Identifier
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
Images
[1] => [[1]] => =>
[2] => [[1,2]] => 0 => 0
[1,1] => [[1],[2]] => 1 => 1
[3] => [[1,2,3]] => 00 => 00
[2,1] => [[1,2],[3]] => 01 => 10
[1,1,1] => [[1],[2],[3]] => 11 => 11
[4] => [[1,2,3,4]] => 000 => 000
[3,1] => [[1,2,3],[4]] => 001 => 100
[2,2] => [[1,2],[3,4]] => 010 => 010
[2,1,1] => [[1,2],[3],[4]] => 011 => 110
[1,1,1,1] => [[1],[2],[3],[4]] => 111 => 111
[5] => [[1,2,3,4,5]] => 0000 => 0000
[4,1] => [[1,2,3,4],[5]] => 0001 => 1000
[3,2] => [[1,2,3],[4,5]] => 0010 => 0100
[3,1,1] => [[1,2,3],[4],[5]] => 0011 => 1100
[2,2,1] => [[1,2],[3,4],[5]] => 0101 => 1010
[2,1,1,1] => [[1,2],[3],[4],[5]] => 0111 => 1110
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => 1111 => 1111
[6] => [[1,2,3,4,5,6]] => 00000 => 00000
[5,1] => [[1,2,3,4,5],[6]] => 00001 => 10000
[4,2] => [[1,2,3,4],[5,6]] => 00010 => 01000
[4,1,1] => [[1,2,3,4],[5],[6]] => 00011 => 11000
[3,3] => [[1,2,3],[4,5,6]] => 00100 => 00100
[3,2,1] => [[1,2,3],[4,5],[6]] => 00101 => 10100
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => 00111 => 11100
[2,2,2] => [[1,2],[3,4],[5,6]] => 01010 => 01010
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => 01011 => 11010
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => 01111 => 11110
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => 11111 => 11111
[7] => [[1,2,3,4,5,6,7]] => 000000 => 000000
[6,1] => [[1,2,3,4,5,6],[7]] => 000001 => 100000
[5,2] => [[1,2,3,4,5],[6,7]] => 000010 => 010000
[5,1,1] => [[1,2,3,4,5],[6],[7]] => 000011 => 110000
[4,3] => [[1,2,3,4],[5,6,7]] => 000100 => 001000
[4,2,1] => [[1,2,3,4],[5,6],[7]] => 000101 => 101000
[4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 000111 => 111000
[3,3,1] => [[1,2,3],[4,5,6],[7]] => 001001 => 100100
[3,2,2] => [[1,2,3],[4,5],[6,7]] => 001010 => 010100
[3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => 001011 => 110100
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => 001111 => 111100
[2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => 010101 => 101010
[2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => 010111 => 111010
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => 011111 => 111110
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => 111111 => 111111
[8] => [[1,2,3,4,5,6,7,8]] => 0000000 => 0000000
[7,1] => [[1,2,3,4,5,6,7],[8]] => 0000001 => 1000000
[6,2] => [[1,2,3,4,5,6],[7,8]] => 0000010 => 0100000
[6,1,1] => [[1,2,3,4,5,6],[7],[8]] => 0000011 => 1100000
[5,3] => [[1,2,3,4,5],[6,7,8]] => 0000100 => 0010000
[5,2,1] => [[1,2,3,4,5],[6,7],[8]] => 0000101 => 1010000
[5,1,1,1] => [[1,2,3,4,5],[6],[7],[8]] => 0000111 => 1110000
[4,4] => [[1,2,3,4],[5,6,7,8]] => 0001000 => 0001000
[4,3,1] => [[1,2,3,4],[5,6,7],[8]] => 0001001 => 1001000
[4,2,2] => [[1,2,3,4],[5,6],[7,8]] => 0001010 => 0101000
[4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => 0001011 => 1101000
[4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => 0001111 => 1111000
[3,3,2] => [[1,2,3],[4,5,6],[7,8]] => 0010010 => 0100100
[3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => 0010011 => 1100100
