Identifier
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
Images
[1] => [[1]] =>
[2] => [[1,2]] => 0
[1,1] => [[1],[2]] => 1
[3] => [[1,2,3]] => 00
[2,1] => [[1,3],[2]] => 10
[1,1,1] => [[1],[2],[3]] => 11
[4] => [[1,2,3,4]] => 000
[3,1] => [[1,3,4],[2]] => 100
[2,2] => [[1,2],[3,4]] => 010
[2,1,1] => [[1,4],[2],[3]] => 110
[1,1,1,1] => [[1],[2],[3],[4]] => 111
[5] => [[1,2,3,4,5]] => 0000
[4,1] => [[1,3,4,5],[2]] => 1000
[3,2] => [[1,2,5],[3,4]] => 0100
[3,1,1] => [[1,4,5],[2],[3]] => 1100
[2,2,1] => [[1,3],[2,5],[4]] => 1010
[2,1,1,1] => [[1,5],[2],[3],[4]] => 1110
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => 1111
[6] => [[1,2,3,4,5,6]] => 00000
[5,1] => [[1,3,4,5,6],[2]] => 10000
[4,2] => [[1,2,5,6],[3,4]] => 01000
[4,1,1] => [[1,4,5,6],[2],[3]] => 11000
[3,3] => [[1,2,3],[4,5,6]] => 00100
[3,2,1] => [[1,3,6],[2,5],[4]] => 10100
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => 11100
[2,2,2] => [[1,2],[3,4],[5,6]] => 01010
[2,2,1,1] => [[1,4],[2,6],[3],[5]] => 11010
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => 11110
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => 11111
[7] => [[1,2,3,4,5,6,7]] => 000000
[6,1] => [[1,3,4,5,6,7],[2]] => 100000
[5,2] => [[1,2,5,6,7],[3,4]] => 010000
[5,1,1] => [[1,4,5,6,7],[2],[3]] => 110000
[4,3] => [[1,2,3,7],[4,5,6]] => 001000
[4,2,1] => [[1,3,6,7],[2,5],[4]] => 101000
[4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => 111000
[3,3,1] => [[1,3,4],[2,6,7],[5]] => 100100
[3,2,2] => [[1,2,7],[3,4],[5,6]] => 010100
[3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => 110100
[3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => 111100
[2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => 101010
[2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => 111010
[2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => 111110
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => 111111
[8] => [[1,2,3,4,5,6,7,8]] => 0000000
[7,1] => [[1,3,4,5,6,7,8],[2]] => 1000000
[6,2] => [[1,2,5,6,7,8],[3,4]] => 0100000
[6,1,1] => [[1,4,5,6,7,8],[2],[3]] => 1100000
[5,3] => [[1,2,3,7,8],[4,5,6]] => 0010000
[5,2,1] => [[1,3,6,7,8],[2,5],[4]] => 1010000
[5,1,1,1] => [[1,5,6,7,8],[2],[3],[4]] => 1110000
[4,4] => [[1,2,3,4],[5,6,7,8]] => 0001000
[4,3,1] => [[1,3,4,8],[2,6,7],[5]] => 1001000
[4,2,2] => [[1,2,7,8],[3,4],[5,6]] => 0101000
[4,2,1,1] => [[1,4,7,8],[2,6],[3],[5]] => 1101000
[4,1,1,1,1] => [[1,6,7,8],[2],[3],[4],[5]] => 1111000
[3,3,2] => [[1,2,5],[3,4,8],[6,7]] => 0100100
[3,3,1,1] => [[1,4,5],[2,7,8],[3],[6]] => 1100100
[3,2,2,1] => [[1,3,8],[2,5],[4,7],[6]] => 1010100
[3,2,1,1,1] => [[1,5,8],[2,7],[3],[4],[6]] => 1110100
[3,1,1,1,1,1] => [[1,7,8],[2],[3],[4],[5],[6]] => 1111100
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => 0101010
[2,2,2,1,1] => [[1,4],[2,6],[3,8],[5],[7]] => 1101010
[2,2,1,1,1,1] => [[1,6],[2,8],[3],[4],[5],[7]] => 1111010
