Identifier
Values
[1] => [[1]] => [[1]] => 0
[2] => [[1,2]] => [[1],[2]] => 1
[1,1] => [[1],[2]] => [[1,2]] => 0
[3] => [[1,2,3]] => [[1],[2],[3]] => 3
[2,1] => [[1,2],[3]] => [[1,3],[2]] => 2
[1,1,1] => [[1],[2],[3]] => [[1,2,3]] => 0
[4] => [[1,2,3,4]] => [[1],[2],[3],[4]] => 6
[3,1] => [[1,2,3],[4]] => [[1,4],[2],[3]] => 5
[2,2] => [[1,2],[3,4]] => [[1,3],[2,4]] => 4
[2,1,1] => [[1,2],[3],[4]] => [[1,3,4],[2]] => 3
[1,1,1,1] => [[1],[2],[3],[4]] => [[1,2,3,4]] => 0
[5] => [[1,2,3,4,5]] => [[1],[2],[3],[4],[5]] => 10
[4,1] => [[1,2,3,4],[5]] => [[1,5],[2],[3],[4]] => 9
[3,2] => [[1,2,3],[4,5]] => [[1,4],[2,5],[3]] => 8
[3,1,1] => [[1,2,3],[4],[5]] => [[1,4,5],[2],[3]] => 7
[2,2,1] => [[1,2],[3,4],[5]] => [[1,3,5],[2,4]] => 6
[2,1,1,1] => [[1,2],[3],[4],[5]] => [[1,3,4,5],[2]] => 4
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [[1,2,3,4,5]] => 0
[6] => [[1,2,3,4,5,6]] => [[1],[2],[3],[4],[5],[6]] => 15
[5,1] => [[1,2,3,4,5],[6]] => [[1,6],[2],[3],[4],[5]] => 14
[4,2] => [[1,2,3,4],[5,6]] => [[1,5],[2,6],[3],[4]] => 13
[4,1,1] => [[1,2,3,4],[5],[6]] => [[1,5,6],[2],[3],[4]] => 12
[3,3] => [[1,2,3],[4,5,6]] => [[1,4],[2,5],[3,6]] => 12
[3,2,1] => [[1,2,3],[4,5],[6]] => [[1,4,6],[2,5],[3]] => 11
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [[1,4,5,6],[2],[3]] => 9
[2,2,2] => [[1,2],[3,4],[5,6]] => [[1,3,5],[2,4,6]] => 9
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [[1,3,5,6],[2,4]] => 8
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [[1,3,4,5,6],[2]] => 5
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [[1,2,3,4,5,6]] => 0
[7] => [[1,2,3,4,5,6,7]] => [[1],[2],[3],[4],[5],[6],[7]] => 21
[6,1] => [[1,2,3,4,5,6],[7]] => [[1,7],[2],[3],[4],[5],[6]] => 20
[5,2] => [[1,2,3,4,5],[6,7]] => [[1,6],[2,7],[3],[4],[5]] => 19
[5,1,1] => [[1,2,3,4,5],[6],[7]] => [[1,6,7],[2],[3],[4],[5]] => 18
[4,3] => [[1,2,3,4],[5,6,7]] => [[1,5],[2,6],[3,7],[4]] => 18
[4,2,1] => [[1,2,3,4],[5,6],[7]] => [[1,5,7],[2,6],[3],[4]] => 17
[4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [[1,5,6,7],[2],[3],[4]] => 15
[3,3,1] => [[1,2,3],[4,5,6],[7]] => [[1,4,7],[2,5],[3,6]] => 16
[3,2,2] => [[1,2,3],[4,5],[6,7]] => [[1,4,6],[2,5,7],[3]] => 15
[3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => [[1,4,6,7],[2,5],[3]] => 14
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [[1,4,5,6,7],[2],[3]] => 11
[2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [[1,3,5,7],[2,4,6]] => 12
[2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [[1,3,5,6,7],[2,4]] => 10
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [[1,3,4,5,6,7],[2]] => 6
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [[1,2,3,4,5,6,7]] => 0
[8] => [[1,2,3,4,5,6,7,8]] => [[1],[2],[3],[4],[5],[6],[7],[8]] => 28
[7,1] => [[1,2,3,4,5,6,7],[8]] => [[1,8],[2],[3],[4],[5],[6],[7]] => 27
[6,2] => [[1,2,3,4,5,6],[7,8]] => [[1,7],[2,8],[3],[4],[5],[6]] => 26
[6,1,1] => [[1,2,3,4,5,6],[7],[8]] => [[1,7,8],[2],[3],[4],[5],[6]] => 25
[5,3] => [[1,2,3,4,5],[6,7,8]] => [[1,6],[2,7],[3,8],[4],[5]] => 25
