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Identifier
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Images
[1] => [[1]] => [1] => [1]
[2] => [[1,2]] => [2] => [2]
[1,1] => [[1],[2]] => [1,1] => [1,1]
[3] => [[1,2,3]] => [3] => [3]
[2,1] => [[1,3],[2]] => [1,2] => [1,2]
[1,1,1] => [[1],[2],[3]] => [1,1,1] => [1,1,1]
[4] => [[1,2,3,4]] => [4] => [4]
[3,1] => [[1,3,4],[2]] => [1,3] => [1,3]
[2,2] => [[1,2],[3,4]] => [2,2] => [2,2]
[2,1,1] => [[1,4],[2],[3]] => [1,1,2] => [1,1,2]
[1,1,1,1] => [[1],[2],[3],[4]] => [1,1,1,1] => [1,1,1,1]
[5] => [[1,2,3,4,5]] => [5] => [5]
[4,1] => [[1,3,4,5],[2]] => [1,4] => [1,4]
[3,2] => [[1,2,5],[3,4]] => [2,3] => [2,3]
[3,1,1] => [[1,4,5],[2],[3]] => [1,1,3] => [1,1,3]
[2,2,1] => [[1,3],[2,5],[4]] => [1,2,2] => [1,2,2]
[2,1,1,1] => [[1,5],[2],[3],[4]] => [1,1,1,2] => [1,1,1,2]
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [1,1,1,1,1] => [1,1,1,1,1]
[6] => [[1,2,3,4,5,6]] => [6] => [6]
[5,1] => [[1,3,4,5,6],[2]] => [1,5] => [1,5]
[4,2] => [[1,2,5,6],[3,4]] => [2,4] => [2,4]
[4,1,1] => [[1,4,5,6],[2],[3]] => [1,1,4] => [1,1,4]
[3,3] => [[1,2,3],[4,5,6]] => [3,3] => [3,3]
[3,2,1] => [[1,3,6],[2,5],[4]] => [1,2,3] => [1,2,3]
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => [1,1,1,3] => [1,1,1,3]
[2,2,2] => [[1,2],[3,4],[5,6]] => [2,2,2] => [2,2,2]
[2,2,1,1] => [[1,4],[2,6],[3],[5]] => [1,1,2,2] => [1,1,2,2]
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [1,1,1,1,2] => [1,1,1,1,2]
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [1,1,1,1,1,1] => [1,1,1,1,1,1]
[7] => [[1,2,3,4,5,6,7]] => [7] => [7]
[6,1] => [[1,3,4,5,6,7],[2]] => [1,6] => [1,6]
[5,2] => [[1,2,5,6,7],[3,4]] => [2,5] => [2,5]
[5,1,1] => [[1,4,5,6,7],[2],[3]] => [1,1,5] => [1,1,5]
[4,3] => [[1,2,3,7],[4,5,6]] => [3,4] => [3,4]
[4,2,1] => [[1,3,6,7],[2,5],[4]] => [1,2,4] => [1,2,4]
[4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => [1,1,1,4] => [1,1,1,4]
[3,3,1] => [[1,3,4],[2,6,7],[5]] => [1,3,3] => [1,3,3]
[3,2,2] => [[1,2,7],[3,4],[5,6]] => [2,2,3] => [2,2,3]
[3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => [1,1,2,3] => [1,1,2,3]
[3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => [1,1,1,1,3] => [1,1,1,1,3]
[2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => [1,2,2,2] => [1,2,2,2]
[2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => [1,1,1,2,2] => [1,1,1,2,2]
