Identifier
-
Mp00177:
Plane partitions
—transpose⟶
Plane partitions
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000329: Dyck paths ⟶ ℤ
Values
[[1]] => [[1]] => [1] => [1,0] => 0
[[1],[1]] => [[1,1]] => [2] => [1,0,1,0] => 0
[[2]] => [[2]] => [2] => [1,0,1,0] => 0
[[1,1]] => [[1],[1]] => [1,1] => [1,1,0,0] => 1
[[1],[1],[1]] => [[1,1,1]] => [3] => [1,0,1,0,1,0] => 0
[[2],[1]] => [[2,1]] => [3] => [1,0,1,0,1,0] => 0
[[1,1],[1]] => [[1,1],[1]] => [2,1] => [1,0,1,1,0,0] => 1
[[3]] => [[3]] => [3] => [1,0,1,0,1,0] => 0
[[2,1]] => [[2],[1]] => [2,1] => [1,0,1,1,0,0] => 1
[[1,1,1]] => [[1],[1],[1]] => [1,1,1] => [1,1,0,1,0,0] => 2
[[1],[1],[1],[1]] => [[1,1,1,1]] => [4] => [1,0,1,0,1,0,1,0] => 0
[[2],[1],[1]] => [[2,1,1]] => [4] => [1,0,1,0,1,0,1,0] => 0
[[2],[2]] => [[2,2]] => [4] => [1,0,1,0,1,0,1,0] => 0
[[1,1],[1],[1]] => [[1,1,1],[1]] => [3,1] => [1,0,1,0,1,1,0,0] => 1
[[1,1],[1,1]] => [[1,1],[1,1]] => [2,2] => [1,1,1,0,0,0] => 1
[[3],[1]] => [[3,1]] => [4] => [1,0,1,0,1,0,1,0] => 0
[[2,1],[1]] => [[2,1],[1]] => [3,1] => [1,0,1,0,1,1,0,0] => 1
[[1,1,1],[1]] => [[1,1],[1],[1]] => [2,1,1] => [1,0,1,1,0,1,0,0] => 2
[[4]] => [[4]] => [4] => [1,0,1,0,1,0,1,0] => 0
[[3,1]] => [[3],[1]] => [3,1] => [1,0,1,0,1,1,0,0] => 1
[[2,2]] => [[2],[2]] => [2,2] => [1,1,1,0,0,0] => 1
[[2,1,1]] => [[2],[1],[1]] => [2,1,1] => [1,0,1,1,0,1,0,0] => 2
[[1,1,1,1]] => [[1],[1],[1],[1]] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 3
[[1],[1],[1],[1],[1]] => [[1,1,1,1,1]] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
[[2],[1],[1],[1]] => [[2,1,1,1]] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
[[2],[2],[1]] => [[2,2,1]] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
[[1,1],[1],[1],[1]] => [[1,1,1,1],[1]] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[[1,1],[1,1],[1]] => [[1,1,1],[1,1]] => [3,2] => [1,0,1,1,1,0,0,0] => 1
[[3],[1],[1]] => [[3,1,1]] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
[[3],[2]] => [[3,2]] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
[[2,1],[1],[1]] => [[2,1,1],[1]] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[[2,1],[2]] => [[2,2],[1]] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[[2,1],[1,1]] => [[2,1],[1,1]] => [3,2] => [1,0,1,1,1,0,0,0] => 1
[[1,1,1],[1],[1]] => [[1,1,1],[1],[1]] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 2
[[1,1,1],[1,1]] => [[1,1],[1,1],[1]] => [2,2,1] => [1,1,1,0,0,1,0,0] => 2
[[4],[1]] => [[4,1]] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
[[3,1],[1]] => [[3,1],[1]] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[[2,2],[1]] => [[2,1],[2]] => [3,2] => [1,0,1,1,1,0,0,0] => 1
[[2,1,1],[1]] => [[2,1],[1],[1]] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 2
[[1,1,1,1],[1]] => [[1,1],[1],[1],[1]] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 3
[[5]] => [[5]] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
[[4,1]] => [[4],[1]] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[[3,2]] => [[3],[2]] => [3,2] => [1,0,1,1,1,0,0,0] => 1
[[3,1,1]] => [[3],[1],[1]] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 2
[[2,2,1]] => [[2],[2],[1]] => [2,2,1] => [1,1,1,0,0,1,0,0] => 2
[[2,1,1,1]] => [[2],[1],[1],[1]] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 3
[[1,1,1,1,1]] => [[1],[1],[1],[1],[1]] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 4
[[1],[1],[1],[1],[1],[1]] => [[1,1,1,1,1,1]] => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[[2],[1],[1],[1],[1]] => [[2,1,1,1,1]] => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[[2],[2],[1],[1]] => [[2,2,1,1]] => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[[2],[2],[2]] => [[2,2,2]] => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[[1,1],[1],[1],[1],[1]] => [[1,1,1,1,1],[1]] => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 1
[[1,1],[1,1],[1],[1]] => [[1,1,1,1],[1,1]] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[[1,1],[1,1],[1,1]] => [[1,1,1],[1,1,1]] => [3,3] => [1,1,1,0,1,0,0,0] => 1
[[3],[1],[1],[1]] => [[3,1,1,1]] => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[[3],[2],[1]] => [[3,2,1]] => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[[3],[3]] => [[3,3]] => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[[2,1],[1],[1],[1]] => [[2,1,1,1],[1]] => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 1
