Identifier
Mp00042: Integer partitions —initial 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] => [[1,2],[3]] => 01 => [2,1]
[1,1,1] => [[1],[2],[3]] => 11 => [1,1,1]
[4] => [[1,2,3,4]] => 000 => [4]
[3,1] => [[1,2,3],[4]] => 001 => [3,1]
[2,2] => [[1,2],[3,4]] => 010 => [2,2]
[2,1,1] => [[1,2],[3],[4]] => 011 => [2,1,1]
[1,1,1,1] => [[1],[2],[3],[4]] => 111 => [1,1,1,1]
[5] => [[1,2,3,4,5]] => 0000 => [5]
[4,1] => [[1,2,3,4],[5]] => 0001 => [4,1]
[3,2] => [[1,2,3],[4,5]] => 0010 => [3,2]
[3,1,1] => [[1,2,3],[4],[5]] => 0011 => [3,1,1]
[2,2,1] => [[1,2],[3,4],[5]] => 0101 => [2,2,1]
[2,1,1,1] => [[1,2],[3],[4],[5]] => 0111 => [2,1,1,1]
[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] => [[1,2,3,4,5],[6]] => 00001 => [5,1]
[4,2] => [[1,2,3,4],[5,6]] => 00010 => [4,2]
[4,1,1] => [[1,2,3,4],[5],[6]] => 00011 => [4,1,1]
[3,3] => [[1,2,3],[4,5,6]] => 00100 => [3,3]
[3,2,1] => [[1,2,3],[4,5],[6]] => 00101 => [3,2,1]
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => 00111 => [3,1,1,1]
[2,2,2] => [[1,2],[3,4],[5,6]] => 01010 => [2,2,2]
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => 01011 => [2,2,1,1]
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => 01111 => [2,1,1,1,1]
[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] => [[1,2,3,4,5,6],[7]] => 000001 => [6,1]
[5,2] => [[1,2,3,4,5],[6,7]] => 000010 => [5,2]
[5,1,1] => [[1,2,3,4,5],[6],[7]] => 000011 => [5,1,1]
[4,3] => [[1,2,3,4],[5,6,7]] => 000100 => [4,3]
[4,2,1] => [[1,2,3,4],[5,6],[7]] => 000101 => [4,2,1]
[4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 000111 => [4,1,1,1]
[3,3,1] => [[1,2,3],[4,5,6],[7]] => 001001 => [3,3,1]
[3,2,2] => [[1,2,3],[4,5],[6,7]] => 001010 => [3,2,2]
[3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => 001011 => [3,2,1,1]
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => 001111 => [3,1,1,1,1]
[2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => 010101 => [2,2,2,1]
[2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => 010111 => [2,2,1,1,1]
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => 011111 => [2,1,1,1,1,1]
[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] => [[1,2,3,4,5,6,7],[8]] => 0000001 => [7,1]
[6,2] => [[1,2,3,4,5,6],[7,8]] => 0000010 => [6,2]
[6,1,1] => [[1,2,3,4,5,6],[7],[8]] => 0000011 => [6,1,1]
[5,3] => [[1,2,3,4,5],[6,7,8]] => 0000100 => [5,3]
[5,2,1] => [[1,2,3,4,5],[6,7],[8]] => 0000101 => [5,2,1]
[5,1,1,1] => [[1,2,3,4,5],[6],[7],[8]] => 0000111 => [5,1,1,1]
[4,4] => [[1,2,3,4],[5,6,7,8]] => 0001000 => [4,4]
[4,3,1] => [[1,2,3,4],[5,6,7],[8]] => 0001001 => [4,3,1]
[4,2,2] => [[1,2,3,4],[5,6],[7,8]] => 0001010 => [4,2,2]
[4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => 0001011 => [4,2,1,1]
