Identifier
Values
{{1}} => [1] => 0
{{1,2}} => [2,1] => 1
{{1},{2}} => [1,2] => 0
{{1,2,3}} => [2,3,1] => 2
{{1,2},{3}} => [2,1,3] => 1
{{1,3},{2}} => [3,2,1] => 2
{{1},{2,3}} => [1,3,2] => 1
{{1},{2},{3}} => [1,2,3] => 0
{{1,2,3,4}} => [2,3,4,1] => 3
{{1,2,3},{4}} => [2,3,1,4] => 2
{{1,2,4},{3}} => [2,4,3,1] => 3
{{1,2},{3,4}} => [2,1,4,3] => 2
{{1,2},{3},{4}} => [2,1,3,4] => 1
{{1,3,4},{2}} => [3,2,4,1] => 3
{{1,3},{2,4}} => [3,4,1,2] => 4
{{1,3},{2},{4}} => [3,2,1,4] => 2
{{1,4},{2,3}} => [4,3,2,1] => 4
{{1},{2,3,4}} => [1,3,4,2] => 2
{{1},{2,3},{4}} => [1,3,2,4] => 1
{{1,4},{2},{3}} => [4,2,3,1] => 3
{{1},{2,4},{3}} => [1,4,3,2] => 2
{{1},{2},{3,4}} => [1,2,4,3] => 1
{{1},{2},{3},{4}} => [1,2,3,4] => 0
{{1,2,3,4,5}} => [2,3,4,5,1] => 4
{{1,2,3,4},{5}} => [2,3,4,1,5] => 3
{{1,2,3,5},{4}} => [2,3,5,4,1] => 4
{{1,2,3},{4,5}} => [2,3,1,5,4] => 3
{{1,2,3},{4},{5}} => [2,3,1,4,5] => 2
{{1,2,4,5},{3}} => [2,4,3,5,1] => 4
{{1,2,4},{3,5}} => [2,4,5,1,3] => 5
{{1,2,4},{3},{5}} => [2,4,3,1,5] => 3
{{1,2,5},{3,4}} => [2,5,4,3,1] => 5
{{1,2},{3,4,5}} => [2,1,4,5,3] => 3
{{1,2},{3,4},{5}} => [2,1,4,3,5] => 2
{{1,2,5},{3},{4}} => [2,5,3,4,1] => 4
{{1,2},{3,5},{4}} => [2,1,5,4,3] => 3
{{1,2},{3},{4,5}} => [2,1,3,5,4] => 2
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => 1
{{1,3,4,5},{2}} => [3,2,4,5,1] => 4
{{1,3,4},{2,5}} => [3,5,4,1,2] => 6
{{1,3,4},{2},{5}} => [3,2,4,1,5] => 3
{{1,3,5},{2,4}} => [3,4,5,2,1] => 6
{{1,3},{2,4,5}} => [3,4,1,5,2] => 5
{{1,3},{2,4},{5}} => [3,4,1,2,5] => 4
{{1,3,5},{2},{4}} => [3,2,5,4,1] => 4
{{1,3},{2,5},{4}} => [3,5,1,4,2] => 5
{{1,3},{2},{4,5}} => [3,2,1,5,4] => 3
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => 2
{{1,4,5},{2,3}} => [4,3,2,5,1] => 5
{{1,4},{2,3,5}} => [4,3,5,1,2] => 6
{{1,4},{2,3},{5}} => [4,3,2,1,5] => 4
{{1,5},{2,3,4}} => [5,3,4,2,1] => 6
{{1},{2,3,4,5}} => [1,3,4,5,2] => 3
{{1},{2,3,4},{5}} => [1,3,4,2,5] => 2
{{1,5},{2,3},{4}} => [5,3,2,4,1] => 5
{{1},{2,3,5},{4}} => [1,3,5,4,2] => 3
{{1},{2,3},{4,5}} => [1,3,2,5,4] => 2
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => 1
{{1,4,5},{2},{3}} => [4,2,3,5,1] => 4
{{1,4},{2,5},{3}} => [4,5,3,1,2] => 6
{{1,4},{2},{3,5}} => [4,2,5,1,3] => 5
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => 3
{{1,5},{2,4},{3}} => [5,4,3,2,1] => 6
{{1},{2,4,5},{3}} => [1,4,3,5,2] => 3
{{1},{2,4},{3,5}} => [1,4,5,2,3] => 4
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => 2
{{1,5},{2},{3,4}} => [5,2,4,3,1] => 5
{{1},{2,5},{3,4}} => [1,5,4,3,2] => 4
