- FindStat

Identifier

Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[5,1]] => [6] => [1,1,1,1,1,1] => 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

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

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

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

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

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

$F_{7} = q + 6\ q^{2} + 14\ q^{3} + 20\ q^{4} + 19\ q^{5} + 11\ q^{6} + 15\ q^{7}$

$F_{8} = q + 8\ q^{2} + 18\ q^{3} + 33\ q^{4} + 32\ q^{5} + 31\ q^{6} + 15\ q^{7} + 22\ q^{8}$

$F_{9} = q + 8\ q^{2} + 25\ q^{3} + 43\ q^{4} + 56\ q^{5} + 54\ q^{6} + 43\ q^{7} + 22\ q^{8} + 30\ q^{9}$

$F_{10} = q + 10\ q^{2} + 29\ q^{3} + 66\ q^{4} + 81\ q^{5} + 99\ q^{6} + 78\ q^{7} + 64\ q^{8} + 30\ q^{9} + 42\ q^{10}$

Description

The length of the partition.

Map

to partition

Description

The underlying integer partition of a plane partition.
This is the partition whose parts are the sums of the individual rows of the plane 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.