Identifier
-
Mp00180:
Integer compositions
—to ribbon⟶
Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤ
Values
[2,1] => [[2,2],[1]] => [1] => [[1]] => 1
[1,2,1] => [[2,2,1],[1]] => [1] => [[1]] => 1
[2,1,1] => [[2,2,2],[1,1]] => [1,1] => [[1],[2]] => 2
[2,2] => [[3,2],[1]] => [1] => [[1]] => 1
[3,1] => [[3,3],[2]] => [2] => [[1,2]] => 1
[1,1,2,1] => [[2,2,1,1],[1]] => [1] => [[1]] => 1
[1,2,1,1] => [[2,2,2,1],[1,1]] => [1,1] => [[1],[2]] => 2
[1,2,2] => [[3,2,1],[1]] => [1] => [[1]] => 1
[1,3,1] => [[3,3,1],[2]] => [2] => [[1,2]] => 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]] => [1,1,1] => [[1],[2],[3]] => 3
[2,1,2] => [[3,2,2],[1,1]] => [1,1] => [[1],[2]] => 2
[2,2,1] => [[3,3,2],[2,1]] => [2,1] => [[1,3],[2]] => 1
[2,3] => [[4,2],[1]] => [1] => [[1]] => 1
[3,1,1] => [[3,3,3],[2,2]] => [2,2] => [[1,2],[3,4]] => 2
[3,2] => [[4,3],[2]] => [2] => [[1,2]] => 1
[4,1] => [[4,4],[3]] => [3] => [[1,2,3]] => 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]] => [1] => [[1]] => 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => [1,1] => [[1],[2]] => 2
[1,1,2,2] => [[3,2,1,1],[1]] => [1] => [[1]] => 1
[1,1,3,1] => [[3,3,1,1],[2]] => [2] => [[1,2]] => 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [1,1,1] => [[1],[2],[3]] => 3
[1,2,1,2] => [[3,2,2,1],[1,1]] => [1,1] => [[1],[2]] => 2
[1,2,2,1] => [[3,3,2,1],[2,1]] => [2,1] => [[1,3],[2]] => 1
[1,2,3] => [[4,2,1],[1]] => [1] => [[1]] => 1
[1,3,1,1] => [[3,3,3,1],[2,2]] => [2,2] => [[1,2],[3,4]] => 2
[1,3,2] => [[4,3,1],[2]] => [2] => [[1,2]] => 1
[1,4,1] => [[4,4,1],[3]] => [3] => [[1,2,3]] => 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => [[1],[2],[3],[4]] => 4
[2,1,1,2] => [[3,2,2,2],[1,1,1]] => [1,1,1] => [[1],[2],[3]] => 3
[2,1,2,1] => [[3,3,2,2],[2,1,1]] => [2,1,1] => [[1,4],[2],[3]] => 1
[2,1,3] => [[4,2,2],[1,1]] => [1,1] => [[1],[2]] => 2
[2,2,1,1] => [[3,3,3,2],[2,2,1]] => [2,2,1] => [[1,3],[2,5],[4]] => 2
[2,2,2] => [[4,3,2],[2,1]] => [2,1] => [[1,3],[2]] => 1
[2,3,1] => [[4,4,2],[3,1]] => [3,1] => [[1,3,4],[2]] => 1
[2,4] => [[5,2],[1]] => [1] => [[1]] => 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]] => [2,2,2] => [[1,2],[3,4],[5,6]] => 3
[3,1,2] => [[4,3,3],[2,2]] => [2,2] => [[1,2],[3,4]] => 2
[3,2,1] => [[4,4,3],[3,2]] => [3,2] => [[1,2,5],[3,4]] => 1
[3,3] => [[5,3],[2]] => [2] => [[1,2]] => 1
[4,1,1] => [[4,4,4],[3,3]] => [3,3] => [[1,2,3],[4,5,6]] => 2
[4,2] => [[5,4],[3]] => [3] => [[1,2,3]] => 1
[5,1] => [[5,5],[4]] => [4] => [[1,2,3,4]] => 1
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]] => [1] => [[1]] => 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => [1,1] => [[1],[2]] => 2
[1,1,1,2,2] => [[3,2,1,1,1],[1]] => [1] => [[1]] => 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]] => [2] => [[1,2]] => 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => [1,1,1] => [[1],[2],[3]] => 3
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => [1,1] => [[1],[2]] => 2
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => [2,1] => [[1,3],[2]] => 1
[1,1,2,3] => [[4,2,1,1],[1]] => [1] => [[1]] => 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => [2,2] => [[1,2],[3,4]] => 2
[1,1,3,2] => [[4,3,1,1],[2]] => [2] => [[1,2]] => 1
[1,1,4,1] => [[4,4,1,1],[3]] => [3] => [[1,2,3]] => 1
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]] => [1,1,1,1] => [[1],[2],[3],[4]] => 4
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => [1,1,1] => [[1],[2],[3]] => 3
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => [2,1,1] => [[1,4],[2],[3]] => 1
