Identifier
Values
[2] => [[1,2]] => [[1],[2]] => 1 => 1
[1,1] => [[1],[2]] => [[1,2]] => 0 => 0
[3] => [[1,2,3]] => [[1],[2],[3]] => 11 => 3
[2,1] => [[1,3],[2]] => [[1,2],[3]] => 01 => 2
[1,1,1] => [[1],[2],[3]] => [[1,2,3]] => 00 => 0
[4] => [[1,2,3,4]] => [[1],[2],[3],[4]] => 111 => 6
[3,1] => [[1,3,4],[2]] => [[1,2],[3],[4]] => 011 => 5
[2,2] => [[1,2],[3,4]] => [[1,3],[2,4]] => 101 => 4
[2,1,1] => [[1,4],[2],[3]] => [[1,2,3],[4]] => 001 => 3
[1,1,1,1] => [[1],[2],[3],[4]] => [[1,2,3,4]] => 000 => 0
[5] => [[1,2,3,4,5]] => [[1],[2],[3],[4],[5]] => 1111 => 10
[4,1] => [[1,3,4,5],[2]] => [[1,2],[3],[4],[5]] => 0111 => 9
[3,2] => [[1,2,5],[3,4]] => [[1,3],[2,4],[5]] => 1011 => 8
[3,1,1] => [[1,4,5],[2],[3]] => [[1,2,3],[4],[5]] => 0011 => 7
[2,2,1] => [[1,3],[2,5],[4]] => [[1,2,4],[3,5]] => 0101 => 6
[2,1,1,1] => [[1,5],[2],[3],[4]] => [[1,2,3,4],[5]] => 0001 => 4
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [[1,2,3,4,5]] => 0000 => 0
[6] => [[1,2,3,4,5,6]] => [[1],[2],[3],[4],[5],[6]] => 11111 => 15
[5,1] => [[1,3,4,5,6],[2]] => [[1,2],[3],[4],[5],[6]] => 01111 => 14
[4,2] => [[1,2,5,6],[3,4]] => [[1,3],[2,4],[5],[6]] => 10111 => 13
[4,1,1] => [[1,4,5,6],[2],[3]] => [[1,2,3],[4],[5],[6]] => 00111 => 12
[3,3] => [[1,2,3],[4,5,6]] => [[1,4],[2,5],[3,6]] => 11011 => 12
[3,2,1] => [[1,3,6],[2,5],[4]] => [[1,2,4],[3,5],[6]] => 01011 => 11
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => [[1,2,3,4],[5],[6]] => 00011 => 9
[2,2,2] => [[1,2],[3,4],[5,6]] => [[1,3,5],[2,4,6]] => 10101 => 9
[2,2,1,1] => [[1,4],[2,6],[3],[5]] => [[1,2,3,5],[4,6]] => 00101 => 8
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [[1,2,3,4,5],[6]] => 00001 => 5
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [[1,2,3,4,5,6]] => 00000 => 0
[7] => [[1,2,3,4,5,6,7]] => [[1],[2],[3],[4],[5],[6],[7]] => 111111 => 21
[6,1] => [[1,3,4,5,6,7],[2]] => [[1,2],[3],[4],[5],[6],[7]] => 011111 => 20
[5,2] => [[1,2,5,6,7],[3,4]] => [[1,3],[2,4],[5],[6],[7]] => 101111 => 19
[5,1,1] => [[1,4,5,6,7],[2],[3]] => [[1,2,3],[4],[5],[6],[7]] => 001111 => 18
[4,3] => [[1,2,3,7],[4,5,6]] => [[1,4],[2,5],[3,6],[7]] => 110111 => 18
[4,2,1] => [[1,3,6,7],[2,5],[4]] => [[1,2,4],[3,5],[6],[7]] => 010111 => 17
[4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => [[1,2,3,4],[5],[6],[7]] => 000111 => 15
[3,3,1] => [[1,3,4],[2,6,7],[5]] => [[1,2,5],[3,6],[4,7]] => 011011 => 16
[3,2,2] => [[1,2,7],[3,4],[5,6]] => [[1,3,5],[2,4,6],[7]] => 101011 => 15
[3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => [[1,2,3,5],[4,6],[7]] => 001011 => 14