[4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => 0001111 => [4,1,1,1,1]
[3,3,2] => [[1,2,3],[4,5,6],[7,8]] => 0010010 => [3,3,2]
[3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => 0010011 => [3,3,1,1]
[3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => 0010101 => [3,2,2,1]
[3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => 0010111 => [3,2,1,1,1]
[3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => 0011111 => [3,1,1,1,1,1]
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => 0101010 => [2,2,2,2]
[2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => 0101011 => [2,2,2,1,1]
[2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => 0101111 => [2,2,1,1,1,1]
[2,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8]] => 0111111 => [2,1,1,1,1,1,1]
[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] => [[1,2,3,4,5,6,7,8],[9]] => 00000001 => [8,1]
[7,2] => [[1,2,3,4,5,6,7],[8,9]] => 00000010 => [7,2]
[7,1,1] => [[1,2,3,4,5,6,7],[8],[9]] => 00000011 => [7,1,1]
[6,3] => [[1,2,3,4,5,6],[7,8,9]] => 00000100 => [6,3]
[6,2,1] => [[1,2,3,4,5,6],[7,8],[9]] => 00000101 => [6,2,1]
[6,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9]] => 00000111 => [6,1,1,1]
[5,4] => [[1,2,3,4,5],[6,7,8,9]] => 00001000 => [5,4]
[5,3,1] => [[1,2,3,4,5],[6,7,8],[9]] => 00001001 => [5,3,1]
[5,2,2] => [[1,2,3,4,5],[6,7],[8,9]] => 00001010 => [5,2,2]
[5,2,1,1] => [[1,2,3,4,5],[6,7],[8],[9]] => 00001011 => [5,2,1,1]
[5,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9]] => 00001111 => [5,1,1,1,1]
[4,4,1] => [[1,2,3,4],[5,6,7,8],[9]] => 00010001 => [4,4,1]
[4,3,2] => [[1,2,3,4],[5,6,7],[8,9]] => 00010010 => [4,3,2]
[4,3,1,1] => [[1,2,3,4],[5,6,7],[8],[9]] => 00010011 => [4,3,1,1]
[4,2,2,1] => [[1,2,3,4],[5,6],[7,8],[9]] => 00010101 => [4,2,2,1]
[4,2,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9]] => 00010111 => [4,2,1,1,1]
[4,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9]] => 00011111 => [4,1,1,1,1,1]
[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => 00100100 => [3,3,3]
[3,3,2,1] => [[1,2,3],[4,5,6],[7,8],[9]] => 00100101 => [3,3,2,1]
[3,3,1,1,1] => [[1,2,3],[4,5,6],[7],[8],[9]] => 00100111 => [3,3,1,1,1]
[3,2,2,2] => [[1,2,3],[4,5],[6,7],[8,9]] => 00101010 => [3,2,2,2]
[3,2,2,1,1] => [[1,2,3],[4,5],[6,7],[8],[9]] => 00101011 => [3,2,2,1,1]
[3,2,1,1,1,1] => [[1,2,3],[4,5],[6],[7],[8],[9]] => 00101111 => [3,2,1,1,1,1]
[3,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9]] => 00111111 => [3,1,1,1,1,1,1]
[2,2,2,2,1] => [[1,2],[3,4],[5,6],[7,8],[9]] => 01010101 => [2,2,2,2,1]
[2,2,2,1,1,1] => [[1,2],[3,4],[5,6],[7],[8],[9]] => 01010111 => [2,2,2,1,1,1]
[2,2,1,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8],[9]] => 01011111 => [2,2,1,1,1,1,1]
