Identifier
-
Mp00045:
Integer partitions
—reading tableau⟶
Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤ
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] => [[1,3],[2]] => [[1,2],[3]] => 2
[1,1,1] => [[1],[2],[3]] => [[1,2,3]] => 0
[4] => [[1,2,3,4]] => [[1],[2],[3],[4]] => 6
[3,1] => [[1,3,4],[2]] => [[1,2],[3],[4]] => 5
[2,2] => [[1,2],[3,4]] => [[1,3],[2,4]] => 4
[2,1,1] => [[1,4],[2],[3]] => [[1,2,3],[4]] => 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] => [[1,3,4,5],[2]] => [[1,2],[3],[4],[5]] => 9
[3,2] => [[1,2,5],[3,4]] => [[1,3],[2,4],[5]] => 8
[3,1,1] => [[1,4,5],[2],[3]] => [[1,2,3],[4],[5]] => 7
[2,2,1] => [[1,3],[2,5],[4]] => [[1,2,4],[3,5]] => 6
[2,1,1,1] => [[1,5],[2],[3],[4]] => [[1,2,3,4],[5]] => 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] => [[1,3,4,5,6],[2]] => [[1,2],[3],[4],[5],[6]] => 14
[4,2] => [[1,2,5,6],[3,4]] => [[1,3],[2,4],[5],[6]] => 13
[4,1,1] => [[1,4,5,6],[2],[3]] => [[1,2,3],[4],[5],[6]] => 12
[3,3] => [[1,2,3],[4,5,6]] => [[1,4],[2,5],[3,6]] => 12
[3,2,1] => [[1,3,6],[2,5],[4]] => [[1,2,4],[3,5],[6]] => 11
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => [[1,2,3,4],[5],[6]] => 9
[2,2,2] => [[1,2],[3,4],[5,6]] => [[1,3,5],[2,4,6]] => 9
[2,2,1,1] => [[1,4],[2,6],[3],[5]] => [[1,2,3,5],[4,6]] => 8
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [[1,2,3,4,5],[6]] => 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] => [[1,3,4,5,6,7],[2]] => [[1,2],[3],[4],[5],[6],[7]] => 20
[5,2] => [[1,2,5,6,7],[3,4]] => [[1,3],[2,4],[5],[6],[7]] => 19
[5,1,1] => [[1,4,5,6,7],[2],[3]] => [[1,2,3],[4],[5],[6],[7]] => 18
[4,3] => [[1,2,3,7],[4,5,6]] => [[1,4],[2,5],[3,6],[7]] => 18
[4,2,1] => [[1,3,6,7],[2,5],[4]] => [[1,2,4],[3,5],[6],[7]] => 17
[4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => [[1,2,3,4],[5],[6],[7]] => 15
[3,3,1] => [[1,3,4],[2,6,7],[5]] => [[1,2,5],[3,6],[4,7]] => 16
[3,2,2] => [[1,2,7],[3,4],[5,6]] => [[1,3,5],[2,4,6],[7]] => 15
[3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => [[1,2,3,5],[4,6],[7]] => 14
[3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => [[1,2,3,4,5],[6],[7]] => 11
[2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => [[1,2,4,6],[3,5,7]] => 12
[2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => [[1,2,3,4,6],[5,7]] => 10
[2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => [[1,2,3,4,5,6],[7]] => 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] => [[1,3,4,5,6,7,8],[2]] => [[1,2],[3],[4],[5],[6],[7],[8]] => 27
[6,2] => [[1,2,5,6,7,8],[3,4]] => [[1,3],[2,4],[5],[6],[7],[8]] => 26
[6,1,1] => [[1,4,5,6,7,8],[2],[3]] => [[1,2,3],[4],[5],[6],[7],[8]] => 25
[5,3] => [[1,2,3,7,8],[4,5,6]] => [[1,4],[2,5],[3,6],[7],[8]] => 25
[5,2,1] => [[1,3,6,7,8],[2,5],[4]] => [[1,2,4],[3,5],[6],[7],[8]] => 24
[5,1,1,1] => [[1,5,6,7,8],[2],[3],[4]] => [[1,2,3,4],[5],[6],[7],[8]] => 22
[4,4] => [[1,2,3,4],[5,6,7,8]] => [[1,5],[2,6],[3,7],[4,8]] => 24
[4,3,1] => [[1,3,4,8],[2,6,7],[5]] => [[1,2,5],[3,6],[4,7],[8]] => 23
[4,2,2] => [[1,2,7,8],[3,4],[5,6]] => [[1,3,5],[2,4,6],[7],[8]] => 22
[4,2,1,1] => [[1,4,7,8],[2,6],[3],[5]] => [[1,2,3,5],[4,6],[7],[8]] => 21
[4,1,1,1,1] => [[1,6,7,8],[2],[3],[4],[5]] => [[1,2,3,4,5],[6],[7],[8]] => 18
[3,3,2] => [[1,2,5],[3,4,8],[6,7]] => [[1,3,6],[2,4,7],[5,8]] => 21
[3,3,1,1] => [[1,4,5],[2,7,8],[3],[6]] => [[1,2,3,6],[4,7],[5,8]] => 20
[3,2,2,1] => [[1,3,8],[2,5],[4,7],[6]] => [[1,2,4,6],[3,5,7],[8]] => 19
