Identifier
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
Images
=>
Cc0002;cc-rep-0Cc0007;cc-rep-1Cc0007;cc-rep-2
[1]=>[[1]]=>[[1]] [2]=>[[1,2]]=>[[1,2]] [1,1]=>[[1],[2]]=>[[1],[2]] [3]=>[[1,2,3]]=>[[1,2,3]] [2,1]=>[[1,3],[2]]=>[[1,2],[3]] [1,1,1]=>[[1],[2],[3]]=>[[1],[2],[3]] [4]=>[[1,2,3,4]]=>[[1,2,3,4]] [3,1]=>[[1,3,4],[2]]=>[[1,2,3],[4]] [2,2]=>[[1,2],[3,4]]=>[[1,2],[3,4]] [2,1,1]=>[[1,4],[2],[3]]=>[[1,2],[3],[4]] [1,1,1,1]=>[[1],[2],[3],[4]]=>[[1],[2],[3],[4]] [5]=>[[1,2,3,4,5]]=>[[1,2,3,4,5]] [4,1]=>[[1,3,4,5],[2]]=>[[1,2,3,4],[5]] [3,2]=>[[1,2,5],[3,4]]=>[[1,2,3],[4,5]] [3,1,1]=>[[1,4,5],[2],[3]]=>[[1,2,3],[4],[5]] [2,2,1]=>[[1,3],[2,5],[4]]=>[[1,2],[3,4],[5]] [2,1,1,1]=>[[1,5],[2],[3],[4]]=>[[1,2],[3],[4],[5]] [1,1,1,1,1]=>[[1],[2],[3],[4],[5]]=>[[1],[2],[3],[4],[5]] [6]=>[[1,2,3,4,5,6]]=>[[1,2,3,4,5,6]] [5,1]=>[[1,3,4,5,6],[2]]=>[[1,2,3,4,5],[6]] [4,2]=>[[1,2,5,6],[3,4]]=>[[1,2,3,4],[5,6]] [4,1,1]=>[[1,4,5,6],[2],[3]]=>[[1,2,3,4],[5],[6]] [3,3]=>[[1,2,3],[4,5,6]]=>[[1,2,3],[4,5,6]] [3,2,1]=>[[1,3,6],[2,5],[4]]=>[[1,2,3],[4,5],[6]] [3,1,1,1]=>[[1,5,6],[2],[3],[4]]=>[[1,2,3],[4],[5],[6]] [2,2,2]=>[[1,2],[3,4],[5,6]]=>[[1,2],[3,4],[5,6]] [2,2,1,1]=>[[1,4],[2,6],[3],[5]]=>[[1,2],[3,4],[5],[6]] [2,1,1,1,1]=>[[1,6],[2],[3],[4],[5]]=>[[1,2],[3],[4],[5],[6]] [1,1,1,1,1,1]=>[[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]] [6,1]=>[[1,3,4,5,6,7],[2]]=>[[1,2,3,4,5,6],[7]] [5,2]=>[[1,2,5,6,7],[3,4]]=>[[1,2,3,4,5],[6,7]] [5,1,1]=>[[1,4,5,6,7],[2],[3]]=>[[1,2,3,4,5],[6],[7]] [4,3]=>[[1,2,3,7],[4,5,6]]=>[[1,2,3,4],[5,6,7]] [4,2,1]=>[[1,3,6,7],[2,5],[4]]=>[[1,2,3,4],[5,6],[7]] [4,1,1,1]=>[[1,5,6,7],[2],[3],[4]]=>[[1,2,3,4],[5],[6],[7]] [3,3,1]=>[[1,3,4],[2,6,7],[5]]=>[[1,2,3],[4,5,6],[7]] [3,2,2]=>[[1,2,7],[3,4],[5,6]]=>[[1,2,3],[4,5],[6,7]] [3,2,1,1]=>[[1,4,7],[2,6],[3],[5]]=>[[1,2,3],[4,5],[6],[7]] [3,1,1,1,1]=>[[1,6,7],[2],[3],[4],[5]]=>[[1,2,3],[4],[5],[6],[7]] [2,2,2,1]=>[[1,3],[2,5],[4,7],[6]]=>[[1,2],[3,4],[5,6],[7]] [2,2,1,1,1]=>[[1,5],[2,7],[3],[4],[6]]=>[[1,2],[3,4],[5],[6],[7]] [2,1,1,1,1,1]=>[[1,7],[2],[3],[4],[5],[6]]=>[[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]] [8]=>[[1,2,3,4,5,6,7,8]]=>[[1,2,3,4,5,6,7,8]] [7,1]=>[[1,3,4,5,6,7,8],[2]]=>[[1,2,3,4,5,6,7],[8]] [6,2]=>[[1,2,5,6,7,8],[3,4]]=>[[1,2,3,4,5,6],[7,8]] [6,1,1]=>[[1,4,5,6,7,8],[2],[3]]=>[[1,2,3,4,5,6],[7],[8]] [5,3]=>[[1,2,3,7,