{7} = 1 + 11\ q + 3\ q^{2}$

$F_{8} = 1 + 14\ q + 7\ q^{2}$

$F_{9} = 1 + 17\ q + 12\ q^{2}$

$F_{10} = 1 + 21\ q + 19\ q^{2} + q^{3}$

$F_{11} = 1 + 25\ q + 28\ q^{2} + 2\ q^{3}$

$F_{12} = 1 + 29\ q + 42\ q^{2} + 5\ q^{3}$

Description

The number of upper covers of a partition in dominance order.

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.