[3,2,1,1,1] => [[1,5,8],[2,7],[3],[4],[6]] => [[1,2,3,4,6],[5,7],[8]] => 17
[3,1,1,1,1,1] => [[1,7,8],[2],[3],[4],[5],[6]] => [[1,2,3,4,5,6],[7],[8]] => 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] => [[1,4],[2,6],[3,8],[5],[7]] => [[1,2,3,5,7],[4,6,8]] => 15
[2,2,1,1,1,1] => [[1,6],[2,8],[3],[4],[5],[7]] => [[1,2,3,4,5,7],[6,8]] => 12
[2,1,1,1,1,1,1] => [[1,8],[2],[3],[4],[5],[6],[7]] => [[1,2,3,4,5,6,7],[8]] => 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] => [[1,3,4,5,6,7,8,9],[2]] => [[1,2],[3],[4],[5],[6],[7],[8],[9]] => 35
[7,2] => [[1,2,5,6,7,8,9],[3,4]] => [[1,3],[2,4],[5],[6],[7],[8],[9]] => 34
[7,1,1] => [[1,4,5,6,7,8,9],[2],[3]] => [[1,2,3],[4],[5],[6],[7],[8],[9]] => 33
[6,3] => [[1,2,3,7,8,9],[4,5,6]] => [[1,4],[2,5],[3,6],[7],[8],[9]] => 33
[6,2,1] => [[1,3,6,7,8,9],[2,5],[4]] => [[1,2,4],[3,5],[6],[7],[8],[9]] => 32
[6,1,1,1] => [[1,5,6,7,8,9],[2],[3],[4]] => [[1,2,3,4],[5],[6],[7],[8],[9]] => 30
[5,4] => [[1,2,3,4,9],[5,6,7,8]] => [[1,5],[2,6],[3,7],[4,8],[9]] => 32
[5,3,1] => [[1,3,4,8,9],[2,6,7],[5]] => [[1,2,5],[3,6],[4,7],[8],[9]] => 31
[5,2,2] => [[1,2,7,8,9],[3,4],[5,6]] => [[1,3,5],[2,4,6],[7],[8],[9]] => 30
[5,2,1,1] => [[1,4,7,8,9],[2,6],[3],[5]] => [[1,2,3,5],[4,6],[7],[8],[9]] => 29
[5,1,1,1,1] => [[1,6,7,8,9],[2],[3],[4],[5]] => [[1,2,3,4,5],[6],[7],[8],[9]] => 26
[4,4,1] => [[1,3,4,5],[2,7,8,9],[6]] => [[1,2,6],[3,7],[4,8],[5,9]] => 30
[4,3,2] => [[1,2,5,9],[3,4,8],[6,7]] => [[1,3,6],[2,4,7],[5,8],[9]] => 29
[4,3,1,1] => [[1,4,5,9],[2,7,8],[3],[6]] => [[1,2,3,6],[4,7],[5,8],[9]] => 28
[4,2,2,1] => [[1,3,8,9],[2,5],[4,7],[6]] => [[1,2,4,6],[3,5,7],[8],[9]] => 27
[4,2,1,1,1] => [[1,5,8,9],[2,7],[3],[4],[6]] => [[1,2,3,4,6],[5,7],[8],[9]] => 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]] => 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] => [[1,3,6],[2,5,9],[4,8],[7]] => [[1,2,4,7],[3,5,8],[6,9]] => 26
[3,3,1,1,1] => [[1,5,6],[2,8,9],[3],[4],[7]] => [[1,2,3,4,7],[5,8],[6,9]] => 24
[3,2,2,2] => [[1,2,9],[3,4],[5,6],[7,8]] => [[1,3,5,7],[2,4,6,8],[9]] => 24
[3,2,2,1,1] => [[1,4,9],[2,6],[3,8],[5],[7]] => [[1,2,3,5,7],[4,6,8],[9]] => 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]] => 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]] => 15
[2,2,2,2,1] => [[1,3],[2,5],[4,7],[6,9],[8]] => [[1,2,4,6,8],[3,5,7,9]] => 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]] => 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]] => 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]] => 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] => [[1,3,4,5,6,7,8,9,10],[2]] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]] => 44
[8,2] => [[1,2,5,6,7,8,9,10],[3,4]] => [[1,3],[2,4],[5],[6],[7],[8],[9],[10]] => 43
[8,1,1] => [[1,4,5,6,7,8,9,10],[2],[3]] => [[1,2,3],[4],[5],[6],[7],[8],[9],[10]] => 42
[7,3] => [[1,2,3,7,8,9,10],[4,5,6]] => [[1,4],[2,5],[3,6],[7],[8],[9],[10]] => 42
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Description
The (standard) major index of a standard tableau.
A descent of a standard tableau T is an index i such that i+1 appears in a row strictly below the row of i. The (standard) major index is the the sum of the descents.
A descent of a standard tableau T is an index i such that i+1 appears in a row strictly below the row of i. The (standard) major index is the the sum of the descents.
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.
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