[2,1,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9]] => 01111111 => [2,1,1,1,1,1,1,1]
[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] => [[1,2,3,4,5,6,7,8,9],[10]] => 000000001 => [9,1]
[8,2] => [[1,2,3,4,5,6,7,8],[9,10]] => 000000010 => [8,2]
[8,1,1] => [[1,2,3,4,5,6,7,8],[9],[10]] => 000000011 => [8,1,1]
[7,3] => [[1,2,3,4,5,6,7],[8,9,10]] => 000000100 => [7,3]
>>> Load all 168 entries. <<<
[7,2,1] => [[1,2,3,4,5,6,7],[8,9],[10]] => 000000101 => [7,2,1]
[7,1,1,1] => [[1,2,3,4,5,6,7],[8],[9],[10]] => 000000111 => [7,1,1,1]
[6,4] => [[1,2,3,4,5,6],[7,8,9,10]] => 000001000 => [6,4]
[6,3,1] => [[1,2,3,4,5,6],[7,8,9],[10]] => 000001001 => [6,3,1]
[6,2,2] => [[1,2,3,4,5,6],[7,8],[9,10]] => 000001010 => [6,2,2]
[6,2,1,1] => [[1,2,3,4,5,6],[7,8],[9],[10]] => 000001011 => [6,2,1,1]
[6,1,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9],[10]] => 000001111 => [6,1,1,1,1]
[5,5] => [[1,2,3,4,5],[6,7,8,9,10]] => 000010000 => [5,5]
[5,4,1] => [[1,2,3,4,5],[6,7,8,9],[10]] => 000010001 => [5,4,1]
[5,3,2] => [[1,2,3,4,5],[6,7,8],[9,10]] => 000010010 => [5,3,2]
[5,3,1,1] => [[1,2,3,4,5],[6,7,8],[9],[10]] => 000010011 => [5,3,1,1]
[5,2,2,1] => [[1,2,3,4,5],[6,7],[8,9],[10]] => 000010101 => [5,2,2,1]
[5,2,1,1,1] => [[1,2,3,4,5],[6,7],[8],[9],[10]] => 000010111 => [5,2,1,1,1]
[5,1,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9],[10]] => 000011111 => [5,1,1,1,1,1]
[4,4,2] => [[1,2,3,4],[5,6,7,8],[9,10]] => 000100010 => [4,4,2]
[4,4,1,1] => [[1,2,3,4],[5,6,7,8],[9],[10]] => 000100011 => [4,4,1,1]
[4,3,3] => [[1,2,3,4],[5,6,7],[8,9,10]] => 000100100 => [4,3,3]
[4,3,2,1] => [[1,2,3,4],[5,6,7],[8,9],[10]] => 000100101 => [4,3,2,1]
[4,3,1,1,1] => [[1,2,3,4],[5,6,7],[8],[9],[10]] => 000100111 => [4,3,1,1,1]
[4,2,2,2] => [[1,2,3,4],[5,6],[7,8],[9,10]] => 000101010 => [4,2,2,2]
[4,2,2,1,1] => [[1,2,3,4],[5,6],[7,8],[9],[10]] => 000101011 => [4,2,2,1,1]
[4,2,1,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9],[10]] => 000101111 => [4,2,1,1,1,1]
[4,1,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9],[10]] => 000111111 => [4,1,1,1,1,1,1]
[3,3,3,1] => [[1,2,3],[4,5,6],[7,8,9],[10]] => 001001001 => [3,3,3,1]
[3,3,2,2] => [[1,2,3],[4,5,6],[7,8],[9,10]] => 001001010 => [3,3,2,2]
[3,3,2,1,1] => [[1,2,3],[4,5,6],[7,8],[9],[10]] => 001001011 => [3,3,2,1,1]
[3,3,1,1,1,1] => [[1,2,3],[4,5,6],[7],[8],[9],[10]] => 001001111 => [3,3,1,1,1,1]
[3,2,2,2,1] => [[1,2,3],[4,5],[6,7],[8,9],[10]] => 001010101 => [3,2,2,2,1]
[3,2,2,1,1,1] => [[1,2,3],[4,5],[6,7],[8],[9],[10]] => 001010111 => [3,2,2,1,1,1]
[3,2,1,1,1,1,1] => [[1,2,3],[4,5],[6],[7],[8],[9],[10]] => 001011111 => [3,2,1,1,1,1,1]
[3,1,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9],[10]] => 001111111 => [3,1,1,1,1,1,1,1]
[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] => [[1,2],[3,4],[5,6],[7,8],[9],[10]] => 010101011 => [2,2,2,2,1,1]