[3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => 0010101 => 1010100
[3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 0010111 => 1110100
[3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => 0011111 => 1111100
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => 0101010 => 0101010
[2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => 0101011 => 1101010
[2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => 0101111 => 1111010
[2,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8]] => 0111111 => 1111110
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => 1111111 => 1111111
[9] => [[1,2,3,4,5,6,7,8,9]] => 00000000 => 00000000
[8,1] => [[1,2,3,4,5,6,7,8],[9]] => 00000001 => 10000000
[7,2] => [[1,2,3,4,5,6,7],[8,9]] => 00000010 => 01000000
[7,1,1] => [[1,2,3,4,5,6,7],[8],[9]] => 00000011 => 11000000
[6,3] => [[1,2,3,4,5,6],[7,8,9]] => 00000100 => 00100000
[6,2,1] => [[1,2,3,4,5,6],[7,8],[9]] => 00000101 => 10100000
[6,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9]] => 00000111 => 11100000
[5,4] => [[1,2,3,4,5],[6,7,8,9]] => 00001000 => 00010000
[5,3,1] => [[1,2,3,4,5],[6,7,8],[9]] => 00001001 => 10010000
[5,2,2] => [[1,2,3,4,5],[6,7],[8,9]] => 00001010 => 01010000
[5,2,1,1] => [[1,2,3,4,5],[6,7],[8],[9]] => 00001011 => 11010000
[5,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9]] => 00001111 => 11110000
[4,4,1] => [[1,2,3,4],[5,6,7,8],[9]] => 00010001 => 10001000
[4,3,2] => [[1,2,3,4],[5,6,7],[8,9]] => 00010010 => 01001000
[4,3,1,1] => [[1,2,3,4],[5,6,7],[8],[9]] => 00010011 => 11001000
[4,2,2,1] => [[1,2,3,4],[5,6],[7,8],[9]] => 00010101 => 10101000
[4,2,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9]] => 00010111 => 11101000
[4,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9]] => 00011111 => 11111000
[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => 00100100 => 00100100
[3,3,2,1] => [[1,2,3],[4,5,6],[7,8],[9]] => 00100101 => 10100100
[3,3,1,1,1] => [[1,2,3],[4,5,6],[7],[8],[9]] => 00100111 => 11100100
[3,2,2,2] => [[1,2,3],[4,5],[6,7],[8,9]] => 00101010 => 01010100
[3,2,2,1,1] => [[1,2,3],[4,5],[6,7],[8],[9]] => 00101011 => 11010100
[3,2,1,1,1,1] => [[1,2,3],[4,5],[6],[7],[8],[9]] => 00101111 => 11110100
[3,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9]] => 00111111 => 11111100
[2,2,2,2,1] => [[1,2],[3,4],[5,6],[7,8],[9]] => 01010101 => 10101010
[2,2,2,1,1,1] => [[1,2],[3,4],[5,6],[7],[8],[9]] => 01010111 => 11101010
[2,2,1,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8],[9]] => 01011111 => 11111010
[2,1,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9]] => 01111111 => 11111110
[1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => 11111111 => 11111111
[10] => [[1,2,3,4,5,6,7,8,9,10]] => 000000000 => 000000000
[9,1] => [[1,2,3,4,5,6,7,8,9],[10]] => 000000001 => 100000000
[8,2] => [[1,2,3,4,5,6,7,8],[9,10]] => 000000010 => 010000000
[8,1,1] => [[1,2,3,4,5,6,7,8],[9],[10]] => 000000011 => 110000000
[7,3] => [[1,2,3,4,5,6,7],[8,9,10]] => 000000100 => 001000000
>>> Load all 178 entries. <<<
[7,2,1] => [[1,2,3,4,5,6,7],[8,9],[10]] => 000000101 => 101000000
