- FindStat

Identifier

Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000148: Integer partitions ⟶ ℤ

Values

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

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

[[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] => 1

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

[[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] => 0

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

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

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

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

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

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

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

[[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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[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] => 0

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

[[2],[2],[1],[1]] => [2,2,1,1] => [4,2] => 0

[[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] => 0

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

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

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

[[3],[3]] => [3,3] => [2,2,2] => 0

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

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

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

[[2,1],[2,1]] => [3,3] => [2,2,2] => 0

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

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

[[1,1,1],[1,1,1]] => [3,3] => [2,2,2] => 0

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

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

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

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

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

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

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

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

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

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

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

[[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] => 2

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

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

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

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

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

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

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

[[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] => 1

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

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

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

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

>>> Load all 1123 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = q$

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

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

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

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

$F_{6} = 6 + 19\ q^{2} + 12\ q^{4} + 11\ q^{6}$

$F_{7} = 21\ q + 32\ q^{3} + 18\ q^{5} + 15\ q^{7}$

$F_{8} = 13 + 47\ q^{2} + 52\ q^{4} + 26\ q^{6} + 22\ q^{8}$

$F_{9} = 48\ q + 89\ q^{3} + 78\ q^{5} + 37\ q^{7} + 30\ q^{9}$

$F_{10} = 24 + 119\ q^{2} + 147\ q^{4} + 116\ q^{6} + 52\ q^{8} + 42\ q^{10}$

Description

The number of odd parts of a 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

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.