Identifier
Mp00042:
Integer partitions
—initial tableau⟶
Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
Images
[1] => [[1]] => [[1]] => [[1]]
[2] => [[1,2]] => [[1],[2]] => [[1],[2]]
[1,1] => [[1],[2]] => [[1,2]] => [[1,2]]
[3] => [[1,2,3]] => [[1],[2],[3]] => [[1],[2],[3]]
[2,1] => [[1,2],[3]] => [[1,3],[2]] => [[1,2],[3]]
[1,1,1] => [[1],[2],[3]] => [[1,2,3]] => [[1,2,3]]
[4] => [[1,2,3,4]] => [[1],[2],[3],[4]] => [[1],[2],[3],[4]]
[3,1] => [[1,2,3],[4]] => [[1,4],[2],[3]] => [[1,2],[3],[4]]
[2,2] => [[1,2],[3,4]] => [[1,3],[2,4]] => [[1,3],[2,4]]
[2,1,1] => [[1,2],[3],[4]] => [[1,3,4],[2]] => [[1,2,3],[4]]
[1,1,1,1] => [[1],[2],[3],[4]] => [[1,2,3,4]] => [[1,2,3,4]]
[5] => [[1,2,3,4,5]] => [[1],[2],[3],[4],[5]] => [[1],[2],[3],[4],[5]]
[4,1] => [[1,2,3,4],[5]] => [[1,5],[2],[3],[4]] => [[1,2],[3],[4],[5]]
[3,2] => [[1,2,3],[4,5]] => [[1,4],[2,5],[3]] => [[1,3],[2,4],[5]]
[3,1,1] => [[1,2,3],[4],[5]] => [[1,4,5],[2],[3]] => [[1,2,3],[4],[5]]
[2,2,1] => [[1,2],[3,4],[5]] => [[1,3,5],[2,4]] => [[1,2,4],[3,5]]
[2,1,1,1] => [[1,2],[3],[4],[5]] => [[1,3,4,5],[2]] => [[1,2,3,4],[5]]
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [[1,2,3,4,5]] => [[1,2,3,4,5]]
[6] => [[1,2,3,4,5,6]] => [[1],[2],[3],[4],[5],[6]] => [[1],[2],[3],[4],[5],[6]]
[5,1] => [[1,2,3,4,5],[6]] => [[1,6],[2],[3],[4],[5]] => [[1,2],[3],[4],[5],[6]]
[4,2] => [[1,2,3,4],[5,6]] => [[1,5],[2,6],[3],[4]] => [[1,3],[2,4],[5],[6]]
[4,1,1] => [[1,2,3,4],[5],[6]] => [[1,5,6],[2],[3],[4]] => [[1,2,3],[4],[5],[6]]
[3,3] => [[1,2,3],[4,5,6]] => [[1,4],[2,5],[3,6]] => [[1,4],[2,5],[3,6]]
[3,2,1] => [[1,2,3],[4,5],[6]] => [[1,4,6],[2,5],[3]] => [[1,2,4],[3,5],[6]]
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [[1,4,5,6],[2],[3]] => [[1,2,3,4],[5],[6]]
[2,2,2] => [[1,2],[3,4],[5,6]] => [[1,3,5],[2,4,6]] => [[1,3,5],[2,4,6]]
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [[1,3,5,6],[2,4]] => [[1,2,3,5],[4,6]]
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [[1,3,4,5,6],[2]] => [[1,2,3,4,5],[6]]
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [[1,2,3,4,5,6]] => [[1,2,3,4,5,6]]
[7] => [[1,2,3,4,5,6,7]] => [[1],[2],[3],[4],[5],[6],[7]] => [[1],[2],[3],[4],[5],[6],[7]]
[6,1] => [[1,2,3,4,5,6],[7]] => [[1,7],[2],[3],[4],[5],[6]] => [[1,2],[3],[4],[5],[6],[7]]
[5,2] => [[1,2,3,4,5],[6,7]] => [[1,6],[2,7],[3],[4],[5]] => [[1,3],[2,4],[5],[6],[7]]
[5,1,1] => [[1,2,3,4,5],[6],[7]] => [[1,6,7],[2],[3],[4],[5]] => [[1,2,3],[4],[5],[6],[7]]
[4,3] => [[1,2,3,4],[5,6,7]] => [[1,5],[2,6],[3,7],[4]] => [[1,4],[2,5],[3,6],[7]]
[4,2,1] => [[1,2,3,4],[5,6],[7]] => [[1,5,7],[2,6],[3],[4]] => [[1,2,4],[3,5],[6],[7]]
[4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [[1,5,6,7],[2],[3],[4]] => [[1,2,3,4],[5],[6],[7]]
[3,3,1] => [[1,2,3],[4,5,6],[7]] => [[1,4,7],[2,5],[3,6]] => [[1,2,5],[3,6],[4,7]]
[3,2,2] => [[1,2,3],[4,5],[6,7]] => [[1,4,6],[2,5,7],[3]] => [[1,3,5],[2,4,6],[7]]
[3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => [[1,4,6,7],[2,5],[3]] => [[1,2,3,5],[4,6],[7]]
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [[1,4,5,6,7],[2],[3]] => [[1,2,3,4,5],[6],[7]]
[2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [[1,3,5,7],[2,4,6]] => [[1,2,4,6],[3,5,7]]