8],[4,5,6]]=>[[1,2,3,4,5],[6,7,8]] [5,2,1]=>[[1,3,6,7,8],[2,5],[4]]=>[[1,2,3,4,5],[6,7],[8]] [5,1,1,1]=>[[1,5,6,7,8],[2],[3],[4]]=>[[1,2,3,4,5],[6],[7],[8]] [4,4]=>[[1,2,3,4],[5,6,7,8]]=>[[1,2,3,4],[5,6,7,8]] [4,3,1]=>[[1,3,4,8],[2,6,7],[5]]=>[[1,2,3,4],[5,6,7],[8]] [4,2,2]=>[[1,2,7,8],[3,4],[5,6]]=>[[1,2,3,4],[5,6],[7,8]] [4,2,1,1]=>[[1,4,7,8],[2,6],[3],[5]]=>[[1,2,3,4],[5,6],[7],[8]] [4,1,1,1,1]=>[[1,6,7,8],[2],[3],[4],[5]]=>[[1,2,3,4],[5],[6],[7],[8]] [3,3,2]=>[[1,2,5],[3,4,8],[6,7]]=>[[1,2,3],[4,5,6],[7,8]] [3,3,1,1]=>[[1,4,5],[2,7,8],[3],[6]]=>[[1,2,3],[4,5,6],[7],[8]] [3,2,2,1]=>[[1,3,8],[2,5],[4,7],[6]]=>[[1,2,3],[4,5],[6,7],[8]] [3,2,1,1,1]=>[[1,5,8],[2,7],[3],[4],[6]]=>[[1,2,3],[4,5],[6],[7],[8]] [3,1,1,1,1,1]=>[[1,7,8],[2],[3],[4],[5],[6]]=>[[1,2,3],[4],[5],[6],[7],[8]] [2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8]]=>[[1,2],[3,4],[5,6],[7,8]] [2,2,2,1,1]=>[[1,4],[2,6],[3,8],[5],[7]]=>[[1,2],[3,4],[5,6],[7],[8]] [2,2,1,1,1,1]=>[[1,6],[2,8],[3],[4],[5],[7]]=>[[1,2],[3,4],[5],[6],[7],[8]] [2,1,1,1,1,1,1]=>[[1,8],[2],[3],[4],[5],[6],[7]]=>[[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]] [9]=>[[1,2,3,4,5,6,7,8,9]]=>[[1,2,3,4,5,6,7,8,9]] [8,1]=>[[1,3,4,5,6,7,8,9],[2]]=>[[1,2,3,4,5,6,7,8],[9]] [7,2]=>[[1,2,5,6,7,8,9],[3,4]]=>[[1,2,3,4,5,6,7],[8,9]] [7,1,1]=>[[1,4,5,6,7,8,9],[2],[3]]=>[[1,2,3,4,5,6,7],[8],[9]] [6,3]=>[[1,2,3,7,8,9],[4,5,6]]=>[[1,2,3,4,5,6],[7,8,9]] [6,2,1]=>[[1,3,6,7,8,9],[2,5],[4]]=>[[1,2,3,4,5,6],[7,8],[9]] [6,1,1,1]=>[[1,5,6,7,8,9],[2],[3],[4]]=>[[1,2,3,4,5,6],[7],[8],[9]] [5,4]=>[[1,2,3,4,9],[5,6,7,8]]=>[[1,2,3,4,5],[6,7,8,9]] [5,3,1]=>[[1,3,4,8,9],[2,6,7],[5]]=>[[1,2,3,4,5],[6,7,8],[9]] [5,2,2]=>[[1,2,7,8,9],[3,4],[5,6]]=>[[1,2,3,4,5],[6,7],[8,9]] [5,2,1,1]=>[[1,4,7,8,9],[2,6],[3],[5]]=>[[1,2,3,4,5],[6,7],[8],[9]] [5,1,1,1,1]=>[[1,6,7,8,9],[2],[3],[4],[5]]=>[[1,2,3,4,5],[6],[7],[8],[9]] [4,4,1]=>[[1,3,4,5],[2,7,8,9],[6]]=>[[1,2,3,4],[5,6,7,8],[9]] [4,3,2]=>[[1,2,5,9],[3,4,8],[6,7]]=>[[1,2,3,4],[5,6,7],[8,9]] [4,3,1,1]=>[[1,4,5,9],[2,7,8],[3],[6]]=>[[1,2,3,4],[5,6,7],[8],[9]] [4,2,2,1]=>[[1,3,8,9],[2,5],[4,7],[6]]=>[[1,2,3,4],[5,6],[7,8],[9]] [4,2,1,1,1]=>[[1,5,8,9],[2,7],[3],[4],[6]]=>[[1,2,3,4],[5,6],[7],[8],[9]] [4,1,1,1,1,1]=>[[1,7,8,9],[2],[3],[4],[5],[6]]=>[[1,2,3,4],[5],[6],[7],[8],[9]] [3,3,3]=>[[1,2,3],[4,5,6],[7,8,9]]=>[[1,2,3],[4,5,6],[7,8,9]] [3,3,2,1]=>[[1,3,6],