{{1},{2},{3,4,5}} => [1,2,4,5,3] => 2
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => 1
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => 4
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => 3
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => 2
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => 1
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => 0
{{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 5
{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => 4
{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => 5
{{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => 4
{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => 3
{{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => 5
{{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => 6
{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => 4
{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => 6
{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => 4
{{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => 3
{{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => 5
{{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => 4
{{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => 3
{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => 2
{{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => 5
{{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => 7
{{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => 4
{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => 7
{{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => 6
{{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => 5
{{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => 5
{{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => 6
{{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => 4
{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => 3
{{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => 6
>>> Load all 278 entries. <<<
{{1,2,5},{3,4,6}} => [2,5,4,6,1,3] => 7
{{1,2,5},{3,4},{6}} => [2,5,4,3,1,6] => 5
{{1,2,6},{3,4,5}} => [2,6,4,5,3,1] => 7
{{1,2},{3,4,5,6}} => [2,1,4,5,6,3] => 4
{{1,2},{3,4,5},{6}} => [2,1,4,5,3,6] => 3
{{1,2,6},{3,4},{5}} => [2,6,4,3,5,1] => 6
{{1,2},{3,4,6},{5}} => [2,1,4,6,5,3] => 4
{{1,2},{3,4},{5,6}} => [2,1,4,3,6,5] => 3
{{1,2},{3,4},{5},{6}} => [2,1,4,3,5,6] => 2
{{1,2,5,6},{3},{4}} => [2,5,3,4,6,1] => 5
{{1,2,5},{3,6},{4}} => [2,5,6,4,1,3] => 7
{{1,2,5},{3},{4,6}} => [2,5,3,6,1,4] => 6
{{1,2,5},{3},{4},{6}} => [2,5,3,4,1,6] => 4