[7,1,1,1] => [[1,2,3,4,5,6,7],[8],[9],[10]] => 000000111 => 111000000
[6,4] => [[1,2,3,4,5,6],[7,8,9,10]] => 000001000 => 000100000
[6,3,1] => [[1,2,3,4,5,6],[7,8,9],[10]] => 000001001 => 100100000
[6,2,2] => [[1,2,3,4,5,6],[7,8],[9,10]] => 000001010 => 010100000
[6,2,1,1] => [[1,2,3,4,5,6],[7,8],[9],[10]] => 000001011 => 110100000
[6,1,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9],[10]] => 000001111 => 111100000
[5,5] => [[1,2,3,4,5],[6,7,8,9,10]] => 000010000 => 000010000
[5,4,1] => [[1,2,3,4,5],[6,7,8,9],[10]] => 000010001 => 100010000
[5,3,2] => [[1,2,3,4,5],[6,7,8],[9,10]] => 000010010 => 010010000
[5,3,1,1] => [[1,2,3,4,5],[6,7,8],[9],[10]] => 000010011 => 110010000
[5,2,2,1] => [[1,2,3,4,5],[6,7],[8,9],[10]] => 000010101 => 101010000
[5,2,1,1,1] => [[1,2,3,4,5],[6,7],[8],[9],[10]] => 000010111 => 111010000
[5,1,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9],[10]] => 000011111 => 111110000
[4,4,2] => [[1,2,3,4],[5,6,7,8],[9,10]] => 000100010 => 010001000
[4,4,1,1] => [[1,2,3,4],[5,6,7,8],[9],[10]] => 000100011 => 110001000
[4,3,3] => [[1,2,3,4],[5,6,7],[8,9,10]] => 000100100 => 001001000
[4,3,2,1] => [[1,2,3,4],[5,6,7],[8,9],[10]] => 000100101 => 101001000
[4,3,1,1,1] => [[1,2,3,4],[5,6,7],[8],[9],[10]] => 000100111 => 111001000
[4,2,2,2] => [[1,2,3,4],[5,6],[7,8],[9,10]] => 000101010 => 010101000
[4,2,2,1,1] => [[1,2,3,4],[5,6],[7,8],[9],[10]] => 000101011 => 110101000
[4,2,1,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9],[10]] => 000101111 => 111101000
[4,1,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9],[10]] => 000111111 => 111111000
[3,3,3,1] => [[1,2,3],[4,5,6],[7,8,9],[10]] => 001001001 => 100100100
[3,3,2,2] => [[1,2,3],[4,5,6],[7,8],[9,10]] => 001001010 => 010100100
[3,3,2,1,1] => [[1,2,3],[4,5,6],[7,8],[9],[10]] => 001001011 => 110100100
[3,3,1,1,1,1] => [[1,2,3],[4,5,6],[7],[8],[9],[10]] => 001001111 => 111100100
[3,2,2,2,1] => [[1,2,3],[4,5],[6,7],[8,9],[10]] => 001010101 => 101010100
[3,2,2,1,1,1] => [[1,2,3],[4,5],[6,7],[8],[9],[10]] => 001010111 => 111010100
[3,2,1,1,1,1,1] => [[1,2,3],[4,5],[6],[7],[8],[9],[10]] => 001011111 => 111110100
[3,1,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9],[10]] => 001111111 => 111111100
[2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => 010101010 => 010101010
[2,2,2,2,1,1] => [[1,2],[3,4],[5,6],[7,8],[9],[10]] => 010101011 => 110101010
[2,2,2,1,1,1,1] => [[1,2],[3,4],[5,6],[7],[8],[9],[10]] => 010101111 => 111101010
[2,2,1,1,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8],[9],[10]] => 010111111 => 111111010
[2,1,1,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]] => 011111111 => 111111110
[1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => 111111111 => 111111111
[5,4,2] => [[1,2,3,4,5],[6,7,8,9],[10,11]] => 0000100010 => 0100010000
[5,3,3] => [[1,2,3,4,5],[6,7,8],[9,10,11]] => 0000100100 => 0010010000
[5,3,2,1] => [[1,2,3,4,5],[6,7,8],[9,10],[11]] => 0000100101 => 1010010000
[5,3,1,1,1] => [[1,2,3,4,5],[6,7,8],[9],[10],[11]] => 0000100111 => 1110010000
[5,2,2,2] => [[1,2,3,4,5],[6,7],[8,9],[10,11]] => 0000101010 => 0101010000
