Identifier
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Images
[1] => [[1]] => => [1]
[2] => [[1,2]] => 0 => [2]
[1,1] => [[1],[2]] => 1 => [1,1]
[3] => [[1,2,3]] => 00 => [3]
[2,1] => [[1,3],[2]] => 10 => [1,2]
[1,1,1] => [[1],[2],[3]] => 11 => [1,1,1]
[4] => [[1,2,3,4]] => 000 => [4]
[3,1] => [[1,3,4],[2]] => 100 => [1,3]
[2,2] => [[1,2],[3,4]] => 010 => [2,2]
[2,1,1] => [[1,4],[2],[3]] => 110 => [1,1,2]
[1,1,1,1] => [[1],[2],[3],[4]] => 111 => [1,1,1,1]
[5] => [[1,2,3,4,5]] => 0000 => [5]
[4,1] => [[1,3,4,5],[2]] => 1000 => [1,4]
[3,2] => [[1,2,5],[3,4]] => 0100 => [2,3]
[3,1,1] => [[1,4,5],[2],[3]] => 1100 => [1,1,3]
[2,2,1] => [[1,3],[2,5],[4]] => 1010 => [1,2,2]
[2,1,1,1] => [[1,5],[2],[3],[4]] => 1110 => [1,1,1,2]
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => 1111 => [1,1,1,1,1]
[6] => [[1,2,3,4,5,6]] => 00000 => [6]
[5,1] => [[1,3,4,5,6],[2]] => 10000 => [1,5]
[4,2] => [[1,2,5,6],[3,4]] => 01000 => [2,4]
[4,1,1] => [[1,4,5,6],[2],[3]] => 11000 => [1,1,4]
[3,3] => [[1,2,3],[4,5,6]] => 00100 => [3,3]
[3,2,1] => [[1,3,6],[2,5],[4]] => 10100 => [1,2,3]
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => 11100 => [1,1,1,3]
[2,2,2] => [[1,2],[3,4],[5,6]] => 01010 => [2,2,2]
[2,2,1,1] => [[1,4],[2,6],[3],[5]] => 11010 => [1,1,2,2]
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => 11110 => [1,1,1,1,2]
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => 11111 => [1,1,1,1,1,1]
[7] => [[1,2,3,4,5,6,7]] => 000000 => [7]
[6,1] => [[1,3,4,5,6,7],[2]] => 100000 => [1,6]
[5,2] => [[1,2,5,6,7],[3,4]] => 010000 => [2,5]
[5,1,1] => [[1,4,5,6,7],[2],[3]] => 110000 => [1,1,5]
[4,3] => [[1,2,3,7],[4,5,6]] => 001000 => [3,4]
[4,2,1] => [[1,3,6,7],[2,5],[4]] => 101000 => [1,2,4]
[4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => 111000 => [1,1,1,4]
[3,3,1] => [[1,3,4],[2,6,7],[5]] => 100100 => [1,3,3]
[3,2,2] => [[1,2,7],[3,4],[5,6]] => 010100 => [2,2,3]
[3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => 110100 => [1,1,2,3]
[3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => 111100 => [1,1,1,1,3]
[2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => 101010 => [1,2,2,2]
[2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => 111010 => [1,1,1,2,2]
[2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => 111110 => [1,1,1,1,1,2]
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => 111111 => [1,1,1,1,1,1,1]
[8] => [[1,2,3,4,5,6,7,8]] => 0000000 => [8]
[7,1] => [[1,3,4,5,6,7,8],[2]] => 1000000 => [1,7]
[6,2] => [[1,2,5,6,7,8],[3,4]] => 0100000 => [2,6]
[6,1,1] => [[1,4,5,6,7,8],[2],[3]] => 1100000 => [1,1,6]
[5,3] => [[1,2,3,7,8],[4,5,6]] => 0010000 => [3,5]
[5,2,1] => [[1,3,6,7,8],[2,5],[4]] => 1010000 => [1,2,5]
[5,1,1,1] => [[1,5,6,7,8],[2],[3],[4]] => 1110000 => [1,1,1,5]