{{1,2,6},{3,5},{4}} => [2,6,5,4,3,1] => 7
{{1,2},{3,5,6},{4}} => [2,1,5,4,6,3] => 4
{{1,2},{3,5},{4,6}} => [2,1,5,6,3,4] => 5
{{1,2},{3,5},{4},{6}} => [2,1,5,4,3,6] => 3
{{1,2,6},{3},{4,5}} => [2,6,3,5,4,1] => 6
{{1,2},{3,6},{4,5}} => [2,1,6,5,4,3] => 5
{{1,2},{3},{4,5,6}} => [2,1,3,5,6,4] => 3
{{1,2},{3},{4,5},{6}} => [2,1,3,5,4,6] => 2
{{1,2,6},{3},{4},{5}} => [2,6,3,4,5,1] => 5
{{1,2},{3,6},{4},{5}} => [2,1,6,4,5,3] => 4
{{1,2},{3},{4,6},{5}} => [2,1,3,6,5,4] => 3
{{1,2},{3},{4},{5,6}} => [2,1,3,4,6,5] => 2
{{1,2},{3},{4},{5},{6}} => [2,1,3,4,5,6] => 1
{{1,3,4,5,6},{2}} => [3,2,4,5,6,1] => 5
{{1,3,4,5},{2,6}} => [3,6,4,5,1,2] => 8
{{1,3,4,5},{2},{6}} => [3,2,4,5,1,6] => 4
{{1,3,4,6},{2,5}} => [3,5,4,6,2,1] => 8
{{1,3,4},{2,5,6}} => [3,5,4,1,6,2] => 7
{{1,3,4},{2,5},{6}} => [3,5,4,1,2,6] => 6
{{1,3,4,6},{2},{5}} => [3,2,4,6,5,1] => 5
{{1,3,4},{2,6},{5}} => [3,6,4,1,5,2] => 7
{{1,3,4},{2},{5,6}} => [3,2,4,1,6,5] => 4
{{1,3,4},{2},{5},{6}} => [3,2,4,1,5,6] => 3
{{1,3,5,6},{2,4}} => [3,4,5,2,6,1] => 7
{{1,3,5},{2,4,6}} => [3,4,5,6,1,2] => 8
{{1,3,5},{2,4},{6}} => [3,4,5,2,1,6] => 6
{{1,3,6},{2,4,5}} => [3,4,6,5,2,1] => 8
{{1,3},{2,4,5,6}} => [3,4,1,5,6,2] => 6
{{1,3},{2,4,5},{6}} => [3,4,1,5,2,6] => 5
{{1,3,6},{2,4},{5}} => [3,4,6,2,5,1] => 7
{{1,3},{2,4,6},{5}} => [3,4,1,6,5,2] => 6
{{1,3},{2,4},{5,6}} => [3,4,1,2,6,5] => 5
{{1,3},{2,4},{5},{6}} => [3,4,1,2,5,6] => 4
{{1,3,5,6},{2},{4}} => [3,2,5,4,6,1] => 5
{{1,3,5},{2,6},{4}} => [3,6,5,4,1,2] => 8
{{1,3,5},{2},{4,6}} => [3,2,5,6,1,4] => 6
{{1,3,5},{2},{4},{6}} => [3,2,5,4,1,6] => 4
{{1,3,6},{2,5},{4}} => [3,5,6,4,2,1] => 8
{{1,3},{2,5,6},{4}} => [3,5,1,4,6,2] => 6
{{1,3},{2,5},{4,6}} => [3,5,1,6,2,4] => 7
{{1,3},{2,5},{4},{6}} => [3,5,1,4,2,6] => 5
{{1,3,6},{2},{4,5}} => [3,2,6,5,4,1] => 6
{{1,3},{2,6},{4,5}} => [3,6,1,5,4,2] => 7
{{1,3},{2},{4,5,6}} => [3,2,1,5,6,4] => 4
{{1,3},{2},{4,5},{6}} => [3,2,1,5,4,6] => 3
{{1,3,6},{2},{4},{5}} => [3,2,6,4,5,1] => 5
{{1,3},{2,6},{4},{5}} => [3,6,1,4,5,2] => 6
{{1,3},{2},{4,6},{5}} => [3,2,1,6,5,4] => 4
{{1,3},{2},{4},{5,6}} => [3,2,1,4,6,5] => 3
{{1,3},{2},{4},{5},{6}} => [3,2,1,4,5,6] => 2
{{1,4,5,6},{2,3}} => [4,3,2,5,6,1] => 6
{{1,4,5},{2,3,6}} => [4,3,6,5,1,2] => 8
{{1,4,5},{2,3},{6}} => [4,3,2,5,1,6] => 5
{{1,4,6},{2,3,5}} => [4,3,5,6,2,1] => 8
{{1,4},{2,3,5,6}} => [4,3,5,1,6,2] => 7
{{1,4},{2,3,5},{6}} => [4,3,5,1,2,6] => 6
{{1,4,6},{2,3},{5}} => [4,3,2,6,5,1] => 6
{{1,4},{2,3,6},{5}} => [4,3,6,1,5,2] => 7
{{1,4},{2,3},{5,6}} => [4,3,2,1,6,5] => 5
