Processing math: 100%

Identifier
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
Images
[1] => [[1]] => [1] =>
[2] => [[1,2]] => [1,2] => 0
[1,1] => [[1],[2]] => [2,1] => 1
[3] => [[1,2,3]] => [1,2,3] => 00
[2,1] => [[1,3],[2]] => [2,1,3] => 10
[1,1,1] => [[1],[2],[3]] => [3,2,1] => 11
[4] => [[1,2,3,4]] => [1,2,3,4] => 000
[3,1] => [[1,3,4],[2]] => [2,1,3,4] => 100
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => 010
[2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 110
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 111
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0000
[4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 1000
[3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 0100
[3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 1100
[2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1010
[2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 1110
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 1111
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => 00000
[5,1] => [[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => 10000
[4,2] => [[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => 01000
[4,1,1] => [[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => 11000
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => 00100
[3,2,1] => [[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => 10100
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => 11100
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => 01010
[2,2,1,1] => [[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => 11010
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => 11110
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => 11111
[7] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => 000000
[6,1] => [[1,3,4,5,6,7],[2]] => [2,1,3,4,5,6,7] => 100000
[5,2] => [[1,2,5,6,7],[3,4]] => [3,4,1,2,5,6,7] => 010000
[5,1,1] => [[1,4,5,6,7],[2],[3]] => [3,2,1,4,5,6,7] => 110000
[4,3] => [[1,2,3,7],[4,5,6]] => [4,5,6,1,2,3,7] => 001000
[4,2,1] => [[1,3,6,7],[2,5],[4]] => [4,2,5,1,3,6,7] => 101000
[4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => [4,3,2,1,5,6,7] => 111000
[3,3,1] => [[1,3,4],[2,6,7],[5]] => [5,2,6,7,1,3,4] => 100100
[3,2,2] => [[1,2,7],[3,4],[5,6]] => [5,6,3,4,1,2,7] => 010100
[3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => [5,3,2,6,1,4,7] => 110100
[3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => [5,4,3,2,1,6,7] => 111100
[2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => [6,4,7,2,5,1,3] => 101010
[2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => [6,4,3,2,7,1,5] => 111010
[2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7] => 111110
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => 111111