[[2,1],[2],[1]] => [[2,2,1],[1]] => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 1
[[2,1],[1,1],[1]] => [[2,1,1],[1,1]] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[[2,1],[2,1]] => [[2,2],[1,1]] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[[1,1,1],[1],[1],[1]] => [[1,1,1,1],[1],[1]] => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 2
[[1,1,1],[1,1],[1]] => [[1,1,1],[1,1],[1]] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 2
[[1,1,1],[1,1,1]] => [[1,1],[1,1],[1,1]] => [2,2,2] => [1,1,1,1,0,0,0,0] => 2
[[4],[1],[1]] => [[4,1,1]] => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[[4],[2]] => [[4,2]] => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[[3,1],[1],[1]] => [[3,1,1],[1]] => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 1
[[3,1],[2]] => [[3,2],[1]] => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 1
[[3,1],[1,1]] => [[3,1],[1,1]] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[[2,2],[1],[1]] => [[2,1,1],[2]] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[[2,2],[2]] => [[2,2],[2]] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[[2,2],[1,1]] => [[2,1],[2,1]] => [3,3] => [1,1,1,0,1,0,0,0] => 1
[[2,1,1],[1],[1]] => [[2,1,1],[1],[1]] => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 2
[[2,1,1],[2]] => [[2,2],[1],[1]] => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 2
[[2,1,1],[1,1]] => [[2,1],[1,1],[1]] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 2
[[1,1,1,1],[1],[1]] => [[1,1,1],[1],[1],[1]] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 3
[[1,1,1,1],[1,1]] => [[1,1],[1,1],[1],[1]] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 3
[[5],[1]] => [[5,1]] => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[[4,1],[1]] => [[4,1],[1]] => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 1
[[3,2],[1]] => [[3,1],[2]] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[[3,1,1],[1]] => [[3,1],[1],[1]] => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 2
[[2,2,1],[1]] => [[2,1],[2],[1]] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 2
[[2,1,1,1],[1]] => [[2,1],[1],[1],[1]] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 3
[[1,1,1,1,1],[1]] => [[1,1],[1],[1],[1],[1]] => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => 4
[[6]] => [[6]] => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[[5,1]] => [[5],[1]] => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 1
[[4,2]] => [[4],[2]] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[[4,1,1]] => [[4],[1],[1]] => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 2
[[3,3]] => [[3],[3]] => [3,3] => [1,1,1,0,1,0,0,0] => 1
[[3,2,1]] => [[3],[2],[1]] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 2
[[3,1,1,1]] => [[3],[1],[1],[1]] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 3
[[2,2,2]] => [[2],[2],[2]] => [2,2,2] => [1,1,1,1,0,0,0,0] => 2
[[2,2,1,1]] => [[2],[2],[1],[1]] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 3
[[2,1,1,1,1]] => [[2],[1],[1],[1],[1]] => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => 4
[[1,1,1,1,1,1]] => [[1],[1],[1],[1],[1],[1]] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 5
[[1,1],[1,1],[1],[1],[1]] => [[1,1,1,1,1],[1,1]] => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => 1
[[1,1],[1,1],[1,1],[1]] => [[1,1,1,1],[1,1,1]] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => 1
[[2,1],[1,1],[1],[1]] => [[2,1,1,1],[1,1]] => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => 1
[[2,1],[1,1],[1,1]] => [[2,1,1],[1,1,1]] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => 1
[[2,1],[2,1],[1]] => [[2,2,1],[1,1]] => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => 1
[[1,1,1],[1,1],[1],[1]] => [[1,1,1,1],[1,1],[1]] => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => 2
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Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.
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.
This is the partition whose parts are the sums of the individual rows of the plane partition.
Map
transpose
Description
The transpose of a plane partition.
This is the mirror symmetry along the vertical axis.
This is the mirror symmetry along the vertical axis.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
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