[2,1,1,1,1,1,1] => [[1,8],[2],[3],[4],[5],[6],[7]] => 1111110
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => 1111111
[9] => [[1,2,3,4,5,6,7,8,9]] => 00000000
[8,1] => [[1,3,4,5,6,7,8,9],[2]] => 10000000
[7,2] => [[1,2,5,6,7,8,9],[3,4]] => 01000000
[7,1,1] => [[1,4,5,6,7,8,9],[2],[3]] => 11000000
[6,3] => [[1,2,3,7,8,9],[4,5,6]] => 00100000
[6,2,1] => [[1,3,6,7,8,9],[2,5],[4]] => 10100000
[6,1,1,1] => [[1,5,6,7,8,9],[2],[3],[4]] => 11100000
[5,4] => [[1,2,3,4,9],[5,6,7,8]] => 00010000
[5,3,1] => [[1,3,4,8,9],[2,6,7],[5]] => 10010000
[5,2,2] => [[1,2,7,8,9],[3,4],[5,6]] => 01010000
[5,2,1,1] => [[1,4,7,8,9],[2,6],[3],[5]] => 11010000
[5,1,1,1,1] => [[1,6,7,8,9],[2],[3],[4],[5]] => 11110000
[4,4,1] => [[1,3,4,5],[2,7,8,9],[6]] => 10001000
[4,3,2] => [[1,2,5,9],[3,4,8],[6,7]] => 01001000
[4,3,1,1] => [[1,4,5,9],[2,7,8],[3],[6]] => 11001000
[4,2,2,1] => [[1,3,8,9],[2,5],[4,7],[6]] => 10101000
[4,2,1,1,1] => [[1,5,8,9],[2,7],[3],[4],[6]] => 11101000
[4,1,1,1,1,1] => [[1,7,8,9],[2],[3],[4],[5],[6]] => 11111000
[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => 00100100
[3,3,2,1] => [[1,3,6],[2,5,9],[4,8],[7]] => 10100100
[3,3,1,1,1] => [[1,5,6],[2,8,9],[3],[4],[7]] => 11100100
[3,2,2,2] => [[1,2,9],[3,4],[5,6],[7,8]] => 01010100
[3,2,2,1,1] => [[1,4,9],[2,6],[3,8],[5],[7]] => 11010100
[3,2,1,1,1,1] => [[1,6,9],[2,8],[3],[4],[5],[7]] => 11110100
[3,1,1,1,1,1,1] => [[1,8,9],[2],[3],[4],[5],[6],[7]] => 11111100
[2,2,2,2,1] => [[1,3],[2,5],[4,7],[6,9],[8]] => 10101010
[2,2,2,1,1,1] => [[1,5],[2,7],[3,9],[4],[6],[8]] => 11101010
[2,2,1,1,1,1,1] => [[1,7],[2,9],[3],[4],[5],[6],[8]] => 11111010
[2,1,1,1,1,1,1,1] => [[1,9],[2],[3],[4],[5],[6],[7],[8]] => 11111110
[1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => 11111111
[10] => [[1,2,3,4,5,6,7,8,9,10]] => 000000000
[9,1] => [[1,3,4,5,6,7,8,9,10],[2]] => 100000000
[8,2] => [[1,2,5,6,7,8,9,10],[3,4]] => 010000000
[8,1,1] => [[1,4,5,6,7,8,9,10],[2],[3]] => 110000000
[7,3] => [[1,2,3,7,8,9,10],[4,5,6]] => 001000000
>>> Load all 201 entries. <<<
[7,2,1] => [[1,3,6,7,8,9,10],[2,5],[4]] => 101000000
[7,1,1,1] => [[1,5,6,7,8,9,10],[2],[3],[4]] => 111000000
[6,4] => [[1,2,3,4,9,10],[5,6,7,8]] => 000100000
[6,3,1] => [[1,3,4,8,9,10],[2,6,7],[5]] => 100100000
[6,2,2] => [[1,2,7,8,9,10],[3,4],[5,6]] => 010100000
[6,2,1,1] => [[1,4,7,8,9,10],[2,6],[3],[5]] => 110100000
[6,1,1,1,1] => [[1,6,7,8,9,10],[2],[3],[4],[5]] => 111100000
[5,5] => [[1,2,3,4,5],[6,7,8,9,10]] => 000010000
[5,4,1] => [[1,3,4,5,10],[2,7,8,9],[6]] => 100010000
[5,3,2] => [[1,2,5,9,10],[3,4,8],[6,7]] => 010010000
[5,3,1,1] => [[1,4,5,9,10],[2,7,8],[3],[6]] => 110010000
[5,2,2,1] => [[1,3,8,9,10],[2,5],[4,7],[6]] => 101010000
[5,2,1,1,1] => [[1,5,8,9,10],[2,7],[3],[4],[6]] => 111010000
[5,1,1,1,1,1] => [[1,7,8,9,10],[2],[3],[4],[5],[6]] => 111110000
[4,4,2] => [[1,2,5,6],[3,4,9,10],[7,8]] => 010001000
[4,4,1,1] => [[1,4,5,6],[2,8,9,10],[3],[7]] => 110001000
