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] => [[1,2],[3]] => [[1,3],[2]] => 1
[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] => [[1,2],[3],[4]] => [[1,3,4],[2]] => 1
[2,2] => [2,2] => [[1,2],[3,4]] => [[1,3],[2,4]] => 2
[2,1,1] => [3,1] => [[1,2,3],[4]] => [[1,4],[2],[3]] => 3
[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] => [[1,2],[3],[4],[5]] => [[1,3,4,5],[2]] => 1
[3,2] => [2,2,1] => [[1,2],[3,4],[5]] => [[1,3,5],[2,4]] => 2
[3,1,1] => [3,1,1] => [[1,2,3],[4],[5]] => [[1,4,5],[2],[3]] => 3
[2,2,1] => [3,2] => [[1,2,3],[4,5]] => [[1,4],[2,5],[3]] => 4
[2,1,1,1] => [4,1] => [[1,2,3,4],[5]] => [[1,5],[2],[3],[4]] => 6
[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] => [[1,2],[3],[4],[5],[6]] => [[1,3,4,5,6],[2]] => 1
[4,2] => [2,2,1,1] => [[1,2],[3,4],[5],[6]] => [[1,3,5,6],[2,4]] => 2
[4,1,1] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => [[1,4,5,6],[2],[3]] => 3
[3,3] => [2,2,2] => [[1,2],[3,4],[5,6]] => [[1,3,5],[2,4,6]] => 3
[3,2,1] => [3,2,1] => [[1,2,3],[4,5],[6]] => [[1,4,6],[2,5],[3]] => 4
[3,1,1,1] => [4,1,1] => [[1,2,3,4],[5],[6]] => [[1,5,6],[2],[3],[4]] => 6
[2,2,2] => [3,3] => [[1,2,3],[4,5,6]] => [[1,4],[2,5],[3,6]] => 6
[2,2,1,1] => [4,2] => [[1,2,3,4],[5,6]] => [[1,5],[2,6],[3],[4]] => 7
[2,1,1,1,1] => [5,1] => [[1,2,3,4,5],[6]] => [[1,6],[2],[3],[4],[5]] => 10
[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] => [[1,2],[3],[4],[5],[6],[7]] => [[1,3,4,5,6,7],[2]] => 1
[5,2] => [2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [[1,3,5,6,7],[2,4]] => 2
[5,1,1] => [3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [[1,4,5,6,7],[2],[3]] => 3
[4,3] => [2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [[1,3,5,7],[2,4,6]] => 3
[4,2,1] => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => [[1,4,6,7],[2,5],[3]] => 4
[4,1,1,1] => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [[1,5,6,7],[2],[3],[4]] => 6
[3,3,1] => [3,2,2] => [[1,2,3],[4,5],[6,7]] => [[1,4,6],[2,5,7],[3]] => 5
[3,2,2] => [3,3,1] => [[1,2,3],[4,5,6],[7]] => [[1,4,7],[2,5],[3,6]] => 6
[3,2,1,1] => [4,2,1] => [[1,2,3,4],[5,6],[7]] => [[1,5,7],[2,6],[3],[4]] => 7
[3,1,1,1,1] => [5,1,1] => [[1,2,3,4,5],[6],[7]] => [[1,6,7],[2],[3],[4],[5]] => 10
[2,2,2,1] => [4,3] => [[1,2,3,4],[5,6,7]] => [[1,5],[2,6],[3,7],[4]] => 9
[2,2,1,1,1] => [5,2] => [[1,2,3,4,5],[6,7]] => [[1,6],[2,7],[3],[4],[5]] => 11
[2,1,1,1,1,1] => [6,1] => [[1,2,3,4,5,6],[7]] => [[1,7],[2],[3],[4],[5],[6]] => 15
[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] => [[1,2],[3],[4],[5],[6],[7],[8]] => [[1,3,4,5,6,7,8],[2]] => 1
[6,2] => [2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => [[1,3,5,6,7,8],[2,4]] => 2
[6,1,1] => [3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => [[1,4,5,6,7,8],[2],[3]] => 3