[5,2,1] => [[1,2,3,4,5],[6,7],[8]] => [[1,6,8],[2,7],[3],[4],[5]] => 24
[5,1,1,1] => [[1,2,3,4,5],[6],[7],[8]] => [[1,6,7,8],[2],[3],[4],[5]] => 22
[4,4] => [[1,2,3,4],[5,6,7,8]] => [[1,5],[2,6],[3,7],[4,8]] => 24
[4,3,1] => [[1,2,3,4],[5,6,7],[8]] => [[1,5,8],[2,6],[3,7],[4]] => 23
[4,2,2] => [[1,2,3,4],[5,6],[7,8]] => [[1,5,7],[2,6,8],[3],[4]] => 22
[4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => [[1,5,7,8],[2,6],[3],[4]] => 21
[4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => [[1,5,6,7,8],[2],[3],[4]] => 18
[3,3,2] => [[1,2,3],[4,5,6],[7,8]] => [[1,4,7],[2,5,8],[3,6]] => 21
[3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => [[1,4,7,8],[2,5],[3,6]] => 20
[3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => [[1,4,6,8],[2,5,7],[3]] => 19
[3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => [[1,4,6,7,8],[2,5],[3]] => 17
[3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => [[1,4,5,6,7,8],[2],[3]] => 13
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [[1,3,5,7],[2,4,6,8]] => 16
[2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => [[1,3,5,7,8],[2,4,6]] => 15
[2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => [[1,3,5,6,7,8],[2,4]] => 12
[2,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8]] => [[1,3,4,5,6,7,8],[2]] => 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]] => 0
[9] => [[1,2,3,4,5,6,7,8,9]] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => 36
[8,1] => [[1,2,3,4,5,6,7,8],[9]] => [[1,9],[2],[3],[4],[5],[6],[7],[8]] => 35
[7,2] => [[1,2,3,4,5,6,7],[8,9]] => [[1,8],[2,9],[3],[4],[5],[6],[7]] => 34
[7,1,1] => [[1,2,3,4,5,6,7],[8],[9]] => [[1,8,9],[2],[3],[4],[5],[6],[7]] => 33
[6,3] => [[1,2,3,4,5,6],[7,8,9]] => [[1,7],[2,8],[3,9],[4],[5],[6]] => 33
[6,2,1] => [[1,2,3,4,5,6],[7,8],[9]] => [[1,7,9],[2,8],[3],[4],[5],[6]] => 32
[6,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9]] => [[1,7,8,9],[2],[3],[4],[5],[6]] => 30
[5,4] => [[1,2,3,4,5],[6,7,8,9]] => [[1,6],[2,7],[3,8],[4,9],[5]] => 32
[5,3,1] => [[1,2,3,4,5],[6,7,8],[9]] => [[1,6,9],[2,7],[3,8],[4],[5]] => 31
[5,2,2] => [[1,2,3,4,5],[6,7],[8,9]] => [[1,6,8],[2,7,9],[3],[4],[5]] => 30
[5,2,1,1] => [[1,2,3,4,5],[6,7],[8],[9]] => [[1,6,8,9],[2,7],[3],[4],[5]] => 29
[5,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9]] => [[1,6,7,8,9],[2],[3],[4],[5]] => 26
[4,4,1] => [[1,2,3,4],[5,6,7,8],[9]] => [[1,5,9],[2,6],[3,7],[4,8]] => 30
[4,3,2] => [[1,2,3,4],[5,6,7],[8,9]] => [[1,5,8],[2,6,9],[3,7],[4]] => 29
[4,3,1,1] => [[1,2,3,4],[5,6,7],[8],[9]] => [[1,5,8,9],[2,6],[3,7],[4]] => 28
[4,2,2,1] => [[1,2,3,4],[5,6],[7,8],[9]] => [[1,5,7,9],[2,6,8],[3],[4]] => 27
[4,2,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9]] => [[1,5,7,8,9],[2,6],[3],[4]] => 25
[4,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9]] => [[1,5,6,7,8,9],[2],[3],[4]] => 21
[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => [[1,4,7],[2,5,8],[3,6,9]] => 27
[3,3,2,1] => [[1,2,3],[4,5,6],[7,8],[9]] => [[1,4,7,9],[2,5,8],[3,6]] => 26