[4,3,3] => [[1,2,3,10],[4,5,6],[7,8,9]] => 001001000
[4,3,2,1] => [[1,3,6,10],[2,5,9],[4,8],[7]] => 101001000
[4,3,1,1,1] => [[1,5,6,10],[2,8,9],[3],[4],[7]] => 111001000
[4,2,2,2] => [[1,2,9,10],[3,4],[5,6],[7,8]] => 010101000
[4,2,2,1,1] => [[1,4,9,10],[2,6],[3,8],[5],[7]] => 110101000
[4,2,1,1,1,1] => [[1,6,9,10],[2,8],[3],[4],[5],[7]] => 111101000
[4,1,1,1,1,1,1] => [[1,8,9,10],[2],[3],[4],[5],[6],[7]] => 111111000
[3,3,3,1] => [[1,3,4],[2,6,7],[5,9,10],[8]] => 100100100
[3,3,2,2] => [[1,2,7],[3,4,10],[5,6],[8,9]] => 010100100
[3,3,2,1,1] => [[1,4,7],[2,6,10],[3,9],[5],[8]] => 110100100
[3,3,1,1,1,1] => [[1,6,7],[2,9,10],[3],[4],[5],[8]] => 111100100
[3,2,2,2,1] => [[1,3,10],[2,5],[4,7],[6,9],[8]] => 101010100
[3,2,2,1,1,1] => [[1,5,10],[2,7],[3,9],[4],[6],[8]] => 111010100
[3,2,1,1,1,1,1] => [[1,7,10],[2,9],[3],[4],[5],[6],[8]] => 111110100
[3,1,1,1,1,1,1,1] => [[1,9,10],[2],[3],[4],[5],[6],[7],[8]] => 111111100
[2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => 010101010
[2,2,2,2,1,1] => [[1,4],[2,6],[3,8],[5,10],[7],[9]] => 110101010
[2,2,2,1,1,1,1] => [[1,6],[2,8],[3,10],[4],[5],[7],[9]] => 111101010
[2,2,1,1,1,1,1,1] => [[1,8],[2,10],[3],[4],[5],[6],[7],[9]] => 111111010
[2,1,1,1,1,1,1,1,1] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]] => 111111110
[1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => 111111111
[5,4,2] => [[1,2,5,6,11],[3,4,9,10],[7,8]] => 0100010000
[5,4,1,1] => [[1,4,5,6,11],[2,8,9,10],[3],[7]] => 1100010000
[5,3,3] => [[1,2,3,10,11],[4,5,6],[7,8,9]] => 0010010000
[5,3,2,1] => [[1,3,6,10,11],[2,5,9],[4,8],[7]] => 1010010000
[5,3,1,1,1] => [[1,5,6,10,11],[2,8,9],[3],[4],[7]] => 1110010000
[5,2,2,2] => [[1,2,9,10,11],[3,4],[5,6],[7,8]] => 0101010000
[5,2,2,1,1] => [[1,4,9,10,11],[2,6],[3,8],[5],[7]] => 1101010000
[4,4,3] => [[1,2,3,7],[4,5,6,11],[8,9,10]] => 0010001000
[4,4,2,1] => [[1,3,6,7],[2,5,10,11],[4,9],[8]] => 1010001000
[4,4,1,1,1] => [[1,5,6,7],[2,9,10,11],[3],[4],[8]] => 1110001000
[4,3,3,1] => [[1,3,4,11],[2,6,7],[5,9,10],[8]] => 1001001000
[4,3,2,2] => [[1,2,7,11],[3,4,10],[5,6],[8,9]] => 0101001000
[4,3,2,1,1] => [[1,4,7,11],[2,6,10],[3,9],[5],[8]] => 1101001000
[4,2,2,2,1] => [[1,3,10,11],[2,5],[4,7],[6,9],[8]] => 1010101000
[3,3,3,2] => [[1,2,5],[3,4,8],[6,7,11],[9,10]] => 0100100100
[3,3,3,1,1] => [[1,4,5],[2,7,8],[3,10,11],[6],[9]] => 1100100100
[3,3,2,2,1] => [[1,3,8],[2,5,11],[4,7],[6,10],[9]] => 1010100100
[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]] => 00000100000
[6,4,2] => [[1,2,5,6,11,12],[3,4,9,10],[7,8]] => 01000100000
[5,5,1,1] => [[1,4,5,6,7],[2,9,10,11,12],[3],[8]] => 11000010000
[5,4,3] => [[1,2,3,7,12],[4,5,6,11],[8,9,10]] => 00100010000
[5,4,2,1] => [[1,3,6,7,12],[2,5,10,11],[4,9],[8]] => 10100010000
[5,4,1,1,1] => [[1,5,6,7,12],[2,9,10,11],[3],[4],[8]] => 11100010000
[5,3,3,1] => [[1,3,4,11,12],[2,6,7],[5,9,10],[8]] => 10010010000
[5,3,2,2] => [[1,2,7,11,12],[3,4,10],[5,6],[8,9]] => 01010010000