[8] => [[1,2,3,4,5,6,7,8]] => [1,2,3,4,5,6,7,8] => 0000000
[7,1] => [[1,3,4,5,6,7,8],[2]] => [2,1,3,4,5,6,7,8] => 1000000
[6,2] => [[1,2,5,6,7,8],[3,4]] => [3,4,1,2,5,6,7,8] => 0100000
[6,1,1] => [[1,4,5,6,7,8],[2],[3]] => [3,2,1,4,5,6,7,8] => 1100000
[5,3] => [[1,2,3,7,8],[4,5,6]] => [4,5,6,1,2,3,7,8] => 0010000
[5,2,1] => [[1,3,6,7,8],[2,5],[4]] => [4,2,5,1,3,6,7,8] => 1010000
[5,1,1,1] => [[1,5,6,7,8],[2],[3],[4]] => [4,3,2,1,5,6,7,8] => 1110000
[4,4] => [[1,2,3,4],[5,6,7,8]] => [5,6,7,8,1,2,3,4] => 0001000
[4,3,1] => [[1,3,4,8],[2,6,7],[5]] => [5,2,6,7,1,3,4,8] => 1001000
[4,2,2] => [[1,2,7,8],[3,4],[5,6]] => [5,6,3,4,1,2,7,8] => 0101000
[4,2,1,1] => [[1,4,7,8],[2,6],[3],[5]] => [5,3,2,6,1,4,7,8] => 1101000
[4,1,1,1,1] => [[1,6,7,8],[2],[3],[4],[5]] => [5,4,3,2,1,6,7,8] => 1111000
[3,3,2] => [[1,2,5],[3,4,8],[6,7]] => [6,7,3,4,8,1,2,5] => 0100100
[3,3,1,1] => [[1,4,5],[2,7,8],[3],[6]] => [6,3,2,7,8,1,4,5] => 1100100
[3,2,2,1] => [[1,3,8],[2,5],[4,7],[6]] => [6,4,7,2,5,1,3,8] => 1010100
[3,2,1,1,1] => [[1,5,8],[2,7],[3],[4],[6]] => [6,4,3,2,7,1,5,8] => 1110100
[3,1,1,1,1,1] => [[1,7,8],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7,8] => 1111100
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [7,8,5,6,3,4,1,2] => 0101010
[2,2,2,1,1] => [[1,4],[2,6],[3,8],[5],[7]] => [7,5,3,8,2,6,1,4] => 1101010
[2,2,1,1,1,1] => [[1,6],[2,8],[3],[4],[5],[7]] => [7,5,4,3,2,8,1,6] => 1111010
[2,1,1,1,1,1,1] => [[1,8],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1,8] => 1111110
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1] => 1111111
[9] => [[1,2,3,4,5,6,7,8,9]] => [1,2,3,4,5,6,7,8,9] => 00000000
[8,1] => [[1,3,4,5,6,7,8,9],[2]] => [2,1,3,4,5,6,7,8,9] => 10000000
[7,2] => [[1,2,5,6,7,8,9],[3,4]] => [3,4,1,2,5,6,7,8,9] => 01000000
[7,1,1] => [[1,4,5,6,7,8,9],[2],[3]] => [3,2,1,4,5,6,7,8,9] => 11000000
[6,3] => [[1,2,3,7,8,9],[4,5,6]] => [4,5,6,1,2,3,7,8,9] => 00100000
[6,2,1] => [[1,3,6,7,8,9],[2,5],[4]] => [4,2,5,1,3,6,7,8,9] => 10100000
[6,1,1,1] => [[1,5,6,7,8,9],[2],[3],[4]] => [4,3,2,1,5,6,7,8,9] => 11100000
[5,4] => [[1,2,3,4,9],[5,6,7,8]] => [5,6,7,8,1,2,3,4,9] => 00010000
[5,3,1] => [[1,3,4,8,9],[2,6,7],[5]] => [5,2,6,7,1,3,4,8,9] => 10010000
[5,2,2] => [[1,2,7,8,9],[3,4],[5,6]] => [5,6,3,4,1,2,7,8,9] => 01010000
[5,2,1,1] => [[1,4,7,8,9],[2,6],[3],[5]] => [5,3,2,6,1,4,7,8,9] => 11010000
[5,1,1,1,1] => [[1,6,7,8,9],[2],[3],[4],[5]] => [5,4,3,2,1,6,7,8,9] => 11110000
[4,4,1] => [[1,3,4,5],[2,7,8,9],[6]] => [6,2,7,8,9,1,3,4,5] => 10001000
[4,3,2] => [[1,2,5,9],[3,4,8],[6,7]] => [6,7,3,4,8,1,2,5,9] => 01001000
[4,3,1,1] => [[1,4,5,9],[2,7,8],[3],[6]] => [6,3,2,7,8,1,4,5,9] => 11001000
[4,2,2,1] => [[1,3,8,9],[2,5],[4,7],[6]] => [6,4,7,2,5,1,3,8,9] => 10101000
[4,2,1,1,1] => [[1,5,8,9],[2,7],[3],[4],[6]] => [6,4,3,2,7,1,5,8,9] => 11101000