[1,2,1,3] => [[4,2,2,1],[1,1]] => [1,1] => [[1],[2]] => 2
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => [2,2,1] => [[1,3],[2,5],[4]] => 2
[1,2,2,2] => [[4,3,2,1],[2,1]] => [2,1] => [[1,3],[2]] => 1
[1,2,3,1] => [[4,4,2,1],[3,1]] => [3,1] => [[1,3,4],[2]] => 1
[1,2,4] => [[5,2,1],[1]] => [1] => [[1]] => 1
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => [2,2,2] => [[1,2],[3,4],[5,6]] => 3
[1,3,1,2] => [[4,3,3,1],[2,2]] => [2,2] => [[1,2],[3,4]] => 2
[1,3,2,1] => [[4,4,3,1],[3,2]] => [3,2] => [[1,2,5],[3,4]] => 1
[1,3,3] => [[5,3,1],[2]] => [2] => [[1,2]] => 1
[1,4,1,1] => [[4,4,4,1],[3,3]] => [3,3] => [[1,2,3],[4,5,6]] => 2
[1,4,2] => [[5,4,1],[3]] => [3] => [[1,2,3]] => 1
[1,5,1] => [[5,5,1],[4]] => [4] => [[1,2,3,4]] => 1
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => 5
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => [[1],[2],[3],[4]] => 4
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => [2,1,1,1] => [[1,5],[2],[3],[4]] => 1
[2,1,1,3] => [[4,2,2,2],[1,1,1]] => [1,1,1] => [[1],[2],[3]] => 3
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => [2,2,1,1] => [[1,4],[2,6],[3],[5]] => 2
[2,1,2,2] => [[4,3,2,2],[2,1,1]] => [2,1,1] => [[1,4],[2],[3]] => 1
[2,1,3,1] => [[4,4,2,2],[3,1,1]] => [3,1,1] => [[1,4,5],[2],[3]] => 1
[2,1,4] => [[5,2,2],[1,1]] => [1,1] => [[1],[2]] => 2
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => [2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => 3
[2,2,1,2] => [[4,3,3,2],[2,2,1]] => [2,2,1] => [[1,3],[2,5],[4]] => 2
[2,2,2,1] => [[4,4,3,2],[3,2,1]] => [3,2,1] => [[1,3,6],[2,5],[4]] => 1
[2,2,3] => [[5,3,2],[2,1]] => [2,1] => [[1,3],[2]] => 1
[2,3,1,1] => [[4,4,4,2],[3,3,1]] => [3,3,1] => [[1,3,4],[2,6,7],[5]] => 2
[2,3,2] => [[5,4,2],[3,1]] => [3,1] => [[1,3,4],[2]] => 1
[2,4,1] => [[5,5,2],[4,1]] => [4,1] => [[1,3,4,5],[2]] => 1
[2,5] => [[6,2],[1]] => [1] => [[1]] => 1
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => [2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => 4
[3,1,1,2] => [[4,3,3,3],[2,2,2]] => [2,2,2] => [[1,2],[3,4],[5,6]] => 3
[3,1,2,1] => [[4,4,3,3],[3,2,2]] => [3,2,2] => [[1,2,7],[3,4],[5,6]] => 1
[3,1,3] => [[5,3,3],[2,2]] => [2,2] => [[1,2],[3,4]] => 2
[3,2,1,1] => [[4,4,4,3],[3,3,2]] => [3,3,2] => [[1,2,5],[3,4,8],[6,7]] => 2
[3,2,2] => [[5,4,3],[3,2]] => [3,2] => [[1,2,5],[3,4]] => 1
[3,3,1] => [[5,5,3],[4,2]] => [4,2] => [[1,2,5,6],[3,4]] => 1
[3,4] => [[6,3],[2]] => [2] => [[1,2]] => 1
[4,1,1,1] => [[4,4,4,4],[3,3,3]] => [3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => 3
[4,1,2] => [[5,4,4],[3,3]] => [3,3] => [[1,2,3],[4,5,6]] => 2
[4,2,1] => [[5,5,4],[4,3]] => [4,3] => [[1,2,3,7],[4,5,6]] => 1
[4,3] => [[6,4],[3]] => [3] => [[1,2,3]] => 1
[5,1,1] => [[5,5,5],[4,4]] => [4,4] => [[1,2,3,4],[5,6,7,8]] => 2
[5,2] => [[6,5],[4]] => [4] => [[1,2,3,4]] => 1
[6,1] => [[6,6],[5]] => [5] => [[1,2,3,4,5]] => 1
[1,1,1,1,1,2,1] => [[2,2,1,1,1,1,1],[1]] => [1] => [[1]] => 1
[1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1],[1,1]] => [1,1] => [[1],[2]] => 2
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Description
The row containing the largest entry of a standard tableau.
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
to ribbon
Description
The ribbon shape corresponding to an integer composition.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
Map
inner shape
Description
The inner shape of a skew partition.
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