[4,4] => [[1,2,3,4],[5,6,7,8]] => 0001000 => [4,4]
[4,3,1] => [[1,3,4,8],[2,6,7],[5]] => 1001000 => [1,3,4]
[4,2,2] => [[1,2,7,8],[3,4],[5,6]] => 0101000 => [2,2,4]
[4,2,1,1] => [[1,4,7,8],[2,6],[3],[5]] => 1101000 => [1,1,2,4]
[4,1,1,1,1] => [[1,6,7,8],[2],[3],[4],[5]] => 1111000 => [1,1,1,1,4]
[3,3,2] => [[1,2,5],[3,4,8],[6,7]] => 0100100 => [2,3,3]
[3,3,1,1] => [[1,4,5],[2,7,8],[3],[6]] => 1100100 => [1,1,3,3]
[3,2,2,1] => [[1,3,8],[2,5],[4,7],[6]] => 1010100 => [1,2,2,3]
[3,2,1,1,1] => [[1,5,8],[2,7],[3],[4],[6]] => 1110100 => [1,1,1,2,3]
[3,1,1,1,1,1] => [[1,7,8],[2],[3],[4],[5],[6]] => 1111100 => [1,1,1,1,1,3]
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => 0101010 => [2,2,2,2]
[2,2,2,1,1] => [[1,4],[2,6],[3,8],[5],[7]] => 1101010 => [1,1,2,2,2]
[2,2,1,1,1,1] => [[1,6],[2,8],[3],[4],[5],[7]] => 1111010 => [1,1,1,1,2,2]
[2,1,1,1,1,1,1] => [[1,8],[2],[3],[4],[5],[6],[7]] => 1111110 => [1,1,1,1,1,1,2]
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => 1111111 => [1,1,1,1,1,1,1,1]
[9] => [[1,2,3,4,5,6,7,8,9]] => 00000000 => [9]
[8,1] => [[1,3,4,5,6,7,8,9],[2]] => 10000000 => [1,8]
[7,2] => [[1,2,5,6,7,8,9],[3,4]] => 01000000 => [2,7]
[7,1,1] => [[1,4,5,6,7,8,9],[2],[3]] => 11000000 => [1,1,7]
[6,3] => [[1,2,3,7,8,9],[4,5,6]] => 00100000 => [3,6]
[6,2,1] => [[1,3,6,7,8,9],[2,5],[4]] => 10100000 => [1,2,6]
[6,1,1,1] => [[1,5,6,7,8,9],[2],[3],[4]] => 11100000 => [1,1,1,6]
[5,4] => [[1,2,3,4,9],[5,6,7,8]] => 00010000 => [4,5]
[5,3,1] => [[1,3,4,8,9],[2,6,7],[5]] => 10010000 => [1,3,5]
[5,2,2] => [[1,2,7,8,9],[3,4],[5,6]] => 01010000 => [2,2,5]
[5,2,1,1] => [[1,4,7,8,9],[2,6],[3],[5]] => 11010000 => [1,1,2,5]
[5,1,1,1,1] => [[1,6,7,8,9],[2],[3],[4],[5]] => 11110000 => [1,1,1,1,5]
[4,4,1] => [[1,3,4,5],[2,7,8,9],[6]] => 10001000 => [1,4,4]
[4,3,2] => [[1,2,5,9],[3,4,8],[6,7]] => 01001000 => [2,3,4]
[4,3,1,1] => [[1,4,5,9],[2,7,8],[3],[6]] => 11001000 => [1,1,3,4]
[4,2,2,1] => [[1,3,8,9],[2,5],[4,7],[6]] => 10101000 => [1,2,2,4]
[4,2,1,1,1] => [[1,5,8,9],[2,7],[3],[4],[6]] => 11101000 => [1,1,1,2,4]
[4,1,1,1,1,1] => [[1,7,8,9],[2],[3],[4],[5],[6]] => 11111000 => [1,1,1,1,1,4]
[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => 00100100 => [3,3,3]
[3,3,2,1] => [[1,3,6],[2,5,9],[4,8],[7]] => 10100100 => [1,2,3,3]
[3,3,1,1,1] => [[1,5,6],[2,8,9],[3],[4],[7]] => 11100100 => [1,1,1,3,3]
[3,2,2,2] => [[1,2,9],[3,4],[5,6],[7,8]] => 01010100 => [2,2,2,3]
[3,2,2,1,1] => [[1,4,9],[2,6],[3,8],[5],[7]] => 11010100 => [1,1,2,2,3]
[3,2,1,1,1,1] => [[1,6,9],[2,8],[3],[4],[5],[7]] => 11110100 => [1,1,1,1,2,3]
[3,1,1,1,1,1,1] => [[1,8,9],[2],[3],[4],[5],[6],[7]] => 11111100 => [1,1,1,1,1,1,3]
[2,2,2,2,1] => [[1,3],[2,5],[4,7],[6,9],[8]] => 10101010 => [1,2,2,2,2]