[5,3] => [2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => [[1,3,5,7,8],[2,4,6]] => 3
[5,2,1] => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => [[1,4,6,7,8],[2,5],[3]] => 4
[5,1,1,1] => [4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => [[1,5,6,7,8],[2],[3],[4]] => 6
[4,4] => [2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [[1,3,5,7],[2,4,6,8]] => 4
[4,3,1] => [3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => [[1,4,6,8],[2,5,7],[3]] => 5
[4,2,2] => [3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => [[1,4,7,8],[2,5],[3,6]] => 6
[4,2,1,1] => [4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => [[1,5,7,8],[2,6],[3],[4]] => 7
[4,1,1,1,1] => [5,1,1,1] => [[1,2,3,4,5],[6],[7],[8]] => [[1,6,7,8],[2],[3],[4],[5]] => 10
[3,3,2] => [3,3,2] => [[1,2,3],[4,5,6],[7,8]] => [[1,4,7],[2,5,8],[3,6]] => 7
[3,3,1,1] => [4,2,2] => [[1,2,3,4],[5,6],[7,8]] => [[1,5,7],[2,6,8],[3],[4]] => 8
[3,2,2,1] => [4,3,1] => [[1,2,3,4],[5,6,7],[8]] => [[1,5,8],[2,6],[3,7],[4]] => 9
[3,2,1,1,1] => [5,2,1] => [[1,2,3,4,5],[6,7],[8]] => [[1,6,8],[2,7],[3],[4],[5]] => 11
[3,1,1,1,1,1] => [6,1,1] => [[1,2,3,4,5,6],[7],[8]] => [[1,7,8],[2],[3],[4],[5],[6]] => 15
[2,2,2,2] => [4,4] => [[1,2,3,4],[5,6,7,8]] => [[1,5],[2,6],[3,7],[4,8]] => 12
[2,2,2,1,1] => [5,3] => [[1,2,3,4,5],[6,7,8]] => [[1,6],[2,7],[3,8],[4],[5]] => 13
[2,2,1,1,1,1] => [6,2] => [[1,2,3,4,5,6],[7,8]] => [[1,7],[2,8],[3],[4],[5],[6]] => 16
[2,1,1,1,1,1,1] => [7,1] => [[1,2,3,4,5,6,7],[8]] => [[1,8],[2],[3],[4],[5],[6],[7]] => 21
[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] => [[1,2],[3],[4],[5],[6],[7],[8],[9]] => [[1,3,4,5,6,7,8,9],[2]] => 1
[7,2] => [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]] => 2
[7,1,1] => [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]] => 3
[6,3] => [2,2,2,1,1,1] => [[1,2],[3,4],[5,6],[7],[8],[9]] => [[1,3,5,7,8,9],[2,4,6]] => 3
[6,2,1] => [3,2,1,1,1,1] => [[1,2,3],[4,5],[6],[7],[8],[9]] => [[1,4,6,7,8,9],[2,5],[3]] => 4
[6,1,1,1] => [4,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9]] => [[1,5,6,7,8,9],[2],[3],[4]] => 6
[5,4] => [2,2,2,2,1] => [[1,2],[3,4],[5,6],[7,8],[9]] => [[1,3,5,7,9],[2,4,6,8]] => 4
[5,3,1] => [3,2,2,1,1] => [[1,2,3],[4,5],[6,7],[8],[9]] => [[1,4,6,8,9],[2,5,7],[3]] => 5
[5,2,2] => [3,3,1,1,1] => [[1,2,3],[4,5,6],[7],[8],[9]] => [[1,4,7,8,9],[2,5],[3,6]] => 6
[5,2,1,1] => [4,2,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9]] => [[1,5,7,8,9],[2,6],[3],[4]] => 7
[5,1,1,1,1] => [5,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9]] => [[1,6,7,8,9],[2],[3],[4],[5]] => 10
[4,4,1] => [3,2,2,2] => [[1,2,3],[4,5],[6,7],[8,9]] => [[1,4,6,8],[2,5,7,9],[3]] => 6
[4,3,2] => [3,3,2,1] => [[1,2,3],[4,5,6],[7,8],[9]] => [[1,4,7,9],[2,5,8],[3,6]] => 7
[4,3,1,1] => [4,2,2,1] => [[1,2,3,4],[5,6],[7,8],[9]] => [[1,5,7,9],[2,6,8],[3],[4]] => 8
