Identifier
-
Mp00045:
Integer partitions
—reading tableau⟶
Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
St000899: Integer compositions ⟶ ℤ
Values
[1] => [[1]] => [1] => [1] => 1
[2] => [[1,2]] => [2] => [2] => 1
[1,1] => [[1],[2]] => [1,1] => [1,1] => 2
[3] => [[1,2,3]] => [3] => [3] => 1
[2,1] => [[1,3],[2]] => [1,2] => [2,1] => 1
[1,1,1] => [[1],[2],[3]] => [1,1,1] => [1,1,1] => 3
[4] => [[1,2,3,4]] => [4] => [4] => 1
[3,1] => [[1,3,4],[2]] => [1,3] => [3,1] => 1
[2,2] => [[1,2],[3,4]] => [2,2] => [2,2] => 2
[2,1,1] => [[1,4],[2],[3]] => [1,1,2] => [2,1,1] => 2
[1,1,1,1] => [[1],[2],[3],[4]] => [1,1,1,1] => [1,1,1,1] => 4
[5] => [[1,2,3,4,5]] => [5] => [5] => 1
[4,1] => [[1,3,4,5],[2]] => [1,4] => [4,1] => 1
[3,2] => [[1,2,5],[3,4]] => [2,3] => [3,2] => 1
[3,1,1] => [[1,4,5],[2],[3]] => [1,1,3] => [3,1,1] => 2
[2,2,1] => [[1,3],[2,5],[4]] => [1,2,2] => [2,1,2] => 1
[2,1,1,1] => [[1,5],[2],[3],[4]] => [1,1,1,2] => [2,1,1,1] => 3
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [1,1,1,1,1] => [1,1,1,1,1] => 5
[6] => [[1,2,3,4,5,6]] => [6] => [6] => 1
[5,1] => [[1,3,4,5,6],[2]] => [1,5] => [5,1] => 1
[4,2] => [[1,2,5,6],[3,4]] => [2,4] => [4,2] => 1
[4,1,1] => [[1,4,5,6],[2],[3]] => [1,1,4] => [4,1,1] => 2
[3,3] => [[1,2,3],[4,5,6]] => [3,3] => [3,3] => 2
[3,2,1] => [[1,3,6],[2,5],[4]] => [1,2,3] => [3,1,2] => 1
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => [1,1,1,3] => [3,1,1,1] => 3
[2,2,2] => [[1,2],[3,4],[5,6]] => [2,2,2] => [2,2,2] => 3
[2,2,1,1] => [[1,4],[2,6],[3],[5]] => [1,1,2,2] => [2,1,1,2] => 2
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [1,1,1,1,2] => [2,1,1,1,1] => 4
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [1,1,1,1,1,1] => [1,1,1,1,1,1] => 6
[7] => [[1,2,3,4,5,6,7]] => [7] => [7] => 1
[6,1] => [[1,3,4,5,6,7],[2]] => [1,6] => [6,1] => 1
[5,2] => [[1,2,5,6,7],[3,4]] => [2,5] => [5,2] => 1
[5,1,1] => [[1,4,5,6,7],[2],[3]] => [1,1,5] => [5,1,1] => 2
[4,3] => [[1,2,3,7],[4,5,6]] => [3,4] => [4,3] => 1
[4,2,1] => [[1,3,6,7],[2,5],[4]] => [1,2,4] => [4,1,2] => 1
[4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => [1,1,1,4] => [4,1,1,1] => 3
[3,3,1] => [[1,3,4],[2,6,7],[5]] => [1,3,3] => [3,1,3] => 1
[3,2,2] => [[1,2,7],[3,4],[5,6]] => [2,2,3] => [3,2,2] => 2
[3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => [1,1,2,3] => [3,1,1,2] => 2
[3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => [1,1,1,1,3] => [3,1,1,1,1] => 4
[2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => [1,2,2,2] => [2,1,2,2] => 2
[2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => [1,1,1,2,2] => [2,1,1,1,2] => 3
[2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => [1,1,1,1,1,2] => [2,1,1,1,1,1] => 5
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 7
[8] => [[1,2,3,4,5,6,7,8]] => [8] => [8] => 1
[7,1] => [[1,3,4,5,6,7,8],[2]] => [1,7] => [7,1] => 1
[6,2] => [[1,2,5,6,7,8],[3,4]] => [2,6] => [6,2] => 1
[6,1,1] => [[1,4,5,6,7,8],[2],[3]] => [1,1,6] => [6,1,1] => 2
[5,3] => [[1,2,3,7,8],[4,5,6]] => [3,5] => [5,3] => 1
[5,2,1] => [[1,3,6,7,8],[2,5],[4]] => [1,2,5] => [5,1,2] => 1
[5,1,1,1] => [[1,5,6,7,8],[2],[3],[4]] => [1,1,1,5] => [5,1,1,1] => 3
[4,4] => [[1,2,3,4],[5,6,7,8]] => [4,4] => [4,4] => 2
[4,3,1] => [[1,3,4,8],[2,6,7],[5]] => [1,3,4] => [4,1,3] => 1
[4,2,2] => [[1,2,7,8],[3,4],[5,6]] => [2,2,4] => [4,2,2] => 2
[4,2,1,1] => [[1,4,7,8],[2,6],[3],[5]] => [1,1,2,4] => [4,1,1,2] => 2
[4,1,1,1,1] => [[1,6,7,8],[2],[3],[4],[5]] => [1,1,1,1,4] => [4,1,1,1,1] => 4
[3,3,2] => [[1,2,5],[3,4,8],[6,7]] => [2,3,3] => [3,2,3] => 1
