Identifier
Values
{{1}} => [1] => [1] => 0
{{1,2}} => [2,1] => [2,1] => 0
{{1},{2}} => [1,2] => [1,2] => 0
{{1,2,3}} => [2,3,1] => [3,2,1] => 1
{{1,2},{3}} => [2,1,3] => [2,1,3] => 0
{{1,3},{2}} => [3,2,1] => [2,3,1] => 0
{{1},{2,3}} => [1,3,2] => [3,1,2] => 0
{{1},{2},{3}} => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}} => [2,3,4,1] => [3,2,4,1] => 1
{{1,2,3},{4}} => [2,3,1,4] => [3,2,1,4] => 1
{{1,2,4},{3}} => [2,4,3,1] => [4,2,3,1] => 2
{{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => 0
{{1,2},{3},{4}} => [2,1,3,4] => [2,1,3,4] => 0
{{1,3,4},{2}} => [3,2,4,1] => [2,3,4,1] => 0
{{1,3},{2,4}} => [3,4,1,2] => [4,3,1,2] => 2
{{1,3},{2},{4}} => [3,2,1,4] => [2,3,1,4] => 0
{{1,4},{2,3}} => [4,3,2,1] => [3,4,2,1] => 1
{{1},{2,3,4}} => [1,3,4,2] => [3,1,4,2] => 0
{{1},{2,3},{4}} => [1,3,2,4] => [3,1,2,4] => 0
{{1,4},{2},{3}} => [4,2,3,1] => [2,4,3,1] => 1
{{1},{2,4},{3}} => [1,4,3,2] => [4,1,3,2] => 1
{{1},{2},{3,4}} => [1,2,4,3] => [1,2,4,3] => 0
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}} => [2,3,4,5,1] => [3,2,4,5,1] => 1
{{1,2,3,4},{5}} => [2,3,4,1,5] => [3,2,4,1,5] => 1
{{1,2,3,5},{4}} => [2,3,5,4,1] => [3,2,5,4,1] => 2
{{1,2,3},{4,5}} => [2,3,1,5,4] => [3,2,1,5,4] => 1
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [3,2,1,4,5] => 1
{{1,2,4,5},{3}} => [2,4,3,5,1] => [4,2,3,5,1] => 2
{{1,2,4},{3,5}} => [2,4,5,1,3] => [4,2,5,1,3] => 1
{{1,2,4},{3},{5}} => [2,4,3,1,5] => [4,2,3,1,5] => 2
{{1,2,5},{3,4}} => [2,5,4,3,1] => [5,2,4,3,1] => 4
{{1,2},{3,4,5}} => [2,1,4,5,3] => [2,1,5,4,3] => 1
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [2,1,4,3,5] => 0
{{1,2,5},{3},{4}} => [2,5,3,4,1] => [5,2,3,4,1] => 3
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [2,1,4,5,3] => 0
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [2,1,5,3,4] => 0
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [2,1,3,4,5] => 0
{{1,3,4,5},{2}} => [3,2,4,5,1] => [2,3,4,5,1] => 0
{{1,3,4},{2,5}} => [3,5,4,1,2] => [5,3,4,1,2] => 4
{{1,3,4},{2},{5}} => [3,2,4,1,5] => [2,3,4,1,5] => 0
{{1,3,5},{2,4}} => [3,4,5,2,1] => [4,3,5,2,1] => 3
{{1,3},{2,4,5}} => [3,4,1,5,2] => [4,3,1,5,2] => 2
{{1,3},{2,4},{5}} => [3,4,1,2,5] => [4,3,1,2,5] => 2
{{1,3,5},{2},{4}} => [3,2,5,4,1] => [2,3,5,4,1] => 1
{{1,3},{2,5},{4}} => [3,5,1,4,2] => [5,3,1,4,2] => 3
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [2,3,1,5,4] => 0
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [2,3,1,4,5] => 0
{{1,4,5},{2,3}} => [4,3,2,5,1] => [3,4,2,5,1] => 1
{{1,4},{2,3,5}} => [4,3,5,1,2] => [3,4,5,1,2] => 0
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [3,4,2,1,5] => 1
{{1,5},{2,3,4}} => [5,3,4,2,1] => [3,5,4,2,1] => 3
{{1},{2,3,4,5}} => [1,3,4,5,2] => [3,1,4,5,2] => 0
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [3,1,4,2,5] => 0
{{1,5},{2,3},{4}} => [5,3,2,4,1] => [3,5,2,4,1] => 2
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [3,1,5,4,2] => 1
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [3,1,2,5,4] => 0
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [3,1,2,4,5] => 0
{{1,4,5},{2},{3}} => [4,2,3,5,1] => [2,4,3,5,1] => 1
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [5,4,3,1,2] => 5
{{1,4