[4,2,2,1] => [4,3,1,1] => [[1,2,3,4],[5,6,7],[8],[9]] => [[1,5,8,9],[2,6],[3,7],[4]] => 9
[4,2,1,1,1] => [5,2,1,1] => [[1,2,3,4,5],[6,7],[8],[9]] => [[1,6,8,9],[2,7],[3],[4],[5]] => 11
[4,1,1,1,1,1] => [6,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9]] => [[1,7,8,9],[2],[3],[4],[5],[6]] => 15
[3,3,3] => [3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => [[1,4,7],[2,5,8],[3,6,9]] => 9
[3,3,2,1] => [4,3,2] => [[1,2,3,4],[5,6,7],[8,9]] => [[1,5,8],[2,6,9],[3,7],[4]] => 10
[3,3,1,1,1] => [5,2,2] => [[1,2,3,4,5],[6,7],[8,9]] => [[1,6,8],[2,7,9],[3],[4],[5]] => 12
[3,2,2,2] => [4,4,1] => [[1,2,3,4],[5,6,7,8],[9]] => [[1,5,9],[2,6],[3,7],[4,8]] => 12
[3,2,2,1,1] => [5,3,1] => [[1,2,3,4,5],[6,7,8],[9]] => [[1,6,9],[2,7],[3,8],[4],[5]] => 13
[3,2,1,1,1,1] => [6,2,1] => [[1,2,3,4,5,6],[7,8],[9]] => [[1,7,9],[2,8],[3],[4],[5],[6]] => 16
[3,1,1,1,1,1,1] => [7,1,1] => [[1,2,3,4,5,6,7],[8],[9]] => [[1,8,9],[2],[3],[4],[5],[6],[7]] => 21
[2,2,2,2,1] => [5,4] => [[1,2,3,4,5],[6,7,8,9]] => [[1,6],[2,7],[3,8],[4,9],[5]] => 16
[2,2,2,1,1,1] => [6,3] => [[1,2,3,4,5,6],[7,8,9]] => [[1,7],[2,8],[3,9],[4],[5],[6]] => 18
[2,2,1,1,1,1,1] => [7,2] => [[1,2,3,4,5,6,7],[8,9]] => [[1,8],[2,9],[3],[4],[5],[6],[7]] => 22
[2,1,1,1,1,1,1,1] => [8,1] => [[1,2,3,4,5,6,7,8],[9]] => [[1,9],[2],[3],[4],[5],[6],[7],[8]] => 28
[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] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]] => [[1,3,4,5,6,7,8,9,10],[2]] => 1
[8,2] => [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]] => 2
[8,1,1] => [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]] => 3
[7,3] => [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]] => 3
>>> Load all 142 entries. <<<
[7,2,1] => [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]] => 4
[7,1,1,1] => [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]] => 6
[6,4] => [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]] => 4
[6,3,1] => [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]] => 5
[6,2,2] => [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]] => 6
[6,2,1,1] => [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]] => 7
[6,1,1,1,1] => [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]] => 10
[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]] => 5
[5,4,1] => [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]] => 6
[5,3,2] => [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]] => 7
[5,3,1,1] => [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]] => 8
[5,2,2,1] => [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]] => 9
[5,2,1,1,1] => [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]] => 11
[5,1,1,1,1,1] => [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]] => 15
[4,4,2] => [3,3,2,2] => [[1,2,3],[4,5,6],[7,8],[9,10]] => [[1,4,7,9],[2,5,8,10],[3,6]] => 8
