Identifier
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00083: Standard tableaux —shape⟶ Integer partitions
Images
=>
Cc0002;cc-rep-0Cc0007;cc-rep-1Cc0002;cc-rep-2
[1]=>[[1]]=>[1] [2]=>[[1,2]]=>[2] [1,1]=>[[1],[2]]=>[1,1] [3]=>[[1,2,3]]=>[3] [2,1]=>[[1,3],[2]]=>[2,1] [1,1,1]=>[[1],[2],[3]]=>[1,1,1] [4]=>[[1,2,3,4]]=>[4] [3,1]=>[[1,3,4],[2]]=>[3,1] [2,2]=>[[1,2],[3,4]]=>[2,2] [2,1,1]=>[[1,4],[2],[3]]=>[2,1,1] [1,1,1,1]=>[[1],[2],[3],[4]]=>[1,1,1,1] [5]=>[[1,2,3,4,5]]=>[5] [4,1]=>[[1,3,4,5],[2]]=>[4,1] [3,2]=>[[1,2,5],[3,4]]=>[3,2] [3,1,1]=>[[1,4,5],[2],[3]]=>[3,1,1] [2,2,1]=>[[1,3],[2,5],[4]]=>[2,2,1] [2,1,1,1]=>[[1,5],[2],[3],[4]]=>[2,1,1,1] [1,1,1,1,1]=>[[1],[2],[3],[4],[5]]=>[1,1,1,1,1] [6]=>[[1,2,3,4,5,6]]=>[6] [5,1]=>[[1,3,4,5,6],[2]]=>[5,1] [4,2]=>[[1,2,5,6],[3,4]]=>[4,2] [4,1,1]=>[[1,4,5,6],[2],[3]]=>[4,1,1] [3,3]=>[[1,2,3],[4,5,6]]=>[3,3] [3,2,1]=>[[1,3,6],[2,5],[4]]=>[3,2,1] [3,1,1,1]=>[[1,5,6],[2],[3],[4]]=>[3,1,1,1] [2,2,2]=>[[1,2],[3,4],[5,6]]=>[2,2,2] [2,2,1,1]=>[[1,4],[2,6],[3],[5]]=>[2,2,1,1] [2,1,1,1,1]=>[[1,6],[2],[3],[4],[5]]=>[2,1,1,1,1] [1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6]]=>[1,1,1,1,1,1] [7]=>[[1,2,3,4,5,6,7]]=>[7] [6,1]=>[[1,3,4,5,6,7],[2]]=>[6,1] [5,2]=>[[1,2,5,6,7],[3,4]]=>[5,2] [5,1,1]=>[[1,4,5,6,7],[2],[3]]=>[5,1,1] [4,3]=>[[1,2,3,7],[4,5,6]]=>[4,3] [4,2,1]=>[[1,3,6,7],[2,5],[4]]=>[4,2,1] [4,1,1,1]=>[[1,5,6,7],[2],[3],[4]]=>[4,1,1,1] [3,3,1]=>[[1,3,4],[2,6,7],[5]]=>[3,3,1] [3,2,2]=>[[1,2,7],[3,4],[5,6]]=>[3,2,2] [3,2,1,1]=>[[1,4,7],[2,6],[3],[5]]=>[3,2,1,1] [3,1,1,1,1]=>[[1,6,7],[2],[3],[4],[5]]=>[3,1,1,1,1] [2,2,2,1]=>[[1,3],[2,5],[4,7],[6]]=>[2,2,2,1] [2,2,1,1,1]=>[[1,5],[2,7],[3],[4],[6]]=>[2,2,1,1,1] [2,1,1,1,1,1]=>[[1,7],[2],[3],[4],[5],[6]]=>[2,1,1,1,1,1] [1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7]]=>[1,1,1,1,1,1,1] [8]=>[[1,2,3,4,5,6,7,8]]=>[8] [7,1]=>[[1,3,4,5,6,7,8],[2]]=>[7,1] [6,2]=>[[1,2,5,6,7,8],[3,4]]=>[6,2] [6,1,1]=>[[1,4,5,6,7,8],[2],[3]]=>[6,1,1] [5,3]=>[[1,2,3,7,8],[4,5,6]]=>[5,3] [5,2,1]=>[[1,3,6,7,8],[2,5],[4]]=>[5,2,1] [5,1,1,1]=>[[1,5,6,7,8],[2],[3],[4]]=>[5,1,1,1] [4,4]=>[[1,2,3,4],[5,6,7,8]]=>[4,4] [4,3,1]=>[[1,3,4,8],[2,6,7],[5]]=>[4,3,1] [4,2,2]=>[[1,2,7,8],[3,4],[5,6]]=>[4,2,2] [4,2,1,1]=>[[1,4,7,8],[2,6],[3],[5]]=>[4,2,1,1] [4,1,1,1,1]=>[[1,6,7,8],[2],[3],[4],[5]]=>[4,1,1,1,1] [3,3,2]=>[[