[2,5,9],[4,8],[7]]=>[[1,2,3],[4,5,6],[7,8],[9]] [3,3,1,1,1]=>[[1,5,6],[2,8,9],[3],[4],[7]]=>[[1,2,3],[4,5,6],[7],[8],[9]] [3,2,2,2]=>[[1,2,9],[3,4],[5,6],[7,8]]=>[[1,2,3],[4,5],[6,7],[8,9]] [3,2,2,1,1]=>[[1,4,9],[2,6],[3,8],[5],[7]]=>[[1,2,3],[4,5],[6,7],[8],[9]] [3,2,1,1,1,1]=>[[1,6,9],[2,8],[3],[4],[5],[7]]=>[[1,2,3],[4,5],[6],[7],[8],[9]] [3,1,1,1,1,1,1]=>[[1,8,9],[2],[3],[4],[5],[6],[7]]=>[[1,2,3],[4],[5],[6],[7],[8],[9]] [2,2,2,2,1]=>[[1,3],[2,5],[4,7],[6,9],[8]]=>[[1,2],[3,4],[5,6],[7,8],[9]] [2,2,2,1,1,1]=>[[1,5],[2,7],[3,9],[4],[6],[8]]=>[[1,2],[3,4],[5,6],[7],[8],[9]] [2,2,1,1,1,1,1]=>[[1,7],[2,9],[3],[4],[5],[6],[8]]=>[[1,2],[3,4],[5],[6],[7],[8],[9]] [2,1,1,1,1,1,1,1]=>[[1,9],[2],[3],[4],[5],[6],[7],[8]]=>[[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]] [10]=>[[1,2,3,4,5,6,7,8,9,10]]=>[[1,2,3,4,5,6,7,8,9,10]] [9,1]=>[[1,3,4,5,6,7,8,9,10],[2]]=>[[1,2,3,4,5,6,7,8,9],[10]] [8,2]=>[[1,2,5,6,7,8,9,10],[3,4]]=>[[1,2,3,4,5,6,7,8],[9,10]] [8,1,1]=>[[1,4,5,6,7,8,9,10],[2],[3]]=>[[1,2,3,4,5,6,7,8],[9],[10]] [7,3]=>[[1,2,3,7,8,9,10],[4,5,6]]=>[[1,2,3,4,5,6,7],[8,9,10]] [7,2,1]=>[[1,3,6,7,8,9,10],[2,5],[4]]=>[[1,2,3,4,5,6,7],[8,9],[10]] [7,1,1,1]=>[[1,5,6,7,8,9,10],[2],[3],[4]]=>[[1,2,3,4,5,6,7],[8],[9],[10]] [6,4]=>[[1,2,3,4,9,10],[5,6,7,8]]=>[[1,2,3,4,5,6],[7,8,9,10]] [6,3,1]=>[[1,3,4,8,9,10],[2,6,7],[5]]=>[[1,2,3,4,5,6],[7,8,9],[10]] [6,2,2]=>[[1,2,7,8,9,10],[3,4],[5,6]]=>[[1,2,3,4,5,6],[7,8],[9,10]] [6,2,1,1]=>[[1,4,7,8,9,10],[2,6],[3],[5]]=>[[1,2,3,4,5,6],[7,8],[9],[10]] [6,1,1,1,1]=>[[1,6,7,8,9,10],[2],[3],[4],[5]]=>[[1,2,3,4,5,6],[7],[8],[9],[10]] [5,5]=>[[1,2,3,4,5],[6,7,8,9,10]]=>[[1,2,3,4,5],[6,7,8,9,10]] [5,4,1]=>[[1,3,4,5,10],[2,7,8,9],[6]]=>[[1,2,3,4,5],[6,7,8,9],[10]] [5,3,2]=>[[1,2,5,9,10],[3,4,8],[6,7]]=>[[1,2,3,4,5],[6,7,8],[9,10]] [5,3,1,1]=>[[1,4,5,9,10],[2,7,8],[3],[6]]=>[[1,2,3,4,5],[6,7,8],[9],[10]] [5,2,2,1]=>[[1,3,8,9,10],[2,5],[4,7],[6]]=>[[1,2,3,4,5],[6,7],[8,9],[10]] [5,2,1,1,1]=>[[1,5,8,9,10],[2,7],[3],[4],[6]]=>[[1,2,3,4,5],[6,7],[8],[9],[10]] [5,1,1,1,1,1]=>[[1,7,8,9,10],[2],[3],[4],[5],[6]]=>[[1,2,3,4,5],[6],[7],[8],[9],[10]] [4,4,2]=>[[1,2,5,6],[3,4,9,10],[7,8]]=>[[1,2,3,4],[5,6,7,8],[9,10]] [4,4,1,1]=>[[1,4,5,6],[2,8,9,10],[3],[7]]=>[[1,2,3,4],[5,6,7,8],[9],[10]] [4,3,3]=>[[1,2,3,10],[4,5,6],[7,8,9]]=>[[1,2,3,4],[5,6,7],[8,9,10]] [4,3,2,1]=>[[1,3,6,10],[2,5,9],[4,8],[7]]=>[[1,2