Identifier
Values
[1] => [1] => [[1]] => [[1]] => 0
[2] => [1,1] => [[1],[2]] => [[1,2]] => 0
[1,1] => [2] => [[1,2]] => [[1],[2]] => 1
[3] => [1,1,1] => [[1],[2],[3]] => [[1,2,3]] => 0
[2,1] => [2,1] => [[1,3],[2]] => [[1,2],[3]] => 2
[1,1,1] => [3] => [[1,2,3]] => [[1],[2],[3]] => 3
[4] => [1,1,1,1] => [[1],[2],[3],[4]] => [[1,2,3,4]] => 0
[3,1] => [2,1,1] => [[1,4],[2],[3]] => [[1,2,3],[4]] => 3
[2,2] => [2,2] => [[1,2],[3,4]] => [[1,3],[2,4]] => 4
[2,1,1] => [3,1] => [[1,3,4],[2]] => [[1,2],[3],[4]] => 5
[1,1,1,1] => [4] => [[1,2,3,4]] => [[1],[2],[3],[4]] => 6
[5] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [[1,2,3,4,5]] => 0
[4,1] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [[1,2,3,4],[5]] => 4
[3,2] => [2,2,1] => [[1,3],[2,5],[4]] => [[1,2,4],[3,5]] => 6
[3,1,1] => [3,1,1] => [[1,4,5],[2],[3]] => [[1,2,3],[4],[5]] => 7
[2,2,1] => [3,2] => [[1,2,5],[3,4]] => [[1,3],[2,4],[5]] => 8
[2,1,1,1] => [4,1] => [[1,3,4,5],[2]] => [[1,2],[3],[4],[5]] => 9
[1,1,1,1,1] => [5] => [[1,2,3,4,5]] => [[1],[2],[3],[4],[5]] => 10
[6] => [1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [[1,2,3,4,5,6]] => 0
[5,1] => [2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [[1,2,3,4,5],[6]] => 5
[4,2] => [2,2,1,1] => [[1,4],[2,6],[3],[5]] => [[1,2,3,5],[4,6]] => 8
[4,1,1] => [3,1,1,1] => [[1,5,6],[2],[3],[4]] => [[1,2,3,4],[5],[6]] => 9
[3,3] => [2,2,2] => [[1,2],[3,4],[5,6]] => [[1,3,5],[2,4,6]] => 9
[3,2,1] => [3,2,1] => [[1,3,6],[2,5],[4]] => [[1,2,4],[3,5],[6]] => 11
[3,1,1,1] => [4,1,1] => [[1,4,5,6],[2],[3]] => [[1,2,3],[4],[5],[6]] => 12
[2,2,2] => [3,3] => [[1,2,3],[4,5,6]] => [[1,4],[2,5],[3,6]] => 12
[2,2,1,1] => [4,2] => [[1,2,5,6],[3,4]] => [[1,3],[2,4],[5],[6]] => 13
[2,1,1,1,1] => [5,1] => [[1,3,4,5,6],[2]] => [[1,2],[3],[4],[5],[6]] => 14
[1,1,1,1,1,1] => [6] => [[1,2,3,4,5,6]] => [[1],[2],[3],[4],[5],[6]] => 15
[7] => [1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [[1,2,3,4,5,6,7]] => 0
[6,1] => [2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => [[1,2,3,4,5,6],[7]] => 6
[5,2] => [2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => [[1,2,3,4,6],[5,7]] => 10
[5,1,1] => [3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => [[1,2,3,4,5],[6],[7]] => 11
[4,3] => [2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => [[1,2,4,6],[3,5,7]] => 12
[4,2,1] => [3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => [[1,2,3,5],[4,6],[7]] => 14
[4,1,1,1] => [4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => [[1,2,3,4],[5],[6],[7]] => 15
[3,3,1] => [3,2,2] => [[1,2,7],[3,4],[5,6]] => [[1,3,5],[2,4,6],[7]] => 15
[3,2,2] => [3,3,1] => [[1,3,4],[2,6,7],[5]] => [[1,2,5],[3,6],[4,7]] => 16
[3,2,1,1] => [4,2,1] => [[1,3,6,7],[2,5],[4]] => [[1,2,4],[3,5],[6],[7]] => 17
[3,1,1,1,1] => [5,1,1] => [[1,4,5,6,7],[2],[3]] => [[1,2,3],[4],[5],[6],[7]] => 18
[2,2,2,1] => [4,3] => [[1,2,3,7],[4,5,6]] => [[1,4],[2,5],[3,6],[7]] => 18
[2,2,1,1,1] => [5,2] => [[1,2,5,6,7],[3,4]] => [[1,3],[2,4],[5],[6],[7]] => 19