[2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => [1,1,1,1,1,2] => [1,1,1,1,1,2]
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
[8] => [[1,2,3,4,5,6,7,8]] => [8] => [8]
[7,1] => [[1,3,4,5,6,7,8],[2]] => [1,7] => [1,7]
[6,2] => [[1,2,5,6,7,8],[3,4]] => [2,6] => [2,6]
[6,1,1] => [[1,4,5,6,7,8],[2],[3]] => [1,1,6] => [1,1,6]
[5,3] => [[1,2,3,7,8],[4,5,6]] => [3,5] => [3,5]
[5,2,1] => [[1,3,6,7,8],[2,5],[4]] => [1,2,5] => [1,2,5]
[5,1,1,1] => [[1,5,6,7,8],[2],[3],[4]] => [1,1,1,5] => [1,1,1,5]
[4,4] => [[1,2,3,4],[5,6,7,8]] => [4,4] => [4,4]
[4,3,1] => [[1,3,4,8],[2,6,7],[5]] => [1,3,4] => [1,3,4]
[4,2,2] => [[1,2,7,8],[3,4],[5,6]] => [2,2,4] => [2,2,4]
[4,2,1,1] => [[1,4,7,8],[2,6],[3],[5]] => [1,1,2,4] => [1,1,2,4]
[4,1,1,1,1] => [[1,6,7,8],[2],[3],[4],[5]] => [1,1,1,1,4] => [1,1,1,1,4]
[3,3,2] => [[1,2,5],[3,4,8],[6,7]] => [2,3,3] => [2,3,3]
[3,3,1,1] => [[1,4,5],[2,7,8],[3],[6]] => [1,1,3,3] => [1,1,3,3]
[3,2,2,1] => [[1,3,8],[2,5],[4,7],[6]] => [1,2,2,3] => [1,2,2,3]
[3,2,1,1,1] => [[1,5,8],[2,7],[3],[4],[6]] => [1,1,1,2,3] => [1,1,1,2,3]
[3,1,1,1,1,1] => [[1,7,8],[2],[3],[4],[5],[6]] => [1,1,1,1,1,3] => [1,1,1,1,1,3]
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [2,2,2,2] => [2,2,2,2]
[2,2,2,1,1] => [[1,4],[2,6],[3,8],[5],[7]] => [1,1,2,2,2] => [1,1,2,2,2]
[2,2,1,1,1,1] => [[1,6],[2,8],[3],[4],[5],[7]] => [1,1,1,1,2,2] => [1,1,1,1,2,2]
[2,1,1,1,1,1,1] => [[1,8],[2],[3],[4],[5],[6],[7]] => [1,1,1,1,1,1,2] => [1,1,1,1,1,1,2]
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
[9] => [[1,2,3,4,5,6,7,8,9]] => [9] => [9]
[8,1] => [[1,3,4,5,6,7,8,9],[2]] => [1,8] => [1,8]
[7,2] => [[1,2,5,6,7,8,9],[3,4]] => [2,7] => [2,7]
[7,1,1] => [[1,4,5,6,7,8,9],[2],[3]] => [1,1,7] => [1,1,7]
[6,3] => [[1,2,3,7,8,9],[4,5,6]] => [3,6] => [3,6]
[6,2,1] => [[1,3,6,7,8,9],[2,5],[4]] => [1,2,6] => [1,2,6]
[6,1,1,1] => [[1,5,6,7,8,9],[2],[3],[4]] => [1,1,1,6] => [1,1,1,6]
[5,4] => [[1,2,3,4,9],[5,6,7,8]] => [4,5] => [4,5]
[5,3,1] => [[1,3,4,8,9],[2,6,7],[5]] => [1,3,5] => [1,3,5]
[5,2,2] => [[1,2,7,8,9],[3,4],[5,6]] => [2,2,5] => [2,2,5]
[5,2,1,1] => [[1,4,7,8,9],[2,6],[3],[5]] => [1,1,2,5] => [1,1,2,5]
[5,1,1,1,1] => [[1,6,7,8,9],[2],[3],[4],[5]] => [1,1,1,1,5] => [1,1,1,1,5]
[4,4,1] => [[1,3,4,5],[2,7,8,9],[6]] => [1,4,4] => [1,4,4]
[4,3,2] => [[1,2,5,9],[3,4,8],[6,7]] => [2,3,4] => [2,3,4]
[4,3,1,1] => [[1,4,5,9],[2,7,8],[3],[6]] => [1,1,3,4] => [1,1,3,4]