1,2,5],[3,4,8],[6,7]]=>[3,3,2] [3,3,1,1]=>[[1,4,5],[2,7,8],[3],[6]]=>[3,3,1,1] [3,2,2,1]=>[[1,3,8],[2,5],[4,7],[6]]=>[3,2,2,1] [3,2,1,1,1]=>[[1,5,8],[2,7],[3],[4],[6]]=>[3,2,1,1,1] [3,1,1,1,1,1]=>[[1,7,8],[2],[3],[4],[5],[6]]=>[3,1,1,1,1,1] [2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8]]=>[2,2,2,2] [2,2,2,1,1]=>[[1,4],[2,6],[3,8],[5],[7]]=>[2,2,2,1,1] [2,2,1,1,1,1]=>[[1,6],[2,8],[3],[4],[5],[7]]=>[2,2,1,1,1,1] [2,1,1,1,1,1,1]=>[[1,8],[2],[3],[4],[5],[6],[7]]=>[2,1,1,1,1,1,1] [1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8]]=>[1,1,1,1,1,1,1,1] [9]=>[[1,2,3,4,5,6,7,8,9]]=>[9] [8,1]=>[[1,3,4,5,6,7,8,9],[2]]=>[8,1] [7,2]=>[[1,2,5,6,7,8,9],[3,4]]=>[7,2] [7,1,1]=>[[1,4,5,6,7,8,9],[2],[3]]=>[7,1,1] [6,3]=>[[1,2,3,7,8,9],[4,5,6]]=>[6,3] [6,2,1]=>[[1,3,6,7,8,9],[2,5],[4]]=>[6,2,1] [6,1,1,1]=>[[1,5,6,7,8,9],[2],[3],[4]]=>[6,1,1,1] [5,4]=>[[1,2,3,4,9],[5,6,7,8]]=>[5,4] [5,3,1]=>[[1,3,4,8,9],[2,6,7],[5]]=>[5,3,1] [5,2,2]=>[[1,2,7,8,9],[3,4],[5,6]]=>[5,2,2] [5,2,1,1]=>[[1,4,7,8,9],[2,6],[3],[5]]=>[5,2,1,1] [5,1,1,1,1]=>[[1,6,7,8,9],[2],[3],[4],[5]]=>[5,1,1,1,1] [4,4,1]=>[[1,3,4,5],[2,7,8,9],[6]]=>[4,4,1] [4,3,2]=>[[1,2,5,9],[3,4,8],[6,7]]=>[4,3,2] [4,3,1,1]=>[[1,4,5,9],[2,7,8],[3],[6]]=>[4,3,1,1] [4,2,2,1]=>[[1,3,8,9],[2,5],[4,7],[6]]=>[4,2,2,1] [4,2,1,1,1]=>[[1,5,8,9],[2,7],[3],[4],[6]]=>[4,2,1,1,1] [4,1,1,1,1,1]=>[[1,7,8,9],[2],[3],[4],[5],[6]]=>[4,1,1,1,1,1] [3,3,3]=>[[1,2,3],[4,5,6],[7,8,9]]=>[3,3,3] [3,3,2,1]=>[[1,3,6],[2,5,9],[4,8],[7]]=>[3,3,2,1] [3,3,1,1,1]=>[[1,5,6],[2,8,9],[3],[4],[7]]=>[3,3,1,1,1] [3,2,2,2]=>[[1,2,9],[3,4],[5,6],[7,8]]=>[3,2,2,2] [3,2,2,1,1]=>[[1,4,9],[2,6],[3,8],[5],[7]]=>[3,2,2,1,1] [3,2,1,1,1,1]=>[[1,6,9],[2,8],[3],[4],[5],[7]]=>[3,2,1,1,1,1] [3,1,1,1,1,1,1]=>[[1,8,9],[2],[3],[4],[5],[6],[7]]=>[3,1,1,1,1,1,1] [2,2,2,2,1]=>[[1,3],[2,5],[4,7],[6,9],[8]]=>[2,2,2,2,1] [2,2,2,1,1,1]=>[[1,5],[2,7],[3,9],[4],[6],[8]]=>[2,2,2,1,1,1] [2,2,1,1,1,1,1]=>[[1,7],[2,9],[3],[4],[5],[6],[8]]=>[2,2,1,1,1,1,1] [2,1,1,1,1,1,1,1]=>[[1,9],[2],[3],[4],[5],[6],[7],[8]]=>[2,1,1,1,1,1,1,1] [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] [10]=>[[1,2,3,4,5,6,7,8,9,10]]=>[10] [9,1]=>[[1,3,4,5,6,7,8,9,10],[2]]=>[9,1] [8,2]=>[[1,2,5,6,7,8,9,10],[3,4]]=>[8,2] [8,1,1]=>[[1,4,5,6,7,8,9,10],[2],[3]]=>[8,1,1] [7,3]=>[[1,2,3,7,8,9,10],[4,5,6]]=>[7,3] [7,2,1]=>[[1,3,6,7,8,9,