[5,2,2,1,1] => [[1,2,3,4,5],[6,7],[8,9],[10],[11]] => 0000101011 => 1101010000
[4,4,3] => [[1,2,3,4],[5,6,7,8],[9,10,11]] => 0001000100 => 0010001000
[4,4,2,1] => [[1,2,3,4],[5,6,7,8],[9,10],[11]] => 0001000101 => 1010001000
[4,4,1,1,1] => [[1,2,3,4],[5,6,7,8],[9],[10],[11]] => 0001000111 => 1110001000
[4,3,2,2] => [[1,2,3,4],[5,6,7],[8,9],[10,11]] => 0001001010 => 0101001000
[4,3,2,1,1] => [[1,2,3,4],[5,6,7],[8,9],[10],[11]] => 0001001011 => 1101001000
[4,2,2,2,1] => [[1,2,3,4],[5,6],[7,8],[9,10],[11]] => 0001010101 => 1010101000
[3,3,3,2] => [[1,2,3],[4,5,6],[7,8,9],[10,11]] => 0010010010 => 0100100100
[3,3,3,1,1] => [[1,2,3],[4,5,6],[7,8,9],[10],[11]] => 0010010011 => 1100100100
[3,3,2,2,1] => [[1,2,3],[4,5,6],[7,8],[9,10],[11]] => 0010010101 => 1010100100
[12] => [[1,2,3,4,5,6,7,8,9,10,11,12]] => 00000000000 => 00000000000
[6,6] => [[1,2,3,4,5,6],[7,8,9,10,11,12]] => 00000100000 => 00000100000
[6,4,2] => [[1,2,3,4,5,6],[7,8,9,10],[11,12]] => 00000100010 => 01000100000
[5,4,3] => [[1,2,3,4,5],[6,7,8,9],[10,11,12]] => 00001000100 => 00100010000
[5,4,2,1] => [[1,2,3,4,5],[6,7,8,9],[10,11],[12]] => 00001000101 => 10100010000
[5,4,1,1,1] => [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]] => 00001000111 => 11100010000
[5,3,3,1] => [[1,2,3,4,5],[6,7,8],[9,10,11],[12]] => 00001001001 => 10010010000
[5,3,2,2] => [[1,2,3,4,5],[6,7,8],[9,10],[11,12]] => 00001001010 => 01010010000
[5,2,2,2,1] => [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]] => 00001010101 => 10101010000
[4,4,3,1] => [[1,2,3,4],[5,6,7,8],[9,10,11],[12]] => 00010001001 => 10010001000
[4,4,2,2] => [[1,2,3,4],[5,6,7,8],[9,10],[11,12]] => 00010001010 => 01010001000
[4,4,2,1,1] => [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]] => 00010001011 => 11010001000
[4,3,3,2] => [[1,2,3,4],[5,6,7],[8,9,10],[11,12]] => 00010010010 => 01001001000
[4,3,2,2,1] => [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]] => 00010010101 => 10101001000
[4,2,2,2,2] => [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]] => 00010101010 => 01010101000
[3,3,3,2,1] => [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]] => 00100100101 => 10100100100
[3,3,2,2,1,1] => [[1,2,3],[4,5,6],[7,8],[9,10],[11],[12]] => 00100101011 => 11010100100
[2,2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]] => 01010101010 => 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 => 11111111111
[5,4,2,1,1] => [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]] => 000010001011 => 110100010000
[5,3,3,1,1] => [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]] => 000010010011 => 110010010000
[5,3,2,2,1] => [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]] => 000010010101 => 101010010000
[4,4,3,1,1] => [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]] => 000100010011 => 110010001000
[4,4,2,2,1] => [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]] => 000100010101 => 101010001000
[] => [] => =>
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
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
reverse
Description
Return the reversal of a binary word.