[2,1,1,1,1,1] => [6,1] => [[1,3,4,5,6,7],[2]] => [[1,2],[3],[4],[5],[6],[7]] => 20
[1,1,1,1,1,1,1] => [7] => [[1,2,3,4,5,6,7]] => [[1],[2],[3],[4],[5],[6],[7]] => 21
[8] => [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
[7,1] => [2,1,1,1,1,1,1] => [[1,8],[2],[3],[4],[5],[6],[7]] => [[1,2,3,4,5,6,7],[8]] => 7
[6,2] => [2,2,1,1,1,1] => [[1,6],[2,8],[3],[4],[5],[7]] => [[1,2,3,4,5,7],[6,8]] => 12
[6,1,1] => [3,1,1,1,1,1] => [[1,7,8],[2],[3],[4],[5],[6]] => [[1,2,3,4,5,6],[7],[8]] => 13
[5,3] => [2,2,2,1,1] => [[1,4],[2,6],[3,8],[5],[7]] => [[1,2,3,5,7],[4,6,8]] => 15
[5,2,1] => [3,2,1,1,1] => [[1,5,8],[2,7],[3],[4],[6]] => [[1,2,3,4,6],[5,7],[8]] => 17
[5,1,1,1] => [4,1,1,1,1] => [[1,6,7,8],[2],[3],[4],[5]] => [[1,2,3,4,5],[6],[7],[8]] => 18
[4,4] => [2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [[1,3,5,7],[2,4,6,8]] => 16
[4,3,1] => [3,2,2,1] => [[1,3,8],[2,5],[4,7],[6]] => [[1,2,4,6],[3,5,7],[8]] => 19
[4,2,2] => [3,3,1,1] => [[1,4,5],[2,7,8],[3],[6]] => [[1,2,3,6],[4,7],[5,8]] => 20
[4,2,1,1] => [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] => [5,1,1,1] => [[1,5,6,7,8],[2],[3],[4]] => [[1,2,3,4],[5],[6],[7],[8]] => 22
[3,3,2] => [3,3,2] => [[1,2,5],[3,4,8],[6,7]] => [[1,3,6],[2,4,7],[5,8]] => 21
[3,3,1,1] => [4,2,2] => [[1,2,7,8],[3,4],[5,6]] => [[1,3,5],[2,4,6],[7],[8]] => 22
[3,2,2,1] => [4,3,1] => [[1,3,4,8],[2,6,7],[5]] => [[1,2,5],[3,6],[4,7],[8]] => 23
[3,2,1,1,1] => [5,2,1] => [[1,3,6,7,8],[2,5],[4]] => [[1,2,4],[3,5],[6],[7],[8]] => 24
[3,1,1,1,1,1] => [6,1,1] => [[1,4,5,6,7,8],[2],[3]] => [[1,2,3],[4],[5],[6],[7],[8]] => 25
[2,2,2,2] => [4,4] => [[1,2,3,4],[5,6,7,8]] => [[1,5],[2,6],[3,7],[4,8]] => 24
[2,2,2,1,1] => [5,3] => [[1,2,3,7,8],[4,5,6]] => [[1,4],[2,5],[3,6],[7],[8]] => 25
[2,2,1,1,1,1] => [6,2] => [[1,2,5,6,7,8],[3,4]] => [[1,3],[2,4],[5],[6],[7],[8]] => 26
[2,1,1,1,1,1,1] => [7,1] => [[1,3,4,5,6,7,8],[2]] => [[1,2],[3],[4],[5],[6],[7],[8]] => 27
[1,1,1,1,1,1,1,1] => [8] => [[1,2,3,4,5,6,7,8]] => [[1],[2],[3],[4],[5],[6],[7],[8]] => 28
[9] => [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
[8,1] => [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
[7,2] => [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
[7,1,1] => [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
[6,3] => [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
[6,2,1] => [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
[6,1,1,1] => [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
[5,4] => [2,2,2,2,1] => [[1,3],[2,5],[4,7],[6,9],[8]] => [[1,2,4,6,8],[3,5,7,9]] => 20
[5,3,1] => [3,2,2,1,1] => [[1,4,9],[2,6],[3,8],[5],[7]] => [[1,2,3,5,7],[4,6,8],[9]] => 23
[5,2,2] => [3,3,1,1,1] => [[1,5,6],[2,8,9],[3],[4],[7]] => [[1,2,3,4,7],[5,8],[6,9]] => 24
[5,2,1,1] => [4,2,1,1,1] => [[1,5,8,9],[2,7],[3],[4],[6]] => [[1,2,3,4,6],[5,7],[8],[9]] => 25
[5,1,1,1,1] => [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] => [3,2,2,2] => [[1,2,9],[3,4],[5,6],[7,8]] => [[1,3,5,7],[2,4,6,8],[9]] => 24