10],[2,5],[4]]=>[7,2,1] [7,1,1,1]=>[[1,5,6,7,8,9,10],[2],[3],[4]]=>[7,1,1,1] [6,4]=>[[1,2,3,4,9,10],[5,6,7,8]]=>[6,4] [6,3,1]=>[[1,3,4,8,9,10],[2,6,7],[5]]=>[6,3,1] [6,2,2]=>[[1,2,7,8,9,10],[3,4],[5,6]]=>[6,2,2] [6,2,1,1]=>[[1,4,7,8,9,10],[2,6],[3],[5]]=>[6,2,1,1] [6,1,1,1,1]=>[[1,6,7,8,9,10],[2],[3],[4],[5]]=>[6,1,1,1,1] [5,5]=>[[1,2,3,4,5],[6,7,8,9,10]]=>[5,5] [5,4,1]=>[[1,3,4,5,10],[2,7,8,9],[6]]=>[5,4,1] [5,3,2]=>[[1,2,5,9,10],[3,4,8],[6,7]]=>[5,3,2] [5,3,1,1]=>[[1,4,5,9,10],[2,7,8],[3],[6]]=>[5,3,1,1] [5,2,2,1]=>[[1,3,8,9,10],[2,5],[4,7],[6]]=>[5,2,2,1] [5,2,1,1,1]=>[[1,5,8,9,10],[2,7],[3],[4],[6]]=>[5,2,1,1,1] [5,1,1,1,1,1]=>[[1,7,8,9,10],[2],[3],[4],[5],[6]]=>[5,1,1,1,1,1] [4,4,2]=>[[1,2,5,6],[3,4,9,10],[7,8]]=>[4,4,2] [4,4,1,1]=>[[1,4,5,6],[2,8,9,10],[3],[7]]=>[4,4,1,1] [4,3,3]=>[[1,2,3,10],[4,5,6],[7,8,9]]=>[4,3,3] [4,3,2,1]=>[[1,3,6,10],[2,5,9],[4,8],[7]]=>[4,3,2,1] [4,3,1,1,1]=>[[1,5,6,10],[2,8,9],[3],[4],[7]]=>[4,3,1,1,1] [4,2,2,2]=>[[1,2,9,10],[3,4],[5,6],[7,8]]=>[4,2,2,2] [4,2,2,1,1]=>[[1,4,9,10],[2,6],[3,8],[5],[7]]=>[4,2,2,1,1] [4,2,1,1,1,1]=>[[1,6,9,10],[2,8],[3],[4],[5],[7]]=>[4,2,1,1,1,1] [4,1,1,1,1,1,1]=>[[1,8,9,10],[2],[3],[4],[5],[6],[7]]=>[4,1,1,1,1,1,1] [3,3,3,1]=>[[1,3,4],[2,6,7],[5,9,10],[8]]=>[3,3,3,1] [3,3,2,2]=>[[1,2,7],[3,4,10],[5,6],[8,9]]=>[3,3,2,2] [3,3,2,1,1]=>[[1,4,7],[2,6,10],[3,9],[5],[8]]=>[3,3,2,1,1] [3,3,1,1,1,1]=>[[1,6,7],[2,9,10],[3],[4],[5],[8]]=>[3,3,1,1,1,1] [3,2,2,2,1]=>[[1,3,10],[2,5],[4,7],[6,9],[8]]=>[3,2,2,2,1] [3,2,2,1,1,1]=>[[1,5,10],[2,7],[3,9],[4],[6],[8]]=>[3,2,2,1,1,1] [3,2,1,1,1,1,1]=>[[1,7,10],[2,9],[3],[4],[5],[6],[8]]=>[3,2,1,1,1,1,1] [3,1,1,1,1,1,1,1]=>[[1,9,10],[2],[3],[4],[5],[6],[7],[8]]=>[3,1,1,1,1,1,1,1] [2,2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8],[9,10]]=>[2,2,2,2,2] [2,2,2,2,1,1]=>[[1,4],[2,6],[3,8],[5,10],[7],[9]]=>[2,2,2,2,1,1] [2,2,2,1,1,1,1]=>[[1,6],[2,8],[3,10],[4],[5],[7],[9]]=>[2,2,2,1,1,1,1] [2,2,1,1,1,1,1,1]=>[[1,8],[2,10],[3],[4],[5],[6],[7],[9]]=>[2,2,1,1,1,1,1,1] [2,1,1,1,1,1,1,1,1]=>[[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]=>[2,1,1,1,1,1,1,1,1] [1,1,1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]=>[1,1,1,1,1,1,1,1,1,1] [6,6]=>[[1,2,3,4,5,6],[7,8,9,10,11,12]]=>[6,6]
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
shape
Description
Sends a tableau to its shape.