[3,3,1,1,1] => [[1,2,3],[4,5,6],[7],[8],[9]] => [[1,4,7,8,9],[2,5],[3,6]] => 24
[3,2,2,2] => [[1,2,3],[4,5],[6,7],[8,9]] => [[1,4,6,8],[2,5,7,9],[3]] => 24
[3,2,2,1,1] => [[1,2,3],[4,5],[6,7],[8],[9]] => [[1,4,6,8,9],[2,5,7],[3]] => 23
[3,2,1,1,1,1] => [[1,2,3],[4,5],[6],[7],[8],[9]] => [[1,4,6,7,8,9],[2,5],[3]] => 20
[3,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9]] => [[1,4,5,6,7,8,9],[2],[3]] => 15
[2,2,2,2,1] => [[1,2],[3,4],[5,6],[7,8],[9]] => [[1,3,5,7,9],[2,4,6,8]] => 20
[2,2,2,1,1,1] => [[1,2],[3,4],[5,6],[7],[8],[9]] => [[1,3,5,7,8,9],[2,4,6]] => 18
[2,2,1,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8],[9]] => [[1,3,5,6,7,8,9],[2,4]] => 14
[2,1,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9]] => [[1,3,4,5,6,7,8,9],[2]] => 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]] => 0
[10] => [[1,2,3,4,5,6,7,8,9,10]] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => 45
[9,1] => [[1,2,3,4,5,6,7,8,9],[10]] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]] => 44
[8,2] => [[1,2,3,4,5,6,7,8],[9,10]] => [[1,9],[2,10],[3],[4],[5],[6],[7],[8]] => 43
[8,1,1] => [[1,2,3,4,5,6,7,8],[9],[10]] => [[1,9,10],[2],[3],[4],[5],[6],[7],[8]] => 42
[7,3] => [[1,2,3,4,5,6,7],[8,9,10]] => [[1,8],[2,9],[3,10],[4],[5],[6],[7]] => 42
>>> Load all 142 entries. <<<
[7,2,1] => [[1,2,3,4,5,6,7],[8,9],[10]] => [[1,8,10],[2,9],[3],[4],[5],[6],[7]] => 41
[7,1,1,1] => [[1,2,3,4,5,6,7],[8],[9],[10]] => [[1,8,9,10],[2],[3],[4],[5],[6],[7]] => 39
[6,4] => [[1,2,3,4,5,6],[7,8,9,10]] => [[1,7],[2,8],[3,9],[4,10],[5],[6]] => 41
[6,3,1] => [[1,2,3,4,5,6],[7,8,9],[10]] => [[1,7,10],[2,8],[3,9],[4],[5],[6]] => 40
[6,2,2] => [[1,2,3,4,5,6],[7,8],[9,10]] => [[1,7,9],[2,8,10],[3],[4],[5],[6]] => 39
[6,2,1,1] => [[1,2,3,4,5,6],[7,8],[9],[10]] => [[1,7,9,10],[2,8],[3],[4],[5],[6]] => 38
[6,1,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9],[10]] => [[1,7,8,9,10],[2],[3],[4],[5],[6]] => 35
[5,5] => [[1,2,3,4,5],[6,7,8,9,10]] => [[1,6],[2,7],[3,8],[4,9],[5,10]] => 40
[5,4,1] => [[1,2,3,4,5],[6,7,8,9],[10]] => [[1,6,10],[2,7],[3,8],[4,9],[5]] => 39
[5,3,2] => [[1,2,3,4,5],[6,7,8],[9,10]] => [[1,6,9],[2,7,10],[3,8],[4],[5]] => 38
[5,3,1,1] => [[1,2,3,4,5],[6,7,8],[9],[10]] => [[1,6,9,10],[2,7],[3,8],[4],[5]] => 37
[5,2,2,1] => [[1,2,3,4,5],[6,7],[8,9],[10]] => [[1,6,8,10],[2,7,9],[3],[4],[5]] => 36
[5,2,1,1,1] => [[1,2,3,4,5],[6,7],[8],[9],[10]] => [[1,6,8,9,10],[2,7],[3],[4],[5]] => 34
[5,1,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9],[10]] => [[1,6,7,8,9,10],[2],[3],[4],[5]] => 30
[4,4,2] => [[1,2,3,4],[5,6,7,8],[9,10]] => [[1,5,9],[2,6,10],[3,7],[4,8]] => 37
[4,4,1,1] => [[1,2,3,4],[5,6,7,8],[9],[10]] => [[1,5,9,10],[2,6],[3,7],[4,8]] => 36
[4,3,3] => [[1,2,3,4],[5,6,7],[8,9,10]] => [[1,5,8],[2,6,9],[3,7,10],[4]] => 36
[4,3,2,1] => [[1,2,3,4],[5,6,7],[8,9],[10]] => [[1,5,8,10],[2,6,9],[3,7],[4]] => 35
[4,3,1,1,1] => [[1,2,3,4],[5,6,7],[8],[9],[10]] => [[1,5,8,9,10],[2,6],[3,7],[4]] => 33
[4,2,2,2] => [[1,2,3,4],[5,6],[7,8],[9,10]] => [[1,5,7,9],[2,6,8,10],[3],[4]] => 33
