Identifier
Values
[1] => [[1]] => [[1]] => [[1]] => 1
[2] => [[1,2]] => [[1,2]] => [[1],[2]] => 2
[1,1] => [[1],[2]] => [[1],[2]] => [[1,2]] => 1
[3] => [[1,2,3]] => [[1,2,3]] => [[1],[2],[3]] => 3
[2,1] => [[1,3],[2]] => [[1,2],[3]] => [[1,3],[2]] => 1
[1,1,1] => [[1],[2],[3]] => [[1],[2],[3]] => [[1,2,3]] => 1
[4] => [[1,2,3,4]] => [[1,2,3,4]] => [[1],[2],[3],[4]] => 4
[3,1] => [[1,3,4],[2]] => [[1,2,4],[3]] => [[1,3],[2],[4]] => 3
[2,2] => [[1,2],[3,4]] => [[1,3],[2,4]] => [[1,2],[3,4]] => 2
[2,1,1] => [[1,4],[2],[3]] => [[1,3],[2],[4]] => [[1,2,4],[3]] => 1
[1,1,1,1] => [[1],[2],[3],[4]] => [[1],[2],[3],[4]] => [[1,2,3,4]] => 1
[5] => [[1,2,3,4,5]] => [[1,2,3,4,5]] => [[1],[2],[3],[4],[5]] => 5
[4,1] => [[1,3,4,5],[2]] => [[1,2,4,5],[3]] => [[1,3],[2],[4],[5]] => 4
[3,2] => [[1,2,5],[3,4]] => [[1,3,4],[2,5]] => [[1,2],[3,5],[4]] => 2
[3,1,1] => [[1,4,5],[2],[3]] => [[1,3,5],[2],[4]] => [[1,2,4],[3],[5]] => 3
[2,2,1] => [[1,3],[2,5],[4]] => [[1,2],[3,4],[5]] => [[1,3,5],[2,4]] => 1
[2,1,1,1] => [[1,5],[2],[3],[4]] => [[1,4],[2],[3],[5]] => [[1,2,3,5],[4]] => 1
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [[1],[2],[3],[4],[5]] => [[1,2,3,4,5]] => 1
[6] => [[1,2,3,4,5,6]] => [[1,2,3,4,5,6]] => [[1],[2],[3],[4],[5],[6]] => 6
[5,1] => [[1,3,4,5,6],[2]] => [[1,2,4,5,6],[3]] => [[1,3],[2],[4],[5],[6]] => 5
[4,2] => [[1,2,5,6],[3,4]] => [[1,3,4,6],[2,5]] => [[1,2],[3,5],[4],[6]] => 4
[4,1,1] => [[1,4,5,6],[2],[3]] => [[1,3,5,6],[2],[4]] => [[1,2,4],[3],[5],[6]] => 4
[3,3] => [[1,2,3],[4,5,6]] => [[1,3,5],[2,4,6]] => [[1,2],[3,4],[5,6]] => 3
[3,2,1] => [[1,3,6],[2,5],[4]] => [[1,2,4],[3,5],[6]] => [[1,3,6],[2,5],[4]] => 1
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => [[1,4,6],[2],[3],[5]] => [[1,2,3,5],[4],[6]] => 3
[2,2,2] => [[1,2],[3,4],[5,6]] => [[1,4],[2,5],[3,6]] => [[1,2,3],[4,5,6]] => 2
[2,2,1,1] => [[1,4],[2,6],[3],[5]] => [[1,3],[2,5],[4],[6]] => [[1,2,4,6],[3,5]] => 1
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [[1,5],[2],[3],[4],[6]] => [[1,2,3,4,6],[5]] => 1
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [[1],[2],[3],[4],[5],[6]] => [[1,2,3,4,5,6]] => 1
[7] => [[1,2,3,4,5,6,7]] => [[1,2,3,4,5,6,7]] => [[1],[2],[3],[4],[5],[6],[7]] => 7
[6,1] => [[1,3,4,5,6,7],[2]] => [[1,2,4,5,6,7],[3]] => [[1,3],[2],[4],[5],[6],[7]] => 6
[5,2] => [[1,2,5,6,7],[3,4]] => [[1,3,4,6,7],[2,5]] => [[1,2],[3,5],[4],[6],[7]] => 5
[5,1,1] => [[1,4,5,6,7],[2],[3]] => [[1,3,5,6,7],[2],[4]] => [[1,2,4],[3],[5],[6],[7]] => 5
[4,3] => [[1,2,3,7],[4,5,6]] => [[1,3,5,6],[2,4,7]] => [[1,2],[3,4],[5,7],[6]] => 3
[4,2,1] => [[1,3,6,7],[2,5],[4]] => [[1,2,4,7],[3,5],[6]] => [[1,3,6],[2,5],[4],[7]] => 4
[4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => [[1,4,6,7],[2],[3],[5]] => [[1,2,3,5],[4],[6],[7]] => 4
[3,3,1] => [[1,3,4],[2,6,7],[5]] => [[1,2,5],[3,6,7],[4]] => [[1,3,4],[2,6],[5,7]] => 3
[3,2,2] => [[1,2,7],[3,4],[5,6]] => [[1,4,6],[2,5],[3,7]] => [[1,2,3],[4,5,7],[6]] => 2
[3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => [[1,3,5],[2,6],[4],[7]] => [[1,2,4,7],[3,6],[5]] => 1