[3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => [[1,2,3,4,5],[6],[7]] => 000011 => 11
[2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => [[1,2,4,6],[3,5,7]] => 010101 => 12
[2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => [[1,2,3,4,6],[5,7]] => 000101 => 10
[2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => [[1,2,3,4,5,6],[7]] => 000001 => 6
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [[1,2,3,4,5,6,7]] => 000000 => 0
[8] => [[1,2,3,4,5,6,7,8]] => [[1],[2],[3],[4],[5],[6],[7],[8]] => 1111111 => 28
[7,1] => [[1,3,4,5,6,7,8],[2]] => [[1,2],[3],[4],[5],[6],[7],[8]] => 0111111 => 27
[6,2] => [[1,2,5,6,7,8],[3,4]] => [[1,3],[2,4],[5],[6],[7],[8]] => 1011111 => 26
[6,1,1] => [[1,4,5,6,7,8],[2],[3]] => [[1,2,3],[4],[5],[6],[7],[8]] => 0011111 => 25
[5,3] => [[1,2,3,7,8],[4,5,6]] => [[1,4],[2,5],[3,6],[7],[8]] => 1101111 => 25
[5,2,1] => [[1,3,6,7,8],[2,5],[4]] => [[1,2,4],[3,5],[6],[7],[8]] => 0101111 => 24
[5,1,1,1] => [[1,5,6,7,8],[2],[3],[4]] => [[1,2,3,4],[5],[6],[7],[8]] => 0001111 => 22
[4,4] => [[1,2,3,4],[5,6,7,8]] => [[1,5],[2,6],[3,7],[4,8]] => 1110111 => 24
[4,3,1] => [[1,3,4,8],[2,6,7],[5]] => [[1,2,5],[3,6],[4,7],[8]] => 0110111 => 23
[4,2,2] => [[1,2,7,8],[3,4],[5,6]] => [[1,3,5],[2,4,6],[7],[8]] => 1010111 => 22
[4,2,1,1] => [[1,4,7,8],[2,6],[3],[5]] => [[1,2,3,5],[4,6],[7],[8]] => 0010111 => 21
[4,1,1,1,1] => [[1,6,7,8],[2],[3],[4],[5]] => [[1,2,3,4,5],[6],[7],[8]] => 0000111 => 18
[3,3,2] => [[1,2,5],[3,4,8],[6,7]] => [[1,3,6],[2,4,7],[5,8]] => 1011011 => 21
[3,3,1,1] => [[1,4,5],[2,7,8],[3],[6]] => [[1,2,3,6],[4,7],[5,8]] => 0011011 => 20
[3,2,2,1] => [[1,3,8],[2,5],[4,7],[6]] => [[1,2,4,6],[3,5,7],[8]] => 0101011 => 19
[3,2,1,1,1] => [[1,5,8],[2,7],[3],[4],[6]] => [[1,2,3,4,6],[5,7],[8]] => 0001011 => 17
[3,1,1,1,1,1] => [[1,7,8],[2],[3],[4],[5],[6]] => [[1,2,3,4,5,6],[7],[8]] => 0000011 => 13
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [[1,3,5,7],[2,4,6,8]] => 1010101 => 16
[2,2,2,1,1] => [[1,4],[2,6],[3,8],[5],[7]] => [[1,2,3,5,7],[4,6,8]] => 0010101 => 15
[2,2,1,1,1,1] => [[1,6],[2,8],[3],[4],[5],[7]] => [[1,2,3,4,5,7],[6,8]] => 0000101 => 12
[2,1,1,1,1,1,1] => [[1,8],[2],[3],[4],[5],[6],[7]] => [[1,2,3,4,5,6,7],[8]] => 0000001 => 7
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [[1,2,3,4,5,6,7,8]] => 0000000 => 0
[9] => [[1,2,3,4,5,6,7,8,9]] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => 11111111 => 36