[5,3,2,1,1] => [[1,4,7,11,12],[2,6,10],[3,9],[5],[8]] => 11010010000
[5,2,2,2,1] => [[1,3,10,11,12],[2,5],[4,7],[6,9],[8]] => 10101010000
[4,4,3,1] => [[1,3,4,8],[2,6,7,12],[5,10,11],[9]] => 10010001000
[4,4,2,2] => [[1,2,7,8],[3,4,11,12],[5,6],[9,10]] => 01010001000
[4,4,2,1,1] => [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]] => 11010001000
[4,3,3,2] => [[1,2,5,12],[3,4,8],[6,7,11],[9,10]] => 01001001000
[4,3,3,1,1] => [[1,4,5,12],[2,7,8],[3,10,11],[6],[9]] => 11001001000
[4,3,2,2,1] => [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]] => 10101001000
[3,3,3,2,1] => [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]] => 10100100100
[3,3,2,2,1,1] => [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]] => 11010100100
[2,2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]] => 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]] => 11111111111
[5,4,3,1] => [[1,3,4,8,13],[2,6,7,12],[5,10,11],[9]] => 100100010000
[5,4,2,2] => [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]] => 010100010000
[5,4,2,1,1] => [[1,4,7,8,13],[2,6,11,12],[3,10],[5],[9]] => 110100010000
[5,3,3,2] => [[1,2,5,12,13],[3,4,8],[6,7,11],[9,10]] => 010010010000
[5,3,3,1,1] => [[1,4,5,12,13],[2,7,8],[3,10,11],[6],[9]] => 110010010000
[5,3,2,2,1] => [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]] => 101010010000
[4,4,3,2] => [[1,2,5,9],[3,4,8,13],[6,7,12],[10,11]] => 010010001000
[4,4,3,1,1] => [[1,4,5,9],[2,7,8,13],[3,11,12],[6],[10]] => 110010001000
[4,4,2,2,1] => [[1,3,8,9],[2,5,12,13],[4,7],[6,11],[10]] => 101010001000
[4,3,3,2,1] => [[1,3,6,13],[2,5,9],[4,8,12],[7,11],[10]] => 101001001000
[7,7] => [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]] => 0000001000000
[5,4,3,2] => [[1,2,5,9,14],[3,4,8,13],[6,7,12],[10,11]] => 0100100010000
[5,4,3,1,1] => [[1,4,5,9,14],[2,7,8,13],[3,11,12],[6],[10]] => 1100100010000
[5,4,2,2,1] => [[1,3,8,9,14],[2,5,12,13],[4,7],[6,11],[10]] => 1010100010000
[5,3,3,2,1] => [[1,3,6,13,14],[2,5,9],[4,8,12],[7,11],[10]] => 1010010010000
[4,4,3,2,1] => [[1,3,6,10],[2,5,9,14],[4,8,13],[7,12],[11]] => 1010010001000
[2,2,2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]] => 0101010101010
[5,4,3,2,1] => [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]] => 10100100010000
[8,8] => [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14,15,16]] => 000000010000000
[2,2,2,2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]] => 010101010101010
[] => [] =>
[2,2,2,2,2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18]] => 01010101010101010
[2,2,2,2,2,2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18],[19,20]] => 0101010101010101010
[10,10] => [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16,17,18,19,20]] => 0000000001000000000
[9,9] => [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16,17,18]] => 00000000100000000
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 $1\leq i < n$.