[3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => [[1,5,7],[2],[3],[4],[6]] => [[1,2,3,4,6],[5],[7]] => 3
[2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => [[1,2],[3,4],[5,6],[7]] => [[1,3,5,7],[2,4,6]] => 1
[2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => [[1,4],[2,6],[3],[5],[7]] => [[1,2,3,5,7],[4,6]] => 1
[2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => [[1,6],[2],[3],[4],[5],[7]] => [[1,2,3,4,5,7],[6]] => 1
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [[1],[2],[3],[4],[5],[6],[7]] => [[1,2,3,4,5,6,7]] => 1
[8] => [[1,2,3,4,5,6,7,8]] => [[1,2,3,4,5,6,7,8]] => [[1],[2],[3],[4],[5],[6],[7],[8]] => 8
[7,1] => [[1,3,4,5,6,7,8],[2]] => [[1,2,4,5,6,7,8],[3]] => [[1,3],[2],[4],[5],[6],[7],[8]] => 7
[6,2] => [[1,2,5,6,7,8],[3,4]] => [[1,3,4,6,7,8],[2,5]] => [[1,2],[3,5],[4],[6],[7],[8]] => 6
[6,1,1] => [[1,4,5,6,7,8],[2],[3]] => [[1,3,5,6,7,8],[2],[4]] => [[1,2,4],[3],[5],[6],[7],[8]] => 6
[5,3] => [[1,2,3,7,8],[4,5,6]] => [[1,3,5,6,8],[2,4,7]] => [[1,2],[3,4],[5,7],[6],[8]] => 5
[5,2,1] => [[1,3,6,7,8],[2,5],[4]] => [[1,2,4,7,8],[3,5],[6]] => [[1,3,6],[2,5],[4],[7],[8]] => 5
[5,1,1,1] => [[1,5,6,7,8],[2],[3],[4]] => [[1,4,6,7,8],[2],[3],[5]] => [[1,2,3,5],[4],[6],[7],[8]] => 5
[4,4] => [[1,2,3,4],[5,6,7,8]] => [[1,3,5,7],[2,4,6,8]] => [[1,2],[3,4],[5,6],[7,8]] => 4
[4,3,1] => [[1,3,4,8],[2,6,7],[5]] => [[1,2,5,7],[3,6,8],[4]] => [[1,3,4],[2,6],[5,8],[7]] => 3
[4,2,2] => [[1,2,7,8],[3,4],[5,6]] => [[1,4,6,8],[2,5],[3,7]] => [[1,2,3],[4,5,7],[6],[8]] => 4
[4,2,1,1] => [[1,4,7,8],[2,6],[3],[5]] => [[1,3,5,8],[2,6],[4],[7]] => [[1,2,4,7],[3,6],[5],[8]] => 4
[4,1,1,1,1] => [[1,6,7,8],[2],[3],[4],[5]] => [[1,5,7,8],[2],[3],[4],[6]] => [[1,2,3,4,6],[5],[7],[8]] => 4
[3,3,2] => [[1,2,5],[3,4,8],[6,7]] => [[1,3,4],[2,6,7],[5,8]] => [[1,2,5],[3,6,8],[4,7]] => 2
[3,3,1,1] => [[1,4,5],[2,7,8],[3],[6]] => [[1,3,6],[2,7,8],[4],[5]] => [[1,2,4,5],[3,7],[6,8]] => 3
[3,2,2,1] => [[1,3,8],[2,5],[4,7],[6]] => [[1,2,6],[3,4],[5,7],[8]] => [[1,3,5,8],[2,4,7],[6]] => 1
[3,2,1,1,1] => [[1,5,8],[2,7],[3],[4],[6]] => [[1,4,6],[2,7],[3],[5],[8]] => [[1,2,3,5,8],[4,7],[6]] => 1
[3,1,1,1,1,1] => [[1,7,8],[2],[3],[4],[5],[6]] => [[1,6,8],[2],[3],[4],[5],[7]] => [[1,2,3,4,5,7],[6],[8]] => 3
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [[1,5],[2,6],[3,7],[4,8]] => [[1,2,3,4],[5,6,7,8]] => 2
[2,2,2,1,1] => [[1,4],[2,6],[3,8],[5],[7]] => [[1,3],[2,5],[4,7],[6],[8]] => [[1,2,4,6,8],[3,5,7]] => 1
[2,2,1,1,1,1] => [[1,6],[2,8],[3],[4],[5],[7]] => [[1,5],[2,7],[3],[4],[6],[8]] => [[1,2,3,4,6,8],[5,7]] => 1
[2,1,1,1,1,1,1] => [[1,8],[2],[3],[4],[5],[6],[7]] => [[1,7],[2],[3],[4],[5],[6],[8]] => [[1,2,3,4,5,6,8],[7]] => 1
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [[1,2,3,4,5,6,7,8]] => 1
[9] => [[1,2,3,4,5,6,7,8,9]] => [[1,2,3,4,5,6,7,8,9]] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => 9
[4,3,1,1] => [[1,4,5,9],[2,7,8],[3],[6]] => [[1,3,6,8],[2,7,9],[4],[5]] => [[1,2,4,5],[3,7],[6,9],[8]] => 3
[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => [[1,4,7],[2,5,8],[3,6,9]] => [[1,2,3],[4,5,6],[7,8,9]] => 3