[4,2,2,1] => [[1,3,8,9],[2,5],[4,7],[6]] => [1,2,2,4] => [1,2,2,4]
[4,2,1,1,1] => [[1,5,8,9],[2,7],[3],[4],[6]] => [1,1,1,2,4] => [1,1,1,2,4]
[4,1,1,1,1,1] => [[1,7,8,9],[2],[3],[4],[5],[6]] => [1,1,1,1,1,4] => [1,1,1,1,1,4]
[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => [3,3,3] => [3,3,3]
[3,3,2,1] => [[1,3,6],[2,5,9],[4,8],[7]] => [1,2,3,3] => [1,2,3,3]
[3,3,1,1,1] => [[1,5,6],[2,8,9],[3],[4],[7]] => [1,1,1,3,3] => [1,1,1,3,3]
[3,2,2,2] => [[1,2,9],[3,4],[5,6],[7,8]] => [2,2,2,3] => [2,2,2,3]
[3,2,2,1,1] => [[1,4,9],[2,6],[3,8],[5],[7]] => [1,1,2,2,3] => [1,1,2,2,3]
[3,2,1,1,1,1] => [[1,6,9],[2,8],[3],[4],[5],[7]] => [1,1,1,1,2,3] => [1,1,1,1,2,3]
[3,1,1,1,1,1,1] => [[1,8,9],[2],[3],[4],[5],[6],[7]] => [1,1,1,1,1,1,3] => [1,1,1,1,1,1,3]
[2,2,2,2,1] => [[1,3],[2,5],[4,7],[6,9],[8]] => [1,2,2,2,2] => [1,2,2,2,2]
[2,2,2,1,1,1] => [[1,5],[2,7],[3,9],[4],[6],[8]] => [1,1,1,2,2,2] => [1,1,1,2,2,2]
[2,2,1,1,1,1,1] => [[1,7],[2,9],[3],[4],[5],[6],[8]] => [1,1,1,1,1,2,2] => [1,1,1,1,1,2,2]
[2,1,1,1,1,1,1,1] => [[1,9],[2],[3],[4],[5],[6],[7],[8]] => [1,1,1,1,1,1,1,2] => [1,1,1,1,1,1,1,2]
[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] => [1,1,1,1,1,1,1,1,1]
[10] => [[1,2,3,4,5,6,7,8,9,10]] => [10] => [10]
[9,1] => [[1,3,4,5,6,7,8,9,10],[2]] => [1,9] => [1,9]
[8,2] => [[1,2,5,6,7,8,9,10],[3,4]] => [2,8] => [2,8]
[8,1,1] => [[1,4,5,6,7,8,9,10],[2],[3]] => [1,1,8] => [1,1,8]
[7,3] => [[1,2,3,7,8,9,10],[4,5,6]] => [3,7] => [3,7]
>>> Load all 138 entries. <<<
[7,2,1] => [[1,3,6,7,8,9,10],[2,5],[4]] => [1,2,7] => [1,2,7]
[7,1,1,1] => [[1,5,6,7,8,9,10],[2],[3],[4]] => [1,1,1,7] => [1,1,1,7]
[6,4] => [[1,2,3,4,9,10],[5,6,7,8]] => [4,6] => [4,6]
[6,3,1] => [[1,3,4,8,9,10],[2,6,7],[5]] => [1,3,6] => [1,3,6]
[6,2,2] => [[1,2,7,8,9,10],[3,4],[5,6]] => [2,2,6] => [2,2,6]
[6,2,1,1] => [[1,4,7,8,9,10],[2,6],[3],[5]] => [1,1,2,6] => [1,1,2,6]
[6,1,1,1,1] => [[1,6,7,8,9,10],[2],[3],[4],[5]] => [1,1,1,1,6] => [1,1,1,1,6]
[5,5] => [[1,2,3,4,5],[6,7,8,9,10]] => [5,5] => [5,5]
[5,4,1] => [[1,3,4,5,10],[2,7,8,9],[6]] => [1,4,5] => [1,4,5]
[5,3,2] => [[1,2,5,9,10],[3,4,8],[6,7]] => [2,3,5] => [2,3,5]
[5,3,1,1] => [[1,4,5,9,10],[2,7,8],[3],[6]] => [1,1,3,5] => [1,1,3,5]
[5,2,2,1] => [[1,3,8,9,10],[2,5],[4,7],[6]] => [1,2,2,5] => [1,2,2,5]
[5,2,1,1,1] => [[1,5,8,9,10],[2,7],[3],[4],[6]] => [1,1,1,2,5] => [1,1,1,2,5]