[8,1] => [[1,3,4,5,6,7,8,9],[2]] => [[1,2],[3],[4],[5],[6],[7],[8],[9]] => 01111111 => 35
[7,2] => [[1,2,5,6,7,8,9],[3,4]] => [[1,3],[2,4],[5],[6],[7],[8],[9]] => 10111111 => 34
[7,1,1] => [[1,4,5,6,7,8,9],[2],[3]] => [[1,2,3],[4],[5],[6],[7],[8],[9]] => 00111111 => 33
[6,3] => [[1,2,3,7,8,9],[4,5,6]] => [[1,4],[2,5],[3,6],[7],[8],[9]] => 11011111 => 33
[6,2,1] => [[1,3,6,7,8,9],[2,5],[4]] => [[1,2,4],[3,5],[6],[7],[8],[9]] => 01011111 => 32
[6,1,1,1] => [[1,5,6,7,8,9],[2],[3],[4]] => [[1,2,3,4],[5],[6],[7],[8],[9]] => 00011111 => 30
[5,4] => [[1,2,3,4,9],[5,6,7,8]] => [[1,5],[2,6],[3,7],[4,8],[9]] => 11101111 => 32
[5,3,1] => [[1,3,4,8,9],[2,6,7],[5]] => [[1,2,5],[3,6],[4,7],[8],[9]] => 01101111 => 31
[5,2,2] => [[1,2,7,8,9],[3,4],[5,6]] => [[1,3,5],[2,4,6],[7],[8],[9]] => 10101111 => 30
[5,2,1,1] => [[1,4,7,8,9],[2,6],[3],[5]] => [[1,2,3,5],[4,6],[7],[8],[9]] => 00101111 => 29
[5,1,1,1,1] => [[1,6,7,8,9],[2],[3],[4],[5]] => [[1,2,3,4,5],[6],[7],[8],[9]] => 00001111 => 26
[4,4,1] => [[1,3,4,5],[2,7,8,9],[6]] => [[1,2,6],[3,7],[4,8],[5,9]] => 01110111 => 30
[4,3,2] => [[1,2,5,9],[3,4,8],[6,7]] => [[1,3,6],[2,4,7],[5,8],[9]] => 10110111 => 29
[4,3,1,1] => [[1,4,5,9],[2,7,8],[3],[6]] => [[1,2,3,6],[4,7],[5,8],[9]] => 00110111 => 28
[4,2,2,1] => [[1,3,8,9],[2,5],[4,7],[6]] => [[1,2,4,6],[3,5,7],[8],[9]] => 01010111 => 27
[4,2,1,1,1] => [[1,5,8,9],[2,7],[3],[4],[6]] => [[1,2,3,4,6],[5,7],[8],[9]] => 00010111 => 25
[4,1,1,1,1,1] => [[1,7,8,9],[2],[3],[4],[5],[6]] => [[1,2,3,4,5,6],[7],[8],[9]] => 00000111 => 21
[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => [[1,4,7],[2,5,8],[3,6,9]] => 11011011 => 27
[3,3,2,1] => [[1,3,6],[2,5,9],[4,8],[7]] => [[1,2,4,7],[3,5,8],[6,9]] => 01011011 => 26
[3,3,1,1,1] => [[1,5,6],[2,8,9],[3],[4],[7]] => [[1,2,3,4,7],[5,8],[6,9]] => 00011011 => 24
[3,2,2,2] => [[1,2,9],[3,4],[5,6],[7,8]] => [[1,3,5,7],[2,4,6,8],[9]] => 10101011 => 24
[3,2,2,1,1] => [[1,4,9],[2,6],[3,8],[5],[7]] => [[1,2,3,5,7],[4,6,8],[9]] => 00101011 => 23
[3,2,1,1,1,1] => [[1,6,9],[2,8],[3],[4],[5],[7]] => [[1,2,3,4,5,7],[6,8],[9]] => 00001011 => 20
[3,1,1,1,1,1,1] => [[1,8,9],[2],[3],[4],[5],[6],[7]] => [[1,2,3,4,5,6,7],[8],[9]] => 00000011 => 15
[2,2,2,2,1] => [[1,3],[2,5],[4,7],[6,9],[8]] => [[1,2,4,6,8],[3,5,7,9]] => 01010101 => 20
[2,2,2,1,1,1] => [[1,5],[2,7],[3,9],[4],[6],[8]] => [[1,2,3,4,6,8],[5,7,9]] => 00010101 => 18
[2,2,1,1,1,1,1] => [[1,7],[2,9],[3],[4],[5],[6],[8]] => [[1,2,3,4,5,6,8],[7,9]] => 00000101 => 14
[2,1,1,1,1,1,1,1] => [[1,9],[2],[3],[4],[5],[6],[7],[8]] => [[1,2,3,4,5,6,7,8],[9]] => 00000001 => 8
