Identifier
-
Mp00311:
Plane partitions
—to partition⟶
Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000091: Integer compositions ⟶ ℤ
Values
[[1]] => [1] => [[1]] => [1] => 0
[[1],[1]] => [1,1] => [[1],[2]] => [1,1] => 0
[[2]] => [2] => [[1,2]] => [2] => 0
[[1,1]] => [2] => [[1,2]] => [2] => 0
[[1],[1],[1]] => [1,1,1] => [[1],[2],[3]] => [1,1,1] => 0
[[2],[1]] => [2,1] => [[1,2],[3]] => [2,1] => 0
[[1,1],[1]] => [2,1] => [[1,2],[3]] => [2,1] => 0
[[3]] => [3] => [[1,2,3]] => [3] => 0
[[2,1]] => [3] => [[1,2,3]] => [3] => 0
[[1,1,1]] => [3] => [[1,2,3]] => [3] => 0
[[1],[1],[1],[1]] => [1,1,1,1] => [[1],[2],[3],[4]] => [1,1,1,1] => 0
[[2],[1],[1]] => [2,1,1] => [[1,2],[3],[4]] => [2,1,1] => 0
[[2],[2]] => [2,2] => [[1,2],[3,4]] => [2,2] => 0
[[1,1],[1],[1]] => [2,1,1] => [[1,2],[3],[4]] => [2,1,1] => 0
[[1,1],[1,1]] => [2,2] => [[1,2],[3,4]] => [2,2] => 0
[[3],[1]] => [3,1] => [[1,2,3],[4]] => [3,1] => 0
[[2,1],[1]] => [3,1] => [[1,2,3],[4]] => [3,1] => 0
[[1,1,1],[1]] => [3,1] => [[1,2,3],[4]] => [3,1] => 0
[[4]] => [4] => [[1,2,3,4]] => [4] => 0
[[3,1]] => [4] => [[1,2,3,4]] => [4] => 0
[[2,2]] => [4] => [[1,2,3,4]] => [4] => 0
[[2,1,1]] => [4] => [[1,2,3,4]] => [4] => 0
[[1,1,1,1]] => [4] => [[1,2,3,4]] => [4] => 0
[[1],[1],[1],[1],[1]] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [1,1,1,1,1] => 0
[[2],[1],[1],[1]] => [2,1,1,1] => [[1,2],[3],[4],[5]] => [2,1,1,1] => 0
[[2],[2],[1]] => [2,2,1] => [[1,2],[3,4],[5]] => [2,2,1] => 0
[[1,1],[1],[1],[1]] => [2,1,1,1] => [[1,2],[3],[4],[5]] => [2,1,1,1] => 0
[[1,1],[1,1],[1]] => [2,2,1] => [[1,2],[3,4],[5]] => [2,2,1] => 0
[[3],[1],[1]] => [3,1,1] => [[1,2,3],[4],[5]] => [3,1,1] => 0
[[3],[2]] => [3,2] => [[1,2,3],[4,5]] => [3,2] => 0
[[2,1],[1],[1]] => [3,1,1] => [[1,2,3],[4],[5]] => [3,1,1] => 0
[[2,1],[2]] => [3,2] => [[1,2,3],[4,5]] => [3,2] => 0
[[2,1],[1,1]] => [3,2] => [[1,2,3],[4,5]] => [3,2] => 0
[[1,1,1],[1],[1]] => [3,1,1] => [[1,2,3],[4],[5]] => [3,1,1] => 0
[[1,1,1],[1,1]] => [3,2] => [[1,2,3],[4,5]] => [3,2] => 0
[[4],[1]] => [4,1] => [[1,2,3,4],[5]] => [4,1] => 0
[[3,1],[1]] => [4,1] => [[1,2,3,4],[5]] => [4,1] => 0
[[2,2],[1]] => [4,1] => [[1,2,3,4],[5]] => [4,1] => 0
[[2,1,1],[1]] => [4,1] => [[1,2,3,4],[5]] => [4,1] => 0
[[1,1,1,1],[1]] => [4,1] => [[1,2,3,4],[5]] => [4,1] => 0
[[5]] => [5] => [[1,2,3,4,5]] => [5] => 0
[[4,1]] => [5] => [[1,2,3,4,5]] => [5] => 0
[[3,2]] => [5] => [[1,2,3,4,5]] => [5] => 0
[[3,1,1]] => [5] => [[1,2,3,4,5]] => [5] => 0
[[2,2,1]] => [5] => [[1,2,3,4,5]] => [5] => 0
[[2,1,1,1]] => [5] => [[1,2,3,4,5]] => [5] => 0
[[1,1,1,1,1]] => [5] => [[1,2,3,4,5]] => [5] => 0
[[1],[1],[1],[1],[1],[1]] => [1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [1,1,1,1,1,1] => 0
[[2],[1],[1],[1],[1]] => [2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [2,1,1,1,1] => 0
[[2],[2],[1],[1]] => [2,2,1,1] => [[1,2],[3,4],[5],[6]] => [2,2,1,1] => 0
[[2],[2],[2]] => [2,2,2] => [[1,2],[3,4],[5,6]] => [2,2,2] => 0
[[1,1],[1],[1],[1],[1]] => [2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [2,1,1,1,1] => 0
[[1,1],[1,1],[1],[1]] => [2,2,1,1] => [[1,2],[3,4],[5],[6]] => [2,2,1,1] => 0
[[1,1],[1,1],[1,1]] => [2,2,2] => [[1,2],[3,4],[5,6]] => [2,2,2] => 0
[[3],[1],[1],[1]] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => [3,1,1,1] => 0
[[3],[2],[1]] => [3,2,1] => [[1,2,3],[4,5],[6]] => [3,2,1] => 0
[[3],[3]] => [3,3] => [[1,2,3],[4,5,6]] => [3,3] => 0
[[2,1],[1],[1],[1]] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => [3,1,1,1] => 0
