- FindStat

Identifier

Mp00311: Plane partitions —to partition⟶ Integer partitions
St000318: Integer partitions ⟶ ℤ (values match St000159The number of distinct parts of the integer partition., St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[6]] => [6] => 2

[[5,1]] => [6] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1123 entries. <<<

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

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

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

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

$F_{5} = 8\ q^{2} + 16\ q^{3}$

$F_{6} = 17\ q^{2} + 27\ q^{3} + 4\ q^{4}$

$F_{7} = 16\ q^{2} + 58\ q^{3} + 12\ q^{4}$

$F_{8} = 30\ q^{2} + 95\ q^{3} + 35\ q^{4}$

$F_{9} = 34\ q^{2} + 158\ q^{3} + 90\ q^{4}$

$F_{10} = 52\ q^{2} + 261\ q^{3} + 177\ q^{4} + 10\ q^{5}$

Description

The number of addable cells of the Ferrers diagram of an integer partition.

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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.