[1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => [[1,2,3,4,5,6,7,8,9]] => 00000000 => 0
[10] => [[1,2,3,4,5,6,7,8,9,10]] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => 111111111 => 45
[9,1] => [[1,3,4,5,6,7,8,9,10],[2]] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]] => 011111111 => 44
[8,2] => [[1,2,5,6,7,8,9,10],[3,4]] => [[1,3],[2,4],[5],[6],[7],[8],[9],[10]] => 101111111 => 43
[8,1,1] => [[1,4,5,6,7,8,9,10],[2],[3]] => [[1,2,3],[4],[5],[6],[7],[8],[9],[10]] => 001111111 => 42
[7,3] => [[1,2,3,7,8,9,10],[4,5,6]] => [[1,4],[2,5],[3,6],[7],[8],[9],[10]] => 110111111 => 42
[7,2,1] => [[1,3,6,7,8,9,10],[2,5],[4]] => [[1,2,4],[3,5],[6],[7],[8],[9],[10]] => 010111111 => 41
>>> Load all 140 entries. <<<
[7,1,1,1] => [[1,5,6,7,8,9,10],[2],[3],[4]] => [[1,2,3,4],[5],[6],[7],[8],[9],[10]] => 000111111 => 39
[6,4] => [[1,2,3,4,9,10],[5,6,7,8]] => [[1,5],[2,6],[3,7],[4,8],[9],[10]] => 111011111 => 41
[6,3,1] => [[1,3,4,8,9,10],[2,6,7],[5]] => [[1,2,5],[3,6],[4,7],[8],[9],[10]] => 011011111 => 40
[6,2,2] => [[1,2,7,8,9,10],[3,4],[5,6]] => [[1,3,5],[2,4,6],[7],[8],[9],[10]] => 101011111 => 39
[6,2,1,1] => [[1,4,7,8,9,10],[2,6],[3],[5]] => [[1,2,3,5],[4,6],[7],[8],[9],[10]] => 001011111 => 38
[6,1,1,1,1] => [[1,6,7,8,9,10],[2],[3],[4],[5]] => [[1,2,3,4,5],[6],[7],[8],[9],[10]] => 000011111 => 35
[5,5] => [[1,2,3,4,5],[6,7,8,9,10]] => [[1,6],[2,7],[3,8],[4,9],[5,10]] => 111101111 => 40
[5,4,1] => [[1,3,4,5,10],[2,7,8,9],[6]] => [[1,2,6],[3,7],[4,8],[5,9],[10]] => 011101111 => 39
[5,3,2] => [[1,2,5,9,10],[3,4,8],[6,7]] => [[1,3,6],[2,4,7],[5,8],[9],[10]] => 101101111 => 38
[5,3,1,1] => [[1,4,5,9,10],[2,7,8],[3],[6]] => [[1,2,3,6],[4,7],[5,8],[9],[10]] => 001101111 => 37
[5,2,2,1] => [[1,3,8,9,10],[2,5],[4,7],[6]] => [[1,2,4,6],[3,5,7],[8],[9],[10]] => 010101111 => 36
[5,2,1,1,1] => [[1,5,8,9,10],[2,7],[3],[4],[6]] => [[1,2,3,4,6],[5,7],[8],[9],[10]] => 000101111 => 34
[5,1,1,1,1,1] => [[1,7,8,9,10],[2],[3],[4],[5],[6]] => [[1,2,3,4,5,6],[7],[8],[9],[10]] => 000001111 => 30
[4,4,2] => [[1,2,5,6],[3,4,9,10],[7,8]] => [[1,3,7],[2,4,8],[5,9],[6,10]] => 101110111 => 37
[4,4,1,1] => [[1,4,5,6],[2,8,9,10],[3],[7]] => [[1,2,3,7],[4,8],[5,9],[6,10]] => 001110111 => 36
[4,3,3] => [[1,2,3,10],[4,5,6],[7,8,9]] => [[1,4,7],[2,5,8],[3,6,9],[10]] => 110110111 => 36
[4,3,2,1] => [[1,3,6,10],[2,5,9],[4,8],[7]] => [[1,2,4,7],[3,5,8],[6,9],[10]] => 010110111 => 35
[4,3,1,1,1] => [[1,5,6,10],[2,8,9],[3],[4],[7]] => [[1,2,3,4,7],[5,8],[6,9],[10]] => 000110111 => 33