[2,2,2,1,1,1,1] => [[1,2],[3,4],[5,6],[7],[8],[9],[10]] => 010101111 => [2,2,2,1,1,1,1]
[2,2,1,1,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8],[9],[10]] => 010111111 => [2,2,1,1,1,1,1,1]
[2,1,1,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]] => 011111111 => [2,1,1,1,1,1,1,1,1]
[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]
[5,3,3] => [[1,2,3,4,5],[6,7,8],[9,10,11]] => 0000100100 => [5,3,3]
[5,3,2,1] => [[1,2,3,4,5],[6,7,8],[9,10],[11]] => 0000100101 => [5,3,2,1]
[5,3,1,1,1] => [[1,2,3,4,5],[6,7,8],[9],[10],[11]] => 0000100111 => [5,3,1,1,1]
[5,2,2,2] => [[1,2,3,4,5],[6,7],[8,9],[10,11]] => 0000101010 => [5,2,2,2]
[5,2,2,1,1] => [[1,2,3,4,5],[6,7],[8,9],[10],[11]] => 0000101011 => [5,2,2,1,1]
[4,4,3] => [[1,2,3,4],[5,6,7,8],[9,10,11]] => 0001000100 => [4,4,3]
[4,3,2,2] => [[1,2,3,4],[5,6,7],[8,9],[10,11]] => 0001001010 => [4,3,2,2]
[4,3,2,1,1] => [[1,2,3,4],[5,6,7],[8,9],[10],[11]] => 0001001011 => [4,3,2,1,1]
[4,2,2,2,1] => [[1,2,3,4],[5,6],[7,8],[9,10],[11]] => 0001010101 => [4,2,2,2,1]
[3,3,3,1,1] => [[1,2,3],[4,5,6],[7,8,9],[10],[11]] => 0010010011 => [3,3,3,1,1]
[3,3,2,2,1] => [[1,2,3],[4,5,6],[7,8],[9,10],[11]] => 0010010101 => [3,3,2,2,1]
[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] => [[1,2,3,4,5,6],[7,8,9,10],[11,12]] => 00000100010 => [6,4,2]
[5,4,3] => [[1,2,3,4,5],[6,7,8,9],[10,11,12]] => 00001000100 => [5,4,3]
[5,2,2,2,1] => [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]] => 00001010101 => [5,2,2,2,1]
[4,4,3,1] => [[1,2,3,4],[5,6,7,8],[9,10,11],[12]] => 00010001001 => [4,4,3,1]
[4,4,2,2] => [[1,2,3,4],[5,6,7,8],[9,10],[11,12]] => 00010001010 => [4,4,2,2]
[4,4,2,1,1] => [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]] => 00010001011 => [4,4,2,1,1]
[4,3,3,2] => [[1,2,3,4],[5,6,7],[8,9,10],[11,12]] => 00010010010 => [4,3,3,2]
[4,3,2,2,1] => [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]] => 00010010101 => [4,3,2,2,1]
[4,2,2,2,2] => [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]] => 00010101010 => [4,2,2,2,2]
[3,3,3,2,1] => [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]] => 00100100101 => [3,3,3,2,1]
[3,3,2,2,1,1] => [[1,2,3],[4,5,6],[7,8],[9,10],[11],[12]] => 00100101011 => [3,3,2,2,1,1]
[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] => [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]] => 000010010011 => [5,3,3,1,1]
[5,3,2,2,1] => [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]] => 000010010101 => [5,3,2,2,1]
[4,4,2,2,1] => [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]] => 000100010101 => [4,4,2,2,1]
[] => [] => => [1]
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
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.