[4,3,2] => [3,3,2,1] => [[1,3,6],[2,5,9],[4,8],[7]] => [[1,2,4,7],[3,5,8],[6,9]] => 26
[4,3,1,1] => [4,2,2,1] => [[1,3,8,9],[2,5],[4,7],[6]] => [[1,2,4,6],[3,5,7],[8],[9]] => 27
[4,2,2,1] => [4,3,1,1] => [[1,4,5,9],[2,7,8],[3],[6]] => [[1,2,3,6],[4,7],[5,8],[9]] => 28
[4,2,1,1,1] => [5,2,1,1] => [[1,4,7,8,9],[2,6],[3],[5]] => [[1,2,3,5],[4,6],[7],[8],[9]] => 29
[4,1,1,1,1,1] => [6,1,1,1] => [[1,5,6,7,8,9],[2],[3],[4]] => [[1,2,3,4],[5],[6],[7],[8],[9]] => 30
[3,3,3] => [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] => [4,3,2] => [[1,2,5,9],[3,4,8],[6,7]] => [[1,3,6],[2,4,7],[5,8],[9]] => 29
[3,3,1,1,1] => [5,2,2] => [[1,2,7,8,9],[3,4],[5,6]] => [[1,3,5],[2,4,6],[7],[8],[9]] => 30
[3,2,2,2] => [4,4,1] => [[1,3,4,5],[2,7,8,9],[6]] => [[1,2,6],[3,7],[4,8],[5,9]] => 30
[3,2,2,1,1] => [5,3,1] => [[1,3,4,8,9],[2,6,7],[5]] => [[1,2,5],[3,6],[4,7],[8],[9]] => 31
[3,2,1,1,1,1] => [6,2,1] => [[1,3,6,7,8,9],[2,5],[4]] => [[1,2,4],[3,5],[6],[7],[8],[9]] => 32
[3,1,1,1,1,1,1] => [7,1,1] => [[1,4,5,6,7,8,9],[2],[3]] => [[1,2,3],[4],[5],[6],[7],[8],[9]] => 33
[2,2,2,2,1] => [5,4] => [[1,2,3,4,9],[5,6,7,8]] => [[1,5],[2,6],[3,7],[4,8],[9]] => 32
[2,2,2,1,1,1] => [6,3] => [[1,2,3,7,8,9],[4,5,6]] => [[1,4],[2,5],[3,6],[7],[8],[9]] => 33
[2,2,1,1,1,1,1] => [7,2] => [[1,2,5,6,7,8,9],[3,4]] => [[1,3],[2,4],[5],[6],[7],[8],[9]] => 34
[2,1,1,1,1,1,1,1] => [8,1] => [[1,3,4,5,6,7,8,9],[2]] => [[1,2],[3],[4],[5],[6],[7],[8],[9]] => 35
[1,1,1,1,1,1,1,1,1] => [9] => [[1,2,3,4,5,6,7,8,9]] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => 36
[10] => [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
[9,1] => [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]] => 9
[8,2] => [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]] => 16
[8,1,1] => [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]] => 17
[7,3] => [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]] => 21
>>> Load all 142 entries. <<<
[7,2,1] => [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]] => 23
[7,1,1,1] => [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]] => 24
[6,4] => [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]] => 24
[6,3,1] => [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]] => 27
[6,2,2] => [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]] => 28
[6,2,1,1] => [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]] => 29
[6,1,1,1,1] => [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]] => 30
[5,5] => [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
[5,4,1] => [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]] => 29
[5,3,2] => [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]] => 31
[5,3,1,1] => [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]] => 32
[5,2,2,1] => [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]] => 33
[5,2,1,1,1] => [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]] => 34
