- FindStat

Identifier

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤ

Values

[1] => [] => [] => 0

[2] => [] => [] => 0

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

[3] => [] => [] => 0

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

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

[4] => [] => [] => 0

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

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

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

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

[5] => [] => [] => 0

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

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

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

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

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

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

[6] => [] => [] => 0

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

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

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

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

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

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

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

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

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

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

[7] => [] => [] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[8] => [] => [] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[9] => [] => [] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[10] => [] => [] => 0

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

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

$F_{4} = 1 + 2\ q + q^{2} + q^{3}$

$F_{5} = 1 + 2\ q + 2\ q^{2} + q^{3} + q^{4}$

$F_{6} = 1 + 3\ q + 3\ q^{2} + 2\ q^{3} + q^{4} + q^{5}$

$F_{7} = 1 + 3\ q + 4\ q^{2} + 3\ q^{3} + 2\ q^{4} + q^{5} + q^{6}$

$F_{8} = 1 + 4\ q + 5\ q^{2} + 5\ q^{3} + 3\ q^{4} + 2\ q^{5} + q^{6} + q^{7}$

$F_{9} = 1 + 4\ q + 7\ q^{2} + 6\ q^{3} + 5\ q^{4} + 3\ q^{5} + 2\ q^{6} + q^{7} + q^{8}$

$F_{10} = 1 + 5\ q + 8\ q^{2} + 9\ q^{3} + 7\ q^{4} + 5\ q^{5} + 3\ q^{6} + 2\ q^{7} + q^{8} + q^{9}$

$F_{11} = 1 + 5\ q + 10\ q^{2} + 11\ q^{3} + 10\ q^{4} + 7\ q^{5} + 5\ q^{6} + 3\ q^{7} + 2\ q^{8} + q^{9} + q^{10}$

$F_{12} = 1 + 6\ q + 12\ q^{2} + 15\ q^{3} + 13\ q^{4} + 11\ q^{5} + 7\ q^{6} + 5\ q^{7} + 3\ q^{8} + 2\ q^{9} + q^{10} + q^{11}$

$F_{13} = 1 + 6\ q + 14\ q^{2} + 18\ q^{3} + 18\ q^{4} + 14\ q^{5} + 11\ q^{6} + 7\ q^{7} + 5\ q^{8} + 3\ q^{9} + 2\ q^{10} + q^{11} + q^{12}$

$F_{14} = 1 + 7\ q + 16\ q^{2} + 23\ q^{3} + 23\ q^{4} + 20\ q^{5} + 15\ q^{6} + 11\ q^{7} + 7\ q^{8} + 5\ q^{9} + 3\ q^{10} + 2\ q^{11} + q^{12} + q^{13}$

$F_{15} = 1 + 7\ q + 19\ q^{2} + 27\ q^{3} + 30\ q^{4} + 26\ q^{5} + 21\ q^{6} + 15\ q^{7} + 11\ q^{8} + 7\ q^{9} + 5\ q^{10} + 3\ q^{11} + 2\ q^{12} + q^{13} + q^{14}$

$F_{16} = 1 + 8\ q + 21\ q^{2} + 34\ q^{3} + 37\ q^{4} + 35\ q^{5} + 28\ q^{6} + 22\ q^{7} + 15\ q^{8} + 11\ q^{9} + 7\ q^{10} + 5\ q^{11} + 3\ q^{12} + 2\ q^{13} + q^{14} + q^{15}$

$F_{17} = 1 + 8\ q + 24\ q^{2} + 39\ q^{3} + 47\ q^{4} + 44\ q^{5} + 38\ q^{6} + 29\ q^{7} + 22\ q^{8} + 15\ q^{9} + 11\ q^{10} + 7\ q^{11} + 5\ q^{12} + 3\ q^{13} + 2\ q^{14} + q^{15} + q^{16}$

Description

The largest part of an integer partition.

Map

first row removal

Description

Removes the first entry of 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.