[3,1,1,1,1,1,1] => [[1,8,9],[2],[3],[4],[5],[6],[7]] => [[1,7,9],[2],[3],[4],[5],[6],[8]] => [[1,2,3,4,5,6,8],[7],[9]] => 3
[2,2,2,2,1] => [[1,3],[2,5],[4,7],[6,9],[8]] => [[1,2],[3,4],[5,6],[7,8],[9]] => [[1,3,5,7,9],[2,4,6,8]] => 1
[2,2,2,1,1,1] => [[1,5],[2,7],[3,9],[4],[6],[8]] => [[1,4],[2,6],[3,8],[5],[7],[9]] => [[1,2,3,5,7,9],[4,6,8]] => 1
[2,2,1,1,1,1,1] => [[1,7],[2,9],[3],[4],[5],[6],[8]] => [[1,6],[2,8],[3],[4],[5],[7],[9]] => [[1,2,3,4,5,7,9],[6,8]] => 1
[2,1,1,1,1,1,1,1] => [[1,9],[2],[3],[4],[5],[6],[7],[8]] => [[1,8],[2],[3],[4],[5],[6],[7],[9]] => [[1,2,3,4,5,6,7,9],[8]] => 1
[1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => [[1,2,3,4,5,6,7,8,9]] => 1
[10] => [[1,2,3,4,5,6,7,8,9,10]] => [[1,2,3,4,5,6,7,8,9,10]] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => 10
[5,5] => [[1,2,3,4,5],[6,7,8,9,10]] => [[1,3,5,7,9],[2,4,6,8,10]] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => 5
[5,3,1,1] => [[1,4,5,9,10],[2,7,8],[3],[6]] => [[1,3,6,8,10],[2,7,9],[4],[5]] => [[1,2,4,5],[3,7],[6,9],[8],[10]] => 5
[4,3,2,1] => [[1,3,6,10],[2,5,9],[4,8],[7]] => [[1,2,4,7],[3,5,8],[6,9],[10]] => [[1,3,6,10],[2,5,9],[4,8],[7]] => 1
[2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => [[1,6],[2,7],[3,8],[4,9],[5,10]] => [[1,2,3,4,5],[6,7,8,9,10]] => 2
[2,2,2,2,1,1] => [[1,4],[2,6],[3,8],[5,10],[7],[9]] => [[1,3],[2,5],[4,7],[6,9],[8],[10]] => [[1,2,4,6,8,10],[3,5,7,9]] => 1
[2,2,2,1,1,1,1] => [[1,6],[2,8],[3,10],[4],[5],[7],[9]] => [[1,5],[2,7],[3,9],[4],[6],[8],[10]] => [[1,2,3,4,6,8,10],[5,7,9]] => 1
[2,2,1,1,1,1,1,1] => [[1,8],[2,10],[3],[4],[5],[6],[7],[9]] => [[1,7],[2,9],[3],[4],[5],[6],[8],[10]] => [[1,2,3,4,5,6,8,10],[7,9]] => 1
[2,1,1,1,1,1,1,1,1] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]] => [[1,9],[2],[3],[4],[5],[6],[7],[8],[10]] => [[1,2,3,4,5,6,7,8,10],[9]] => 1
[1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => [[1,2,3,4,5,6,7,8,9,10]] => 1
[6,6] => [[1,2,3,4,5,6],[7,8,9,10,11,12]] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]] => 6
[2,2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]] => [[1,7],[2,8],[3,9],[4,10],[5,11],[6,12]] => [[1,2,3,4,5,6],[7,8,9,10,11,12]] => 2
[1,1,1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]] => [[1,2,3,4,5,6,7,8,9,10,11,12]] => 1
[5,4,3,2,1] => [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]] => [[1,2,4,7,11],[3,5,8,12],[6,9,13],[10,14],[15]] => [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]] => 1
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Description
The row containing the largest entry of a standard tableau.
Map
row-to-column-descents
Description
Return a standard tableau whose column descent set equals the row descent set of the original tableau.
A column descent in a standard tableau is an entry $i$ such that $i+1$ appears in a column to the left of the cell containing $i$, in English notation.
A row descent is an entry $i$ such that $i+1$ appears in a row above of the cell containing $i$.
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
conjugate
Description
Sends a standard tableau to its conjugate tableau.