- FindStat

Identifier

Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000759: Integer partitions ⟶ ℤ

Values

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

[[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] => 3

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

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

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

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

[[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] => 1

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

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

[[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] => 2

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

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

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

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

[[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] => 1

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

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

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

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

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

[[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] => 2

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

[[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] => 2

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

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

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

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

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

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

[[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] => 1

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

[[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] => 1

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

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

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

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

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

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

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

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

[[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] => 4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[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] => 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] => 2

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

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

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

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

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

$F_{5} = 3\ q + 12\ q^{2} + 9\ q^{3}$

$F_{6} = 8\ q + 21\ q^{2} + 15\ q^{3} + 4\ q^{4}$

$F_{7} = 8\ q + 36\ q^{2} + 34\ q^{3} + 8\ q^{4}$

$F_{8} = 19\ q + 62\ q^{2} + 60\ q^{3} + 19\ q^{4}$

$F_{9} = 22\ q + 101\ q^{2} + 116\ q^{3} + 43\ q^{4}$

$F_{10} = 43\ q + 170\ q^{2} + 196\ q^{3} + 81\ q^{4} + 10\ q^{5}$

Description

The smallest missing part in an integer partition.
In [3], this is referred to as the mex, the minimal excluded part of the partition.
For compositions, this is studied in [sec.3.2., 1].

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.