- FindStat

Identifier

Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[3,1]] => [4] => [1,1,1,1,0,0,0,0,1,0] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[4],[1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => 3

[[3,1],[1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => 3

[[2,2],[1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => 3

[[2,1,1],[1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => 3

[[1,1,1,1],[1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => 3

[[5]] => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2

[[4,1]] => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2

[[3,2]] => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2

[[3,1,1]] => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2

[[2,2,1]] => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2

[[2,1,1,1]] => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2

[[1,1,1,1,1]] => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2

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

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

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

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

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

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

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

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

[[3],[2],[1]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 4

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

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

[[2,1],[2],[1]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 4

[[2,1],[1,1],[1]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 4

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

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

[[1,1,1],[1,1],[1]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 4

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

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

[[4],[2]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 3

[[3,1],[1],[1]] => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => 2

[[3,1],[2]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 3

[[3,1],[1,1]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 3

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

[[2,2],[2]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 3

[[2,2],[1,1]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 3

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

[[2,1,1],[2]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 3

[[2,1,1],[1,1]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 3

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

[[1,1,1,1],[1,1]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 3

[[5],[1]] => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => 3

[[4,1],[1]] => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => 3

[[3,2],[1]] => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => 3

[[3,1,1],[1]] => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => 3

[[2,2,1],[1]] => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => 3

[[2,1,1,1],[1]] => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => 3

[[1,1,1,1,1],[1]] => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => 3

[[6]] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 2

[[5,1]] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 2

[[4,2]] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 2

[[4,1,1]] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 2

[[3,3]] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 2

[[3,2,1]] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 2

[[3,1,1,1]] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 2

[[2,2,2]] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 2

[[2,2,1,1]] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 2

[[2,1,1,1,1]] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 2

[[1,1,1,1,1,1]] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 2

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

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

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

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

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

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

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

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

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

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

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

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

Description

The number of up steps after the last double rise of a Dyck path.

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.

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.