[4,4,1,1] => [4,2,2,2] => [[1,2,3,4],[5,6],[7,8],[9,10]] => [[1,5,7,9],[2,6,8,10],[3],[4]] => 9
[4,3,3] => [3,3,3,1] => [[1,2,3],[4,5,6],[7,8,9],[10]] => [[1,4,7,10],[2,5,8],[3,6,9]] => 9
[4,3,2,1] => [4,3,2,1] => [[1,2,3,4],[5,6,7],[8,9],[10]] => [[1,5,8,10],[2,6,9],[3,7],[4]] => 10
[4,3,1,1,1] => [5,2,2,1] => [[1,2,3,4,5],[6,7],[8,9],[10]] => [[1,6,8,10],[2,7,9],[3],[4],[5]] => 12
[4,2,2,2] => [4,4,1,1] => [[1,2,3,4],[5,6,7,8],[9],[10]] => [[1,5,9,10],[2,6],[3,7],[4,8]] => 12
[4,2,2,1,1] => [5,3,1,1] => [[1,2,3,4,5],[6,7,8],[9],[10]] => [[1,6,9,10],[2,7],[3,8],[4],[5]] => 13
[4,2,1,1,1,1] => [6,2,1,1] => [[1,2,3,4,5,6],[7,8],[9],[10]] => [[1,7,9,10],[2,8],[3],[4],[5],[6]] => 16
[4,1,1,1,1,1,1] => [7,1,1,1] => [[1,2,3,4,5,6,7],[8],[9],[10]] => [[1,8,9,10],[2],[3],[4],[5],[6],[7]] => 21
[3,3,3,1] => [4,3,3] => [[1,2,3,4],[5,6,7],[8,9,10]] => [[1,5,8],[2,6,9],[3,7,10],[4]] => 12
[3,3,2,2] => [4,4,2] => [[1,2,3,4],[5,6,7,8],[9,10]] => [[1,5,9],[2,6,10],[3,7],[4,8]] => 13
[3,3,2,1,1] => [5,3,2] => [[1,2,3,4,5],[6,7,8],[9,10]] => [[1,6,9],[2,7,10],[3,8],[4],[5]] => 14
[3,3,1,1,1,1] => [6,2,2] => [[1,2,3,4,5,6],[7,8],[9,10]] => [[1,7,9],[2,8,10],[3],[4],[5],[6]] => 17
[3,2,2,2,1] => [5,4,1] => [[1,2,3,4,5],[6,7,8,9],[10]] => [[1,6,10],[2,7],[3,8],[4,9],[5]] => 16
[3,2,2,1,1,1] => [6,3,1] => [[1,2,3,4,5,6],[7,8,9],[10]] => [[1,7,10],[2,8],[3,9],[4],[5],[6]] => 18
[3,2,1,1,1,1,1] => [7,2,1] => [[1,2,3,4,5,6,7],[8,9],[10]] => [[1,8,10],[2,9],[3],[4],[5],[6],[7]] => 22
[3,1,1,1,1,1,1,1] => [8,1,1] => [[1,2,3,4,5,6,7,8],[9],[10]] => [[1,9,10],[2],[3],[4],[5],[6],[7],[8]] => 28
[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]] => 20
[2,2,2,2,1,1] => [6,4] => [[1,2,3,4,5,6],[7,8,9,10]] => [[1,7],[2,8],[3,9],[4,10],[5],[6]] => 21
[2,2,2,1,1,1,1] => [7,3] => [[1,2,3,4,5,6,7],[8,9,10]] => [[1,8],[2,9],[3,10],[4],[5],[6],[7]] => 24
[2,2,1,1,1,1,1,1] => [8,2] => [[1,2,3,4,5,6,7,8],[9,10]] => [[1,9],[2,10],[3],[4],[5],[6],[7],[8]] => 29
[2,1,1,1,1,1,1,1,1] => [9,1] => [[1,2,3,4,5,6,7,8,9],[10]] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]] => 36
[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]] => 6
[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]] => 30
[] => [] => [] => [] => 0
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Description
The inversion number of a standard tableau as defined by Haglund and Stevens.
Their inversion number is the total number of inversion pairs for the tableau. An inversion pair is defined as a pair of cells (a,b), (x,y) such that the content of (x,y) is greater than the content of (a,b) and (x,y) is north of the inversion path of (a,b), where the inversion path is defined in detail in [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
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.