[2,2,2,1,1,1] => [[1,5],[2,7],[3,9],[4],[6],[8]] => 11101010 => [1,1,1,2,2,2]
[2,2,1,1,1,1,1] => [[1,7],[2,9],[3],[4],[5],[6],[8]] => 11111010 => [1,1,1,1,1,2,2]
[2,1,1,1,1,1,1,1] => [[1,9],[2],[3],[4],[5],[6],[7],[8]] => 11111110 => [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]] => 11111111 => [1,1,1,1,1,1,1,1,1]
[10] => [[1,2,3,4,5,6,7,8,9,10]] => 000000000 => [10]
[9,1] => [[1,3,4,5,6,7,8,9,10],[2]] => 100000000 => [1,9]
[8,2] => [[1,2,5,6,7,8,9,10],[3,4]] => 010000000 => [2,8]
[8,1,1] => [[1,4,5,6,7,8,9,10],[2],[3]] => 110000000 => [1,1,8]
[7,3] => [[1,2,3,7,8,9,10],[4,5,6]] => 001000000 => [3,7]
>>> Load all 161 entries. <<<
[7,2,1] => [[1,3,6,7,8,9,10],[2,5],[4]] => 101000000 => [1,2,7]
[7,1,1,1] => [[1,5,6,7,8,9,10],[2],[3],[4]] => 111000000 => [1,1,1,7]
[6,4] => [[1,2,3,4,9,10],[5,6,7,8]] => 000100000 => [4,6]
[6,3,1] => [[1,3,4,8,9,10],[2,6,7],[5]] => 100100000 => [1,3,6]
[6,2,2] => [[1,2,7,8,9,10],[3,4],[5,6]] => 010100000 => [2,2,6]
[6,2,1,1] => [[1,4,7,8,9,10],[2,6],[3],[5]] => 110100000 => [1,1,2,6]
[6,1,1,1,1] => [[1,6,7,8,9,10],[2],[3],[4],[5]] => 111100000 => [1,1,1,1,6]
[5,5] => [[1,2,3,4,5],[6,7,8,9,10]] => 000010000 => [5,5]
[5,4,1] => [[1,3,4,5,10],[2,7,8,9],[6]] => 100010000 => [1,4,5]
[5,3,2] => [[1,2,5,9,10],[3,4,8],[6,7]] => 010010000 => [2,3,5]
[5,3,1,1] => [[1,4,5,9,10],[2,7,8],[3],[6]] => 110010000 => [1,1,3,5]
[5,2,2,1] => [[1,3,8,9,10],[2,5],[4,7],[6]] => 101010000 => [1,2,2,5]
[5,2,1,1,1] => [[1,5,8,9,10],[2,7],[3],[4],[6]] => 111010000 => [1,1,1,2,5]
[5,1,1,1,1,1] => [[1,7,8,9,10],[2],[3],[4],[5],[6]] => 111110000 => [1,1,1,1,1,5]
[4,4,2] => [[1,2,5,6],[3,4,9,10],[7,8]] => 010001000 => [2,4,4]
[4,4,1,1] => [[1,4,5,6],[2,8,9,10],[3],[7]] => 110001000 => [1,1,4,4]
[4,3,3] => [[1,2,3,10],[4,5,6],[7,8,9]] => 001001000 => [3,3,4]
[4,3,2,1] => [[1,3,6,10],[2,5,9],[4,8],[7]] => 101001000 => [1,2,3,4]
[4,3,1,1,1] => [[1,5,6,10],[2,8,9],[3],[4],[7]] => 111001000 => [1,1,1,3,4]
[4,2,2,2] => [[1,2,9,10],[3,4],[5,6],[7,8]] => 010101000 => [2,2,2,4]
[4,2,2,1,1] => [[1,4,9,10],[2,6],[3,8],[5],[7]] => 110101000 => [1,1,2,2,4]
[4,2,1,1,1,1] => [[1,6,9,10],[2,8],[3],[4],[5],[7]] => 111101000 => [1,1,1,1,2,4]
[4,1,1,1,1,1,1] => [[1,8,9,10],[2],[3],[4],[5],[6],[7]] => 111111000 => [1,1,1,1,1,1,4]
[3,3,3,1] => [[1,3,4],[2,6,7],[5,9,10],[8]] => 100100100 => [1,3,3,3]
[3,3,2,2] => [[1,2,7],[3,4,10],[5,6],[8,9]] => 010100100 => [2,2,3,3]
[3,3,2,1,1] => [[1,4,7],[2,6,10],[3,9],[5],[8]] => 110100100 => [1,1,2,3,3]
[3,3,1,1,1,1] => [[1,6,7],[2,9,10],[3],[4],[5],[8]] => 111100100 => [1,1,1,1,3,3]
[3,2,2,2,1] => [[1,3,10],[2,5],[4,7],[6,9],[8]] => 101010100 => [1,2,2,2,3]
[3,2,2,1,1,1] => [[1,5,10],[2,7],[3,9],[4],[6],[8]] => 111010100 => [1,1,1,2,2,3]