[5,1,1,1,1,1] => [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]] => 35
[4,4,2] => [3,3,2,2] => [[1,2,7],[3,4,10],[5,6],[8,9]] => [[1,3,5,8],[2,4,6,9],[7,10]] => 32
[4,4,1,1] => [4,2,2,2] => [[1,2,9,10],[3,4],[5,6],[7,8]] => [[1,3,5,7],[2,4,6,8],[9],[10]] => 33
[4,3,3] => [3,3,3,1] => [[1,3,4],[2,6,7],[5,9,10],[8]] => [[1,2,5,8],[3,6,9],[4,7,10]] => 33
[4,3,2,1] => [4,3,2,1] => [[1,3,6,10],[2,5,9],[4,8],[7]] => [[1,2,4,7],[3,5,8],[6,9],[10]] => 35
[4,3,1,1,1] => [5,2,2,1] => [[1,3,8,9,10],[2,5],[4,7],[6]] => [[1,2,4,6],[3,5,7],[8],[9],[10]] => 36
[4,2,2,2] => [4,4,1,1] => [[1,4,5,6],[2,8,9,10],[3],[7]] => [[1,2,3,7],[4,8],[5,9],[6,10]] => 36
[4,2,2,1,1] => [5,3,1,1] => [[1,4,5,9,10],[2,7,8],[3],[6]] => [[1,2,3,6],[4,7],[5,8],[9],[10]] => 37
[4,2,1,1,1,1] => [6,2,1,1] => [[1,4,7,8,9,10],[2,6],[3],[5]] => [[1,2,3,5],[4,6],[7],[8],[9],[10]] => 38
[4,1,1,1,1,1,1] => [7,1,1,1] => [[1,5,6,7,8,9,10],[2],[3],[4]] => [[1,2,3,4],[5],[6],[7],[8],[9],[10]] => 39
[3,3,3,1] => [4,3,3] => [[1,2,3,10],[4,5,6],[7,8,9]] => [[1,4,7],[2,5,8],[3,6,9],[10]] => 36
[3,3,2,2] => [4,4,2] => [[1,2,5,6],[3,4,9,10],[7,8]] => [[1,3,7],[2,4,8],[5,9],[6,10]] => 37
[3,3,2,1,1] => [5,3,2] => [[1,2,5,9,10],[3,4,8],[6,7]] => [[1,3,6],[2,4,7],[5,8],[9],[10]] => 38
[3,3,1,1,1,1] => [6,2,2] => [[1,2,7,8,9,10],[3,4],[5,6]] => [[1,3,5],[2,4,6],[7],[8],[9],[10]] => 39
[3,2,2,2,1] => [5,4,1] => [[1,3,4,5,10],[2,7,8,9],[6]] => [[1,2,6],[3,7],[4,8],[5,9],[10]] => 39
[3,2,2,1,1,1] => [6,3,1] => [[1,3,4,8,9,10],[2,6,7],[5]] => [[1,2,5],[3,6],[4,7],[8],[9],[10]] => 40
[3,2,1,1,1,1,1] => [7,2,1] => [[1,3,6,7,8,9,10],[2,5],[4]] => [[1,2,4],[3,5],[6],[7],[8],[9],[10]] => 41
[3,1,1,1,1,1,1,1] => [8,1,1] => [[1,4,5,6,7,8,9,10],[2],[3]] => [[1,2,3],[4],[5],[6],[7],[8],[9],[10]] => 42
[2,2,2,2,2] => [5,5] => [[1,2,3,4,5],[6,7,8,9,10]] => [[1,6],[2,7],[3,8],[4,9],[5,10]] => 40
[2,2,2,2,1,1] => [6,4] => [[1,2,3,4,9,10],[5,6,7,8]] => [[1,5],[2,6],[3,7],[4,8],[9],[10]] => 41
[2,2,2,1,1,1,1] => [7,3] => [[1,2,3,7,8,9,10],[4,5,6]] => [[1,4],[2,5],[3,6],[7],[8],[9],[10]] => 42
[2,2,1,1,1,1,1,1] => [8,2] => [[1,2,5,6,7,8,9,10],[3,4]] => [[1,3],[2,4],[5],[6],[7],[8],[9],[10]] => 43
[2,1,1,1,1,1,1,1,1] => [9,1] => [[1,3,4,5,6,7,8,9,10],[2]] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]] => 44
[1,1,1,1,1,1,1,1,1,1] => [10] => [[1,2,3,4,5,6,7,8,9,10]] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => 45
[12] => [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
[6,6] => [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
[2,2,2,2,2,2] => [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
[] => [] => [] => [] => 0
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
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.
Map
conjugate
Description
Return the conjugate partition of the partition.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
Map
conjugate
Description
Sends a standard tableau to its conjugate tableau.
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.