{{1,4},{2,3},{5},{6}} => [4,3,2,1,5,6] => 4
{{1,5,6},{2,3,4}} => [5,3,4,2,6,1] => 7
{{1,5},{2,3,4,6}} => [5,3,4,6,1,2] => 8
{{1,5},{2,3,4},{6}} => [5,3,4,2,1,6] => 6
{{1,6},{2,3,4,5}} => [6,3,4,5,2,1] => 8
{{1},{2,3,4,5,6}} => [1,3,4,5,6,2] => 4
{{1},{2,3,4,5},{6}} => [1,3,4,5,2,6] => 3
{{1,6},{2,3,4},{5}} => [6,3,4,2,5,1] => 7
{{1},{2,3,4,6},{5}} => [1,3,4,6,5,2] => 4
{{1},{2,3,4},{5,6}} => [1,3,4,2,6,5] => 3
{{1},{2,3,4},{5},{6}} => [1,3,4,2,5,6] => 2
{{1,5,6},{2,3},{4}} => [5,3,2,4,6,1] => 6
{{1,5},{2,3,6},{4}} => [5,3,6,4,1,2] => 8
{{1,5},{2,3},{4,6}} => [5,3,2,6,1,4] => 7
{{1,5},{2,3},{4},{6}} => [5,3,2,4,1,6] => 5
{{1,6},{2,3,5},{4}} => [6,3,5,4,2,1] => 8
{{1},{2,3,5,6},{4}} => [1,3,5,4,6,2] => 4
{{1},{2,3,5},{4,6}} => [1,3,5,6,2,4] => 5
{{1},{2,3,5},{4},{6}} => [1,3,5,4,2,6] => 3
{{1,6},{2,3},{4,5}} => [6,3,2,5,4,1] => 7
{{1},{2,3,6},{4,5}} => [1,3,6,5,4,2] => 5
{{1},{2,3},{4,5,6}} => [1,3,2,5,6,4] => 3
{{1},{2,3},{4,5},{6}} => [1,3,2,5,4,6] => 2
{{1,6},{2,3},{4},{5}} => [6,3,2,4,5,1] => 6
{{1},{2,3,6},{4},{5}} => [1,3,6,4,5,2] => 4
{{1},{2,3},{4,6},{5}} => [1,3,2,6,5,4] => 3
{{1},{2,3},{4},{5,6}} => [1,3,2,4,6,5] => 2
{{1},{2,3},{4},{5},{6}} => [1,3,2,4,5,6] => 1
{{1,4,5,6},{2},{3}} => [4,2,3,5,6,1] => 5
{{1,4,5},{2,6},{3}} => [4,6,3,5,1,2] => 8
{{1,4,5},{2},{3,6}} => [4,2,6,5,1,3] => 7
{{1,4,5},{2},{3},{6}} => [4,2,3,5,1,6] => 4
{{1,4,6},{2,5},{3}} => [4,5,3,6,2,1] => 8
{{1,4},{2,5,6},{3}} => [4,5,3,1,6,2] => 7
{{1,4},{2,5},{3,6}} => [4,5,6,1,2,3] => 9
{{1,4},{2,5},{3},{6}} => [4,5,3,1,2,6] => 6
{{1,4,6},{2},{3,5}} => [4,2,5,6,3,1] => 7
{{1,4},{2,6},{3,5}} => [4,6,5,1,3,2] => 9
{{1,4},{2},{3,5,6}} => [4,2,5,1,6,3] => 6
{{1,4},{2},{3,5},{6}} => [4,2,5,1,3,6] => 5
{{1,4,6},{2},{3},{5}} => [4,2,3,6,5,1] => 5
{{1,4},{2,6},{3},{5}} => [4,6,3,1,5,2] => 7
{{1,4},{2},{3,6},{5}} => [4,2,6,1,5,3] => 6
{{1,4},{2},{3},{5,6}} => [4,2,3,1,6,5] => 4
{{1,4},{2},{3},{5},{6}} => [4,2,3,1,5,6] => 3
{{1,5,6},{2,4},{3}} => [5,4,3,2,6,1] => 7
{{1,5},{2,4,6},{3}} => [5,4,3,6,1,2] => 8
{{1,5},{2,4},{3,6}} => [5,4,6,2,1,3] => 9
{{1,5},{2,4},{3},{6}} => [5,4,3,2,1,6] => 6
{{1,6},{2,4,5},{3}} => [6,4,3,5,2,1] => 8
{{1},{2,4,5,6},{3}} => [1,4,3,5,6,2] => 4
{{1},{2,4,5},{3,6}} => [1,4,6,5,2,3] => 6
{{1},{2,4,5},{3},{6}} => [1,4,3,5,2,6] => 3
{{1,6},{2,4},{3,5}} => [6,4,5,2,3,1] => 9
{{1},{2,4,6},{3,5}} => [1,4,5,6,3,2] => 6
{{1},{2,4},{3,5,6}} => [1,4,5,2,6,3] => 5
{{1},{2,4},{3,5},{6}} => [1,4,5,2,3,6] => 4
{{1,6},{2,4},{3},{5}} => [6,4,3,2,5,1] => 7
{{1},{2,4,6},{3},{5}} => [1,4,3,6,5,2] => 4