[4,1,1,1,1,1] => [[1,7,8,9],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7,8,9] => 11111000
[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => [7,8,9,4,5,6,1,2,3] => 00100100
[3,3,2,1] => [[1,3,6],[2,5,9],[4,8],[7]] => [7,4,8,2,5,9,1,3,6] => 10100100
[3,3,1,1,1] => [[1,5,6],[2,8,9],[3],[4],[7]] => [7,4,3,2,8,9,1,5,6] => 11100100
[3,2,2,2] => [[1,2,9],[3,4],[5,6],[7,8]] => [7,8,5,6,3,4,1,2,9] => 01010100
[3,2,2,1,1] => [[1,4,9],[2,6],[3,8],[5],[7]] => [7,5,3,8,2,6,1,4,9] => 11010100
[3,2,1,1,1,1] => [[1,6,9],[2,8],[3],[4],[5],[7]] => [7,5,4,3,2,8,1,6,9] => 11110100
[3,1,1,1,1,1,1] => [[1,8,9],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1,8,9] => 11111100
[2,2,2,2,1] => [[1,3],[2,5],[4,7],[6,9],[8]] => [8,6,9,4,7,2,5,1,3] => 10101010
[2,2,2,1,1,1] => [[1,5],[2,7],[3,9],[4],[6],[8]] => [8,6,4,3,9,2,7,1,5] => 11101010
[2,2,1,1,1,1,1] => [[1,7],[2,9],[3],[4],[5],[6],[8]] => [8,6,5,4,3,2,9,1,7] => 11111010
[2,1,1,1,1,1,1,1] => [[1,9],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1,9] => 11111110
[1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,3,2,1] => 11111111
[10] => [[1,2,3,4,5,6,7,8,9,10]] => [1,2,3,4,5,6,7,8,9,10] => 000000000
[9,1] => [[1,3,4,5,6,7,8,9,10],[2]] => [2,1,3,4,5,6,7,8,9,10] => 100000000
[8,2] => [[1,2,5,6,7,8,9,10],[3,4]] => [3,4,1,2,5,6,7,8,9,10] => 010000000
[8,1,1] => [[1,4,5,6,7,8,9,10],[2],[3]] => [3,2,1,4,5,6,7,8,9,10] => 110000000
[7,3] => [[1,2,3,7,8,9,10],[4,5,6]] => [4,5,6,1,2,3,7,8,9,10] => 001000000
>>> Load all 142 entries. <<<
[7,2,1] => [[1,3,6,7,8,9,10],[2,5],[4]] => [4,2,5,1,3,6,7,8,9,10] => 101000000
[7,1,1,1] => [[1,5,6,7,8,9,10],[2],[3],[4]] => [4,3,2,1,5,6,7,8,9,10] => 111000000
[6,4] => [[1,2,3,4,9,10],[5,6,7,8]] => [5,6,7,8,1,2,3,4,9,10] => 000100000
[6,3,1] => [[1,3,4,8,9,10],[2,6,7],[5]] => [5,2,6,7,1,3,4,8,9,10] => 100100000
[6,2,2] => [[1,2,7,8,9,10],[3,4],[5,6]] => [5,6,3,4,1,2,7,8,9,10] => 010100000
[6,2,1,1] => [[1,4,7,8,9,10],[2,6],[3],[5]] => [5,3,2,6,1,4,7,8,9,10] => 110100000
[6,1,1,1,1] => [[1,6,7,8,9,10],[2],[3],[4],[5]] => [5,4,3,2,1,6,7,8,9,10] => 111100000
[5,5] => [[1,2,3,4,5],[6,7,8,9,10]] => [6,7,8,9,10,1,2,3,4,5] => 000010000
[5,4,1] => [[1,3,4,5,10],[2,7,8,9],[6]] => [6,2,7,8,9,1,3,4,5,10] => 100010000
[5,3,2] => [[1,2,5,9,10],[3,4,8],[6,7]] => [6,7,3,4,8,1,2,5,9,10] => 010010000
[5,3,1,1] => [[1,4,5,9,10],[2,7,8],[3],[6]] => [6,3,2,7,8,1,4,5,9,10] => 110010000
[5,2,2,1] => [[1,3,8,9,10],[2,5],[4,7],[6]] => [6,4,7,2,5,1,3,8,9,10] => 101010000
[5,2,1,1,1] => [[1,5,8,9,10],[2,7],[3],[4],[6]] => [6,4,3,2,7,1,5,8,9,10] => 111010000
[5,1,1,1,1,1] => [[1,7,8,9,10],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7,8,9,10] => 111110000
[4,4,2] => [[1,2,5,6],[3,4,9,10],[7,8]] => [7,8,3,4,9,10,1,2,5,6] => 010001000