[4,2,2,1,1] => [[1,2,3,4],[5,6],[7,8],[9],[10]] => [[1,5,7,9,10],[2,6,8],[3],[4]] => 32
[4,2,1,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9],[10]] => [[1,5,7,8,9,10],[2,6],[3],[4]] => 29
[4,1,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9],[10]] => [[1,5,6,7,8,9,10],[2],[3],[4]] => 24
[3,3,3,1] => [[1,2,3],[4,5,6],[7,8,9],[10]] => [[1,4,7,10],[2,5,8],[3,6,9]] => 33
[3,3,2,2] => [[1,2,3],[4,5,6],[7,8],[9,10]] => [[1,4,7,9],[2,5,8,10],[3,6]] => 32
[3,3,2,1,1] => [[1,2,3],[4,5,6],[7,8],[9],[10]] => [[1,4,7,9,10],[2,5,8],[3,6]] => 31
[3,3,1,1,1,1] => [[1,2,3],[4,5,6],[7],[8],[9],[10]] => [[1,4,7,8,9,10],[2,5],[3,6]] => 28
[3,2,2,2,1] => [[1,2,3],[4,5],[6,7],[8,9],[10]] => [[1,4,6,8,10],[2,5,7,9],[3]] => 29
[3,2,2,1,1,1] => [[1,2,3],[4,5],[6,7],[8],[9],[10]] => [[1,4,6,8,9,10],[2,5,7],[3]] => 27
[3,2,1,1,1,1,1] => [[1,2,3],[4,5],[6],[7],[8],[9],[10]] => [[1,4,6,7,8,9,10],[2,5],[3]] => 23
[3,1,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9],[10]] => [[1,4,5,6,7,8,9,10],[2],[3]] => 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]] => 25
[2,2,2,2,1,1] => [[1,2],[3,4],[5,6],[7,8],[9],[10]] => [[1,3,5,7,9,10],[2,4,6,8]] => 24
[2,2,2,1,1,1,1] => [[1,2],[3,4],[5,6],[7],[8],[9],[10]] => [[1,3,5,7,8,9,10],[2,4,6]] => 21
[2,2,1,1,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8],[9],[10]] => [[1,3,5,6,7,8,9,10],[2,4]] => 16
[2,1,1,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]] => [[1,3,4,5,6,7,8,9,10],[2]] => 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]] => 0
[6,6] => [[1,2,3,4,5,6],[7,8,9,10,11,12]] => [[1,7],[2,8],[3,9],[4,10],[5,11],[6,12]] => 60
[2,2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 36
[1,1,1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]] => [[1,2,3,4,5,6,7,8,9,10,11,12]] => 0
[] => [] => [] => 0
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Description
The cocharge of a standard tableau.
The cocharge of a standard tableau $T$, denoted $\mathrm{cc}(T)$, is defined to be the cocharge of the reading word of the tableau. The cocharge of a permutation $w_1 w_2\cdots w_n$ can be computed by the following algorithm:
1) Starting from $w_n$, scan the entries right-to-left until finding the entry $1$ with a superscript $0$.
2) Continue scanning until the $2$ is found, and label this with a superscript $1$. Then scan until the $3$ is found, labeling with a $2$, and so on, incrementing the label each time, until the beginning of the word is reached. Then go back to the end and scan again from right to left, and *do not* increment the superscript label for the first number found in the next scan. Then continue scanning and labeling, each time incrementing the superscript only if we have not cycled around the word since the last labeling.
3) The cocharge is defined as the sum of the superscript labels on the letters.
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
Map
conjugate
Description
Sends a standard tableau to its conjugate tableau.