{{1},{2,4},{3,6},{5}} => [1,4,6,2,5,3] => 5
{{1},{2,4},{3},{5,6}} => [1,4,3,2,6,5] => 3
{{1},{2,4},{3},{5},{6}} => [1,4,3,2,5,6] => 2
{{1,5,6},{2},{3,4}} => [5,2,4,3,6,1] => 6
{{1,5},{2,6},{3,4}} => [5,6,4,3,1,2] => 9
{{1,5},{2},{3,4,6}} => [5,2,4,6,1,3] => 7
{{1,5},{2},{3,4},{6}} => [5,2,4,3,1,6] => 5
{{1,6},{2,5},{3,4}} => [6,5,4,3,2,1] => 9
{{1},{2,5,6},{3,4}} => [1,5,4,3,6,2] => 5
{{1},{2,5},{3,4,6}} => [1,5,4,6,2,3] => 6
{{1},{2,5},{3,4},{6}} => [1,5,4,3,2,6] => 4
{{1,6},{2},{3,4,5}} => [6,2,4,5,3,1] => 7
{{1},{2,6},{3,4,5}} => [1,6,4,5,3,2] => 6
{{1},{2},{3,4,5,6}} => [1,2,4,5,6,3] => 3
{{1},{2},{3,4,5},{6}} => [1,2,4,5,3,6] => 2
{{1,6},{2},{3,4},{5}} => [6,2,4,3,5,1] => 6
{{1},{2,6},{3,4},{5}} => [1,6,4,3,5,2] => 5
{{1},{2},{3,4,6},{5}} => [1,2,4,6,5,3] => 3
{{1},{2},{3,4},{5,6}} => [1,2,4,3,6,5] => 2
{{1},{2},{3,4},{5},{6}} => [1,2,4,3,5,6] => 1
{{1,5,6},{2},{3},{4}} => [5,2,3,4,6,1] => 5
{{1,5},{2,6},{3},{4}} => [5,6,3,4,1,2] => 8
{{1,5},{2},{3,6},{4}} => [5,2,6,4,1,3] => 7
{{1,5},{2},{3},{4,6}} => [5,2,3,6,1,4] => 6
{{1,5},{2},{3},{4},{6}} => [5,2,3,4,1,6] => 4
{{1,6},{2,5},{3},{4}} => [6,5,3,4,2,1] => 8
{{1},{2,5,6},{3},{4}} => [1,5,3,4,6,2] => 4
{{1},{2,5},{3,6},{4}} => [1,5,6,4,2,3] => 6
{{1},{2,5},{3},{4,6}} => [1,5,3,6,2,4] => 5
{{1},{2,5},{3},{4},{6}} => [1,5,3,4,2,6] => 3
{{1,6},{2},{3,5},{4}} => [6,2,5,4,3,1] => 7
{{1},{2,6},{3,5},{4}} => [1,6,5,4,3,2] => 6
{{1},{2},{3,5,6},{4}} => [1,2,5,4,6,3] => 3
{{1},{2},{3,5},{4,6}} => [1,2,5,6,3,4] => 4
{{1},{2},{3,5},{4},{6}} => [1,2,5,4,3,6] => 2
{{1,6},{2},{3},{4,5}} => [6,2,3,5,4,1] => 6
{{1},{2,6},{3},{4,5}} => [1,6,3,5,4,2] => 5
{{1},{2},{3,6},{4,5}} => [1,2,6,5,4,3] => 4
{{1},{2},{3},{4,5,6}} => [1,2,3,5,6,4] => 2
{{1},{2},{3},{4,5},{6}} => [1,2,3,5,4,6] => 1
{{1,6},{2},{3},{4},{5}} => [6,2,3,4,5,1] => 5
{{1},{2,6},{3},{4},{5}} => [1,6,3,4,5,2] => 4
{{1},{2},{3,6},{4},{5}} => [1,2,6,4,5,3] => 3
{{1},{2},{3},{4,6},{5}} => [1,2,3,6,5,4] => 2
{{1},{2},{3},{4},{5,6}} => [1,2,3,4,6,5] => 1
{{1},{2},{3},{4},{5},{6}} => [1,2,3,4,5,6] => 0
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Description
The depth of a permutation.
This is given by
$$\operatorname{dp}(\sigma) = \sum_{\sigma_i>i} (\sigma_i-i) = |\{ i \leq j : \sigma_i > j\}|.$$
The depth is half of the total displacement [4], Problem 5.1.1.28, or Spearman’s disarray [3] $\sum_i |\sigma_i-i|$.
Permutations with depth at most $1$ are called almost-increasing in [5].
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.