[4,2,2,2] => [[1,2,9,10],[3,4],[5,6],[7,8]] => [[1,3,5,7],[2,4,6,8],[9],[10]] => 101010111 => 33
[4,2,2,1,1] => [[1,4,9,10],[2,6],[3,8],[5],[7]] => [[1,2,3,5,7],[4,6,8],[9],[10]] => 001010111 => 32
[4,2,1,1,1,1] => [[1,6,9,10],[2,8],[3],[4],[5],[7]] => [[1,2,3,4,5,7],[6,8],[9],[10]] => 000010111 => 29
[4,1,1,1,1,1,1] => [[1,8,9,10],[2],[3],[4],[5],[6],[7]] => [[1,2,3,4,5,6,7],[8],[9],[10]] => 000000111 => 24
[3,3,3,1] => [[1,3,4],[2,6,7],[5,9,10],[8]] => [[1,2,5,8],[3,6,9],[4,7,10]] => 011011011 => 33
[3,3,2,2] => [[1,2,7],[3,4,10],[5,6],[8,9]] => [[1,3,5,8],[2,4,6,9],[7,10]] => 101011011 => 32
[3,3,2,1,1] => [[1,4,7],[2,6,10],[3,9],[5],[8]] => [[1,2,3,5,8],[4,6,9],[7,10]] => 001011011 => 31
[3,3,1,1,1,1] => [[1,6,7],[2,9,10],[3],[4],[5],[8]] => [[1,2,3,4,5,8],[6,9],[7,10]] => 000011011 => 28
[3,2,2,2,1] => [[1,3,10],[2,5],[4,7],[6,9],[8]] => [[1,2,4,6,8],[3,5,7,9],[10]] => 010101011 => 29
[3,2,2,1,1,1] => [[1,5,10],[2,7],[3,9],[4],[6],[8]] => [[1,2,3,4,6,8],[5,7,9],[10]] => 000101011 => 27
[3,2,1,1,1,1,1] => [[1,7,10],[2,9],[3],[4],[5],[6],[8]] => [[1,2,3,4,5,6,8],[7,9],[10]] => 000001011 => 23
[3,1,1,1,1,1,1,1] => [[1,9,10],[2],[3],[4],[5],[6],[7],[8]] => [[1,2,3,4,5,6,7,8],[9],[10]] => 000000011 => 17
[2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => [[1,3,5,7,9],[2,4,6,8,10]] => 101010101 => 25
[2,2,2,2,1,1] => [[1,4],[2,6],[3,8],[5,10],[7],[9]] => [[1,2,3,5,7,9],[4,6,8,10]] => 001010101 => 24
[2,2,2,1,1,1,1] => [[1,6],[2,8],[3,10],[4],[5],[7],[9]] => [[1,2,3,4,5,7,9],[6,8,10]] => 000010101 => 21
[2,2,1,1,1,1,1,1] => [[1,8],[2,10],[3],[4],[5],[6],[7],[9]] => [[1,2,3,4,5,6,7,9],[8,10]] => 000000101 => 16
[2,1,1,1,1,1,1,1,1] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]] => [[1,2,3,4,5,6,7,8,9],[10]] => 000000001 => 9
[1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => [[1,2,3,4,5,6,7,8,9,10]] => 000000000 => 0
[5,3,1,1,1] => [[1,5,6,10,11],[2,8,9],[3],[4],[7]] => [[1,2,3,4,7],[5,8],[6,9],[10],[11]] => 0001101111 => 43
[5,2,2,1,1] => [[1,4,9,10,11],[2,6],[3,8],[5],[7]] => [[1,2,3,5,7],[4,6,8],[9],[10],[11]] => 0010101111 => 42
[4,4,1,1,1] => [[1,5,6,7],[2,9,10,11],[3],[4],[8]] => [[1,2,3,4,8],[5,9],[6,10],[7,11]] => 0001110111 => 42
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Description
The sum of the positions of the ones in a binary word.
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
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
conjugate
Description
Sends a standard tableau to its conjugate tableau.