[[2,1],[2],[1]] => [3,2,1] => [[1,2,3],[4,5],[6]] => [3,2,1] => 0
[[2,1],[1,1],[1]] => [3,2,1] => [[1,2,3],[4,5],[6]] => [3,2,1] => 0
[[2,1],[2,1]] => [3,3] => [[1,2,3],[4,5,6]] => [3,3] => 0
[[1,1,1],[1],[1],[1]] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => [3,1,1,1] => 0
[[1,1,1],[1,1],[1]] => [3,2,1] => [[1,2,3],[4,5],[6]] => [3,2,1] => 0
[[1,1,1],[1,1,1]] => [3,3] => [[1,2,3],[4,5,6]] => [3,3] => 0
[[4],[1],[1]] => [4,1,1] => [[1,2,3,4],[5],[6]] => [4,1,1] => 0
[[4],[2]] => [4,2] => [[1,2,3,4],[5,6]] => [4,2] => 0
[[3,1],[1],[1]] => [4,1,1] => [[1,2,3,4],[5],[6]] => [4,1,1] => 0
[[3,1],[2]] => [4,2] => [[1,2,3,4],[5,6]] => [4,2] => 0
[[3,1],[1,1]] => [4,2] => [[1,2,3,4],[5,6]] => [4,2] => 0
[[2,2],[1],[1]] => [4,1,1] => [[1,2,3,4],[5],[6]] => [4,1,1] => 0
[[2,2],[2]] => [4,2] => [[1,2,3,4],[5,6]] => [4,2] => 0
[[2,2],[1,1]] => [4,2] => [[1,2,3,4],[5,6]] => [4,2] => 0
[[2,1,1],[1],[1]] => [4,1,1] => [[1,2,3,4],[5],[6]] => [4,1,1] => 0
[[2,1,1],[2]] => [4,2] => [[1,2,3,4],[5,6]] => [4,2] => 0
[[2,1,1],[1,1]] => [4,2] => [[1,2,3,4],[5,6]] => [4,2] => 0
[[1,1,1,1],[1],[1]] => [4,1,1] => [[1,2,3,4],[5],[6]] => [4,1,1] => 0
[[1,1,1,1],[1,1]] => [4,2] => [[1,2,3,4],[5,6]] => [4,2] => 0
[[5],[1]] => [5,1] => [[1,2,3,4,5],[6]] => [5,1] => 0
[[4,1],[1]] => [5,1] => [[1,2,3,4,5],[6]] => [5,1] => 0
[[3,2],[1]] => [5,1] => [[1,2,3,4,5],[6]] => [5,1] => 0
[[3,1,1],[1]] => [5,1] => [[1,2,3,4,5],[6]] => [5,1] => 0
[[2,2,1],[1]] => [5,1] => [[1,2,3,4,5],[6]] => [5,1] => 0
[[2,1,1,1],[1]] => [5,1] => [[1,2,3,4,5],[6]] => [5,1] => 0
[[1,1,1,1,1],[1]] => [5,1] => [[1,2,3,4,5],[6]] => [5,1] => 0
[[6]] => [6] => [[1,2,3,4,5,6]] => [6] => 0
[[5,1]] => [6] => [[1,2,3,4,5,6]] => [6] => 0
[[4,2]] => [6] => [[1,2,3,4,5,6]] => [6] => 0
[[4,1,1]] => [6] => [[1,2,3,4,5,6]] => [6] => 0
[[3,3]] => [6] => [[1,2,3,4,5,6]] => [6] => 0
[[3,2,1]] => [6] => [[1,2,3,4,5,6]] => [6] => 0
[[3,1,1,1]] => [6] => [[1,2,3,4,5,6]] => [6] => 0
[[2,2,2]] => [6] => [[1,2,3,4,5,6]] => [6] => 0
[[2,2,1,1]] => [6] => [[1,2,3,4,5,6]] => [6] => 0
[[2,1,1,1,1]] => [6] => [[1,2,3,4,5,6]] => [6] => 0
[[1,1,1,1,1,1]] => [6] => [[1,2,3,4,5,6]] => [6] => 0
[[1],[1],[1],[1],[1],[1],[1]] => [1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [1,1,1,1,1,1,1] => 0
[[2],[1],[1],[1],[1],[1]] => [2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [2,1,1,1,1,1] => 0
[[2],[2],[1],[1],[1]] => [2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [2,2,1,1,1] => 0
[[2],[2],[2],[1]] => [2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [2,2,2,1] => 0
[[1,1],[1],[1],[1],[1],[1]] => [2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [2,1,1,1,1,1] => 0
[[1,1],[1,1],[1],[1],[1]] => [2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [2,2,1,1,1] => 0
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Description
The descent variation of a composition.
Defined in [1].
Defined in [1].
Map
horizontal strip sizes
Description
The composition of horizontal strip sizes.
We associate to a standard Young tableau $T$ the composition $(c_1,\dots,c_k)$, such that $k$ is minimal and the numbers $c_1+\dots+c_i + 1,\dots,c_1+\dots+c_{i+1}$ form a horizontal strip in $T$ for all $i$.
We associate to a standard Young tableau $T$ the composition $(c_1,\dots,c_k)$, such that $k$ is minimal and the numbers $c_1+\dots+c_i + 1,\dots,c_1+\dots+c_{i+1}$ form a horizontal strip in $T$ for all $i$.
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
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.
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