[3,3,1,1] => [[1,4,5],[2,7,8],[3],[6]] => [1,1,3,3] => [3,1,1,3] => 2
[3,2,2,1] => [[1,3,8],[2,5],[4,7],[6]] => [1,2,2,3] => [3,1,2,2] => 2
[3,2,1,1,1] => [[1,5,8],[2,7],[3],[4],[6]] => [1,1,1,2,3] => [3,1,1,1,2] => 3
[3,1,1,1,1,1] => [[1,7,8],[2],[3],[4],[5],[6]] => [1,1,1,1,1,3] => [3,1,1,1,1,1] => 5
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [2,2,2,2] => [2,2,2,2] => 4
[2,2,2,1,1] => [[1,4],[2,6],[3,8],[5],[7]] => [1,1,2,2,2] => [2,1,1,2,2] => 2
[2,2,1,1,1,1] => [[1,6],[2,8],[3],[4],[5],[7]] => [1,1,1,1,2,2] => [2,1,1,1,1,2] => 4
[2,1,1,1,1,1,1] => [[1,8],[2],[3],[4],[5],[6],[7]] => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1] => 6
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 8
[9] => [[1,2,3,4,5,6,7,8,9]] => [9] => [9] => 1
[8,1] => [[1,3,4,5,6,7,8,9],[2]] => [1,8] => [8,1] => 1
[7,2] => [[1,2,5,6,7,8,9],[3,4]] => [2,7] => [7,2] => 1
[7,1,1] => [[1,4,5,6,7,8,9],[2],[3]] => [1,1,7] => [7,1,1] => 2
[6,3] => [[1,2,3,7,8,9],[4,5,6]] => [3,6] => [6,3] => 1
[6,2,1] => [[1,3,6,7,8,9],[2,5],[4]] => [1,2,6] => [6,1,2] => 1
[6,1,1,1] => [[1,5,6,7,8,9],[2],[3],[4]] => [1,1,1,6] => [6,1,1,1] => 3
[5,4] => [[1,2,3,4,9],[5,6,7,8]] => [4,5] => [5,4] => 1
[5,3,1] => [[1,3,4,8,9],[2,6,7],[5]] => [1,3,5] => [5,1,3] => 1
[5,2,2] => [[1,2,7,8,9],[3,4],[5,6]] => [2,2,5] => [5,2,2] => 2
[5,2,1,1] => [[1,4,7,8,9],[2,6],[3],[5]] => [1,1,2,5] => [5,1,1,2] => 2
[5,1,1,1,1] => [[1,6,7,8,9],[2],[3],[4],[5]] => [1,1,1,1,5] => [5,1,1,1,1] => 4
[4,4,1] => [[1,3,4,5],[2,7,8,9],[6]] => [1,4,4] => [4,1,4] => 1
[4,3,2] => [[1,2,5,9],[3,4,8],[6,7]] => [2,3,4] => [4,2,3] => 1
[4,3,1,1] => [[1,4,5,9],[2,7,8],[3],[6]] => [1,1,3,4] => [4,1,1,3] => 2
[4,2,2,1] => [[1,3,8,9],[2,5],[4,7],[6]] => [1,2,2,4] => [4,1,2,2] => 2
[4,2,1,1,1] => [[1,5,8,9],[2,7],[3],[4],[6]] => [1,1,1,2,4] => [4,1,1,1,2] => 3
[4,1,1,1,1,1] => [[1,7,8,9],[2],[3],[4],[5],[6]] => [1,1,1,1,1,4] => [4,1,1,1,1,1] => 5
[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => [3,3,3] => [3,3,3] => 3
[3,3,2,1] => [[1,3,6],[2,5,9],[4,8],[7]] => [1,2,3,3] => [3,1,2,3] => 1
[3,3,1,1,1] => [[1,5,6],[2,8,9],[3],[4],[7]] => [1,1,1,3,3] => [3,1,1,1,3] => 3
[3,2,2,2] => [[1,2,9],[3,4],[5,6],[7,8]] => [2,2,2,3] => [3,2,2,2] => 3
[3,2,2,1,1] => [[1,4,9],[2,6],[3,8],[5],[7]] => [1,1,2,2,3] => [3,1,1,2,2] => 2
[3,2,1,1,1,1] => [[1,6,9],[2,8],[3],[4],[5],[7]] => [1,1,1,1,2,3] => [3,1,1,1,1,2] => 4
[3,1,1,1,1,1,1] => [[1,8,9],[2],[3],[4],[5],[6],[7]] => [1,1,1,1,1,1,3] => [3,1,1,1,1,1,1] => 6
[2,2,2,2,1] => [[1,3],[2,5],[4,7],[6,9],[8]] => [1,2,2,2,2] => [2,1,2,2,2] => 3
[2,2,2,1,1,1] => [[1,5],[2,7],[3,9],[4],[6],[8]] => [1,1,1,2,2,2] => [2,1,1,1,2,2] => 3
[2,2,1,1,1,1,1] => [[1,7],[2,9],[3],[4],[5],[6],[8]] => [1,1,1,1,1,2,2] => [2,1,1,1,1,1,2] => 5
[2,1,1,1,1,1,1,1] => [[1,9],[2],[3],[4],[5],[6],[7],[8]] => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1] => 7
[1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 9
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Description
The maximal number of repetitions of an integer composition.
This is the maximal part of the composition obtained by applying the delta morphism.
This is the maximal part of the composition obtained by applying the delta morphism.
Map
rotate back to front
Description
The back to front rotation of an integer composition.
Map
reading tableau
Description
Return the RSK recording tableau of the reading word of the (standard) tableau $T$ labeled down (in English convention) each column to the shape of a partition.
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 $i$.
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