,3,4],[5,6,7],[8,9],[10]] [4,3,1,1,1]=>[[1,5,6,10],[2,8,9],[3],[4],[7]]=>[[1,2,3,4],[5,6,7],[8],[9],[10]] [4,2,2,2]=>[[1,2,9,10],[3,4],[5,6],[7,8]]=>[[1,2,3,4],[5,6],[7,8],[9,10]] [4,2,2,1,1]=>[[1,4,9,10],[2,6],[3,8],[5],[7]]=>[[1,2,3,4],[5,6],[7,8],[9],[10]] [4,2,1,1,1,1]=>[[1,6,9,10],[2,8],[3],[4],[5],[7]]=>[[1,2,3,4],[5,6],[7],[8],[9],[10]] [4,1,1,1,1,1,1]=>[[1,8,9,10],[2],[3],[4],[5],[6],[7]]=>[[1,2,3,4],[5],[6],[7],[8],[9],[10]] [3,3,3,1]=>[[1,3,4],[2,6,7],[5,9,10],[8]]=>[[1,2,3],[4,5,6],[7,8,9],[10]] [3,3,2,2]=>[[1,2,7],[3,4,10],[5,6],[8,9]]=>[[1,2,3],[4,5,6],[7,8],[9,10]] [3,3,2,1,1]=>[[1,4,7],[2,6,10],[3,9],[5],[8]]=>[[1,2,3],[4,5,6],[7,8],[9],[10]] [3,3,1,1,1,1]=>[[1,6,7],[2,9,10],[3],[4],[5],[8]]=>[[1,2,3],[4,5,6],[7],[8],[9],[10]] [3,2,2,2,1]=>[[1,3,10],[2,5],[4,7],[6,9],[8]]=>[[1,2,3],[4,5],[6,7],[8,9],[10]] [3,2,2,1,1,1]=>[[1,5,10],[2,7],[3,9],[4],[6],[8]]=>[[1,2,3],[4,5],[6,7],[8],[9],[10]] [3,2,1,1,1,1,1]=>[[1,7,10],[2,9],[3],[4],[5],[6],[8]]=>[[1,2,3],[4,5],[6],[7],[8],[9],[10]] [3,1,1,1,1,1,1,1]=>[[1,9,10],[2],[3],[4],[5],[6],[7],[8]]=>[[1,2,3],[4],[5],[6],[7],[8],[9],[10]] [2,2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8],[9,10]]=>[[1,2],[3,4],[5,6],[7,8],[9,10]] [2,2,2,2,1,1]=>[[1,4],[2,6],[3,8],[5,10],[7],[9]]=>[[1,2],[3,4],[5,6],[7,8],[9],[10]] [2,2,2,1,1,1,1]=>[[1,6],[2,8],[3,10],[4],[5],[7],[9]]=>[[1,2],[3,4],[5,6],[7],[8],[9],[10]] [2,2,1,1,1,1,1,1]=>[[1,8],[2,10],[3],[4],[5],[6],[7],[9]]=>[[1,2],[3,4],[5],[6],[7],[8],[9],[10]] [2,1,1,1,1,1,1,1,1]=>[[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]=>[[1,2],[3],[4],[5],[6],[7],[8],[9],[10]] [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]] [10,1]=>[[1,3,4,5,6,7,8,9,10,11],[2]]=>[[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]] [6,6]=>[[1,2,3,4,5,6],[7,8,9,10,11,12]]=>[[1,2,3,4,5,6],[7,8,9,10,11,12]] [5,5,1,1]=>[[1,4,5,6,7],[2,9,10,11,12],[3],[8]]=>[[1,2,3,4,5],[6,7,8,9,10],[11],[12]] [2,2,2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]=>[[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]] [1,1,1,1,1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]=>[[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]] []=>[]=>[]
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
Schützenberger involution
Description
Sends a standard tableau to the standard tableau obtained via the Schützenberger involution.