[5,1,1,1,1,1] => [[1,7,8,9,10],[2],[3],[4],[5],[6]] => [1,1,1,1,1,5] => [1,1,1,1,1,5]
[4,4,2] => [[1,2,5,6],[3,4,9,10],[7,8]] => [2,4,4] => [2,4,4]
[4,4,1,1] => [[1,4,5,6],[2,8,9,10],[3],[7]] => [1,1,4,4] => [1,1,4,4]
[4,3,3] => [[1,2,3,10],[4,5,6],[7,8,9]] => [3,3,4] => [3,3,4]
[4,3,2,1] => [[1,3,6,10],[2,5,9],[4,8],[7]] => [1,2,3,4] => [1,2,3,4]
[4,3,1,1,1] => [[1,5,6,10],[2,8,9],[3],[4],[7]] => [1,1,1,3,4] => [1,1,1,3,4]
[4,2,2,2] => [[1,2,9,10],[3,4],[5,6],[7,8]] => [2,2,2,4] => [2,2,2,4]
[4,2,2,1,1] => [[1,4,9,10],[2,6],[3,8],[5],[7]] => [1,1,2,2,4] => [1,1,2,2,4]
[4,2,1,1,1,1] => [[1,6,9,10],[2,8],[3],[4],[5],[7]] => [1,1,1,1,2,4] => [1,1,1,1,2,4]
[4,1,1,1,1,1,1] => [[1,8,9,10],[2],[3],[4],[5],[6],[7]] => [1,1,1,1,1,1,4] => [1,1,1,1,1,1,4]
[3,3,3,1] => [[1,3,4],[2,6,7],[5,9,10],[8]] => [1,3,3,3] => [1,3,3,3]
[3,3,2,2] => [[1,2,7],[3,4,10],[5,6],[8,9]] => [2,2,3,3] => [2,2,3,3]
[3,3,2,1,1] => [[1,4,7],[2,6,10],[3,9],[5],[8]] => [1,1,2,3,3] => [1,1,2,3,3]
[3,3,1,1,1,1] => [[1,6,7],[2,9,10],[3],[4],[5],[8]] => [1,1,1,1,3,3] => [1,1,1,1,3,3]
[3,2,2,2,1] => [[1,3,10],[2,5],[4,7],[6,9],[8]] => [1,2,2,2,3] => [1,2,2,2,3]
[3,2,2,1,1,1] => [[1,5,10],[2,7],[3,9],[4],[6],[8]] => [1,1,1,2,2,3] => [1,1,1,2,2,3]
[3,2,1,1,1,1,1] => [[1,7,10],[2,9],[3],[4],[5],[6],[8]] => [1,1,1,1,1,2,3] => [1,1,1,1,1,2,3]
[3,1,1,1,1,1,1,1] => [[1,9,10],[2],[3],[4],[5],[6],[7],[8]] => [1,1,1,1,1,1,1,3] => [1,1,1,1,1,1,1,3]
[2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => [2,2,2,2,2] => [2,2,2,2,2]
[2,2,2,2,1,1] => [[1,4],[2,6],[3,8],[5,10],[7],[9]] => [1,1,2,2,2,2] => [1,1,2,2,2,2]
[2,2,2,1,1,1,1] => [[1,6],[2,8],[3,10],[4],[5],[7],[9]] => [1,1,1,1,2,2,2] => [1,1,1,1,2,2,2]
[2,2,1,1,1,1,1,1] => [[1,8],[2,10],[3],[4],[5],[6],[7],[9]] => [1,1,1,1,1,1,2,2] => [1,1,1,1,1,1,2,2]
[2,1,1,1,1,1,1,1,1] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]] => [1,1,1,1,1,1,1,1,2] => [1,1,1,1,1,1,1,1,2]
[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] => [1,1,1,1,1,1,1,1,1,1]
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
horizontal strip sizes
Description
The composition of horizontal strip sizes.
We associate to a standard Young tableau T the composition (c1,…,ck), such that k is minimal and the numbers c1+⋯+ci+1,…,c1+⋯+ci+1 form a horizontal strip in T for all i.
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00314Foata bijection.