[3,2,1,1,1,1,1] => [[1,7,10],[2,9],[3],[4],[5],[6],[8]] => 111110100 => [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]] => 111111100 => [1,1,1,1,1,1,1,3]
[2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => 010101010 => [2,2,2,2,2]
[2,2,2,2,1,1] => [[1,4],[2,6],[3,8],[5,10],[7],[9]] => 110101010 => [1,1,2,2,2,2]
[2,2,2,1,1,1,1] => [[1,6],[2,8],[3,10],[4],[5],[7],[9]] => 111101010 => [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]] => 111111010 => [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]] => 111111110 => [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]] => 111111111 => [1,1,1,1,1,1,1,1,1,1]
[5,4,2] => [[1,2,5,6,11],[3,4,9,10],[7,8]] => 0100010000 => [2,4,5]
[5,3,3] => [[1,2,3,10,11],[4,5,6],[7,8,9]] => 0010010000 => [3,3,5]
[5,3,1,1,1] => [[1,5,6,10,11],[2,8,9],[3],[4],[7]] => 1110010000 => [1,1,1,3,5]
[5,2,2,2] => [[1,2,9,10,11],[3,4],[5,6],[7,8]] => 0101010000 => [2,2,2,5]
[4,3,2,2] => [[1,2,7,11],[3,4,10],[5,6],[8,9]] => 0101001000 => [2,2,3,4]
[4,3,2,1,1] => [[1,4,7,11],[2,6,10],[3,9],[5],[8]] => 1101001000 => [1,1,2,3,4]
[4,2,2,2,1] => [[1,3,10,11],[2,5],[4,7],[6,9],[8]] => 1010101000 => [1,2,2,2,4]
[3,3,3,1,1] => [[1,4,5],[2,7,8],[3,10,11],[6],[9]] => 1100100100 => [1,1,3,3,3]
[3,3,2,2,1] => [[1,3,8],[2,5,11],[4,7],[6,10],[9]] => 1010100100 => [1,2,2,3,3]
[12] => [[1,2,3,4,5,6,7,8,9,10,11,12]] => 00000000000 => [12]
[6,6] => [[1,2,3,4,5,6],[7,8,9,10,11,12]] => 00000100000 => [6,6]
[6,4,2] => [[1,2,5,6,11,12],[3,4,9,10],[7,8]] => 01000100000 => [2,4,6]
[5,4,3] => [[1,2,3,7,12],[4,5,6,11],[8,9,10]] => 00100010000 => [3,4,5]
[4,4,3,1] => [[1,3,4,8],[2,6,7,12],[5,10,11],[9]] => 10010001000 => [1,3,4,4]
[4,4,2,2] => [[1,2,7,8],[3,4,11,12],[5,6],[9,10]] => 01010001000 => [2,2,4,4]
[4,4,2,1,1] => [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]] => 11010001000 => [1,1,2,4,4]
[4,3,3,2] => [[1,2,5,12],[3,4,8],[6,7,11],[9,10]] => 01001001000 => [2,3,3,4]
[3,3,3,2,1] => [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]] => 10100100100 => [1,2,3,3,3]
[3,3,2,2,1,1] => [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]] => 11010100100 => [1,1,2,2,3,3]
[2,2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]] => 01010101010 => [2,2,2,2,2,2]
[1,1,1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]] => 11111111111 => [1,1,1,1,1,1,1,1,1,1,1,1]
[5,3,3,1,1] => [[1,4,5,12,13],[2,7,8],[3,10,11],[6],[9]] => 110010010000 => [1,1,3,3,5]
[] => [] => => [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
descent word
Description
The descent word of a standard Young tableau.
For a standard Young tableau of size $n$ we set $w_i=1$ if $i+1$ is in a lower row than $i$, and $0$ otherwise, for $1\leq i < n$.
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.