[2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [[1,3,5,6,7],[2,4]] => [[1,2,3,4,6],[5,7]]
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [[1,3,4,5,6,7],[2]] => [[1,2,3,4,5,6],[7]]
[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]]
[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]]
[7,1] => [[1,2,3,4,5,6,7],[8]] => [[1,8],[2],[3],[4],[5],[6],[7]] => [[1,2],[3],[4],[5],[6],[7],[8]]
[6,2] => [[1,2,3,4,5,6],[7,8]] => [[1,7],[2,8],[3],[4],[5],[6]] => [[1,3],[2,4],[5],[6],[7],[8]]
[6,1,1] => [[1,2,3,4,5,6],[7],[8]] => [[1,7,8],[2],[3],[4],[5],[6]] => [[1,2,3],[4],[5],[6],[7],[8]]
[5,3] => [[1,2,3,4,5],[6,7,8]] => [[1,6],[2,7],[3,8],[4],[5]] => [[1,4],[2,5],[3,6],[7],[8]]
[5,2,1] => [[1,2,3,4,5],[6,7],[8]] => [[1,6,8],[2,7],[3],[4],[5]] => [[1,2,4],[3,5],[6],[7],[8]]
[5,1,1,1] => [[1,2,3,4,5],[6],[7],[8]] => [[1,6,7,8],[2],[3],[4],[5]] => [[1,2,3,4],[5],[6],[7],[8]]
[4,4] => [[1,2,3,4],[5,6,7,8]] => [[1,5],[2,6],[3,7],[4,8]] => [[1,5],[2,6],[3,7],[4,8]]
[4,3,1] => [[1,2,3,4],[5,6,7],[8]] => [[1,5,8],[2,6],[3,7],[4]] => [[1,2,5],[3,6],[4,7],[8]]
[4,2,2] => [[1,2,3,4],[5,6],[7,8]] => [[1,5,7],[2,6,8],[3],[4]] => [[1,3,5],[2,4,6],[7],[8]]
[4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => [[1,5,7,8],[2,6],[3],[4]] => [[1,2,3,5],[4,6],[7],[8]]
[4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => [[1,5,6,7,8],[2],[3],[4]] => [[1,2,3,4,5],[6],[7],[8]]
[3,3,2] => [[1,2,3],[4,5,6],[7,8]] => [[1,4,7],[2,5,8],[3,6]] => [[1,3,6],[2,4,7],[5,8]]
[3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => [[1,4,7,8],[2,5],[3,6]] => [[1,2,3,6],[4,7],[5,8]]
[3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => [[1,4,6,8],[2,5,7],[3]] => [[1,2,4,6],[3,5,7],[8]]
[3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => [[1,4,6,7,8],[2,5],[3]] => [[1,2,3,4,6],[5,7],[8]]
[3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => [[1,4,5,6,7,8],[2],[3]] => [[1,2,3,4,5,6],[7],[8]]
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [[1,3,5,7],[2,4,6,8]] => [[1,3,5,7],[2,4,6,8]]
[2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => [[1,3,5,7,8],[2,4,6]] => [[1,2,3,5,7],[4,6,8]]
[2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => [[1,3,5,6,7,8],[2,4]] => [[1,2,3,4,5,7],[6,8]]
[2,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8]] => [[1,3,4,5,6,7,8],[2]] => [[1,2,3,4,5,6,7],[8]]
[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]]
[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]]
[8,1] => [[1,2,3,4,5,6,7,8],[9]] => [[1,9],[2],[3],[4],[5],[6],[7],[8]] => [[1,2],[3],[4],[5],[6],[7],[8],[9]]
[7,2] => [[1,2,3,4,5,6,7],[8,9]] => [[1,8],[2,9],[3],[4],[5],[6],[7]] => [[1,3],[2,4],[5],[6],[7],[8],[9]]
[7,1,1] => [[1,2,3,4,5,6,7],[8],[9]] => [[1,8,9],[2],[3],[4],[5],[6],[7]] => [[1,2,3],[4],[5],[6],[7],[8],[9]]
[6,3] => [[1,2,3,4,5,6],[7,8,9]] => [[1,7],[2,8],[3,9],[4],[5],[6]] => [[1,4],[2,5],[3,6],[7],[8],[9]]
[6,2,1] => [[1,2,3,4,5,6],[7,8],[9]] => [[1,7,9],[2,8],[3],[4],[5],[6]] => [[1,2,4],[3,5],[6],[7],[8],[9]]
[6,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9]] => [[1,7,8,9],[2],[3],[4],[5],[6]] => [[1,2,3,4],[5],[6],[7],[8],[9]]
[5,4] => [[1,2,3,4,5],[6,7,8,9]] => [[1,6],[2,7],[3,8],[4,9],[5]] => [[1,5],[2,6],[3,7],[4,8],[9]]