[4,4,1,1] => [[1,4,5,6],[2,8,9,10],[3],[7]] => [7,3,2,8,9,10,1,4,5,6] => 110001000
[4,3,3] => [[1,2,3,10],[4,5,6],[7,8,9]] => [7,8,9,4,5,6,1,2,3,10] => 001001000
[4,3,2,1] => [[1,3,6,10],[2,5,9],[4,8],[7]] => [7,4,8,2,5,9,1,3,6,10] => 101001000
[4,3,1,1,1] => [[1,5,6,10],[2,8,9],[3],[4],[7]] => [7,4,3,2,8,9,1,5,6,10] => 111001000
[4,2,2,2] => [[1,2,9,10],[3,4],[5,6],[7,8]] => [7,8,5,6,3,4,1,2,9,10] => 010101000
[4,2,2,1,1] => [[1,4,9,10],[2,6],[3,8],[5],[7]] => [7,5,3,8,2,6,1,4,9,10] => 110101000
[4,2,1,1,1,1] => [[1,6,9,10],[2,8],[3],[4],[5],[7]] => [7,5,4,3,2,8,1,6,9,10] => 111101000
[4,1,1,1,1,1,1] => [[1,8,9,10],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1,8,9,10] => 111111000
[3,3,3,1] => [[1,3,4],[2,6,7],[5,9,10],[8]] => [8,5,9,10,2,6,7,1,3,4] => 100100100
[3,3,2,2] => [[1,2,7],[3,4,10],[5,6],[8,9]] => [8,9,5,6,3,4,10,1,2,7] => 010100100
[3,3,2,1,1] => [[1,4,7],[2,6,10],[3,9],[5],[8]] => [8,5,3,9,2,6,10,1,4,7] => 110100100
[3,3,1,1,1,1] => [[1,6,7],[2,9,10],[3],[4],[5],[8]] => [8,5,4,3,2,9,10,1,6,7] => 111100100
[3,2,2,2,1] => [[1,3,10],[2,5],[4,7],[6,9],[8]] => [8,6,9,4,7,2,5,1,3,10] => 101010100
[3,2,2,1,1,1] => [[1,5,10],[2,7],[3,9],[4],[6],[8]] => [8,6,4,3,9,2,7,1,5,10] => 111010100
[3,2,1,1,1,1,1] => [[1,7,10],[2,9],[3],[4],[5],[6],[8]] => [8,6,5,4,3,2,9,1,7,10] => 111110100
[3,1,1,1,1,1,1,1] => [[1,9,10],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1,9,10] => 111111100
[2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => [9,10,7,8,5,6,3,4,1,2] => 010101010
[2,2,2,2,1,1] => [[1,4],[2,6],[3,8],[5,10],[7],[9]] => [9,7,5,10,3,8,2,6,1,4] => 110101010
[2,2,2,1,1,1,1] => [[1,6],[2,8],[3,10],[4],[5],[7],[9]] => [9,7,5,4,3,10,2,8,1,6] => 111101010
[2,2,1,1,1,1,1,1] => [[1,8],[2,10],[3],[4],[5],[6],[7],[9]] => [9,7,6,5,4,3,2,10,1,8] => 111111010
[2,1,1,1,1,1,1,1,1] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,3,2,1,10] => 111111110
[1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,4,3,2,1] => 111111111
[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] => 00000000000
[6,6] => [[1,2,3,4,5,6],[7,8,9,10,11,12]] => [7,8,9,10,11,12,1,2,3,4,5,6] => 00000100000
[2,2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]] => [11,12,9,10,7,8,5,6,3,4,1,2] => 01010101010
[1,1,1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]] => [12,11,10,9,8,7,6,5,4,3,2,1] => 11111111111
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
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.
Map
descent word
Description
The descent positions of a permutation as a binary word.
For a permutation π of n letters and each 1≤i≤n−1 such that π(i)>π(i+1) we set wi=1, otherwise wi=0.
Thus, the length of the word is one less the size of the permutation. In particular, the descent word is undefined for the empty permutation.