- FindStat

Identifier

Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤ

Values

[1] => [1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

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

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

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

$F_{5} = q + 2\ q^{2} + 2\ q^{3} + q^{4} + q^{5}$

$F_{6} = q + 3\ q^{2} + 3\ q^{3} + 2\ q^{4} + q^{5} + q^{6}$

$F_{7} = q + 3\ q^{2} + 4\ q^{3} + 3\ q^{4} + 2\ q^{5} + q^{6} + q^{7}$

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

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

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

$F_{11} = q + 5\ q^{2} + 10\ q^{3} + 11\ q^{4} + 10\ q^{5} + 7\ q^{6} + 5\ q^{7} + 3\ q^{8} + 2\ q^{9} + q^{10} + q^{11}$

$F_{12} = q + 6\ q^{2} + 12\ q^{3} + 15\ q^{4} + 13\ q^{5} + 11\ q^{6} + 7\ q^{7} + 5\ q^{8} + 3\ q^{9} + 2\ q^{10} + q^{11} + q^{12}$

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

$F_{14} = q + 7\ q^{2} + 16\ q^{3} + 23\ q^{4} + 23\ q^{5} + 20\ q^{6} + 15\ q^{7} + 11\ q^{8} + 7\ q^{9} + 5\ q^{10} + 3\ q^{11} + 2\ q^{12} + q^{13} + q^{14}$

$F_{15} = q + 7\ q^{2} + 19\ q^{3} + 27\ q^{4} + 30\ q^{5} + 26\ q^{6} + 21\ q^{7} + 15\ q^{8} + 11\ q^{9} + 7\ q^{10} + 5\ q^{11} + 3\ q^{12} + 2\ q^{13} + q^{14} + q^{15}$

$F_{16} = q + 8\ q^{2} + 21\ q^{3} + 34\ q^{4} + 37\ q^{5} + 35\ q^{6} + 28\ q^{7} + 22\ q^{8} + 15\ q^{9} + 11\ q^{10} + 7\ q^{11} + 5\ q^{12} + 3\ q^{13} + 2\ q^{14} + q^{15} + q^{16}$

$F_{17} = q + 8\ q^{2} + 24\ q^{3} + 39\ q^{4} + 47\ q^{5} + 44\ q^{6} + 38\ q^{7} + 29\ q^{8} + 22\ q^{9} + 15\ q^{10} + 11\ q^{11} + 7\ q^{12} + 5\ q^{13} + 3\ q^{14} + 2\ q^{15} + q^{16} + q^{17}$

Description

The length of the 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.