[5,3,1] => [[1,2,3,4,5],[6,7,8],[9]] => [[1,6,9],[2,7],[3,8],[4],[5]] => [[1,2,5],[3,6],[4,7],[8],[9]]
[5,2,2] => [[1,2,3,4,5],[6,7],[8,9]] => [[1,6,8],[2,7,9],[3],[4],[5]] => [[1,3,5],[2,4,6],[7],[8],[9]]
[5,2,1,1] => [[1,2,3,4,5],[6,7],[8],[9]] => [[1,6,8,9],[2,7],[3],[4],[5]] => [[1,2,3,5],[4,6],[7],[8],[9]]
[5,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9]] => [[1,6,7,8,9],[2],[3],[4],[5]] => [[1,2,3,4,5],[6],[7],[8],[9]]
[4,4,1] => [[1,2,3,4],[5,6,7,8],[9]] => [[1,5,9],[2,6],[3,7],[4,8]] => [[1,2,6],[3,7],[4,8],[5,9]]
[4,3,2] => [[1,2,3,4],[5,6,7],[8,9]] => [[1,5,8],[2,6,9],[3,7],[4]] => [[1,3,6],[2,4,7],[5,8],[9]]
[4,3,1,1] => [[1,2,3,4],[5,6,7],[8],[9]] => [[1,5,8,9],[2,6],[3,7],[4]] => [[1,2,3,6],[4,7],[5,8],[9]]
[4,2,2,1] => [[1,2,3,4],[5,6],[7,8],[9]] => [[1,5,7,9],[2,6,8],[3],[4]] => [[1,2,4,6],[3,5,7],[8],[9]]
[4,2,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9]] => [[1,5,7,8,9],[2,6],[3],[4]] => [[1,2,3,4,6],[5,7],[8],[9]]
[4,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9]] => [[1,5,6,7,8,9],[2],[3],[4]] => [[1,2,3,4,5,6],[7],[8],[9]]
[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => [[1,4,7],[2,5,8],[3,6,9]] => [[1,4,7],[2,5,8],[3,6,9]]
[3,3,2,1] => [[1,2,3],[4,5,6],[7,8],[9]] => [[1,4,7,9],[2,5,8],[3,6]] => [[1,2,4,7],[3,5,8],[6,9]]
[3,3,1,1,1] => [[1,2,3],[4,5,6],[7],[8],[9]] => [[1,4,7,8,9],[2,5],[3,6]] => [[1,2,3,4,7],[5,8],[6,9]]
[3,2,2,2] => [[1,2,3],[4,5],[6,7],[8,9]] => [[1,4,6,8],[2,5,7,9],[3]] => [[1,3,5,7],[2,4,6,8],[9]]
[3,2,2,1,1] => [[1,2,3],[4,5],[6,7],[8],[9]] => [[1,4,6,8,9],[2,5,7],[3]] => [[1,2,3,5,7],[4,6,8],[9]]
[3,2,1,1,1,1] => [[1,2,3],[4,5],[6],[7],[8],[9]] => [[1,4,6,7,8,9],[2,5],[3]] => [[1,2,3,4,5,7],[6,8],[9]]
[3,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9]] => [[1,4,5,6,7,8,9],[2],[3]] => [[1,2,3,4,5,6,7],[8],[9]]
[2,2,2,2,1] => [[1,2],[3,4],[5,6],[7,8],[9]] => [[1,3,5,7,9],[2,4,6,8]] => [[1,2,4,6,8],[3,5,7,9]]
[2,2,2,1,1,1] => [[1,2],[3,4],[5,6],[7],[8],[9]] => [[1,3,5,7,8,9],[2,4,6]] => [[1,2,3,4,6,8],[5,7,9]]
[2,2,1,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8],[9]] => [[1,3,5,6,7,8,9],[2,4]] => [[1,2,3,4,5,6,8],[7,9]]
[2,1,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9]] => [[1,3,4,5,6,7,8,9],[2]] => [[1,2,3,4,5,6,7,8],[9]]
[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]]
[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]]
[9,1] => [[1,2,3,4,5,6,7,8,9],[10]] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]
[8,2] => [[1,2,3,4,5,6,7,8],[9,10]] => [[1,9],[2,10],[3],[4],[5],[6],[7],[8]] => [[1,3],[2,4],[5],[6],[7],[8],[9],[10]]
[8,1,1] => [[1,2,3,4,5,6,7,8],[9],[10]] => [[1,9,10],[2],[3],[4],[5],[6],[7],[8]] => [[1,2,3],[4],[5],[6],[7],[8],[9],[10]]
[7,3] => [[1,2,3,4,5,6,7],[8,9,10]] => [[1,8],[2,9],[3,10],[4],[5],[6],[7]] => [[1,4],[2,5],[3,6],[7],[8],[9],[10]]
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initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
Map
conjugate
Description
Sends a standard tableau to its conjugate tableau.
Map
Schützenberger involution
Description
Sends a standard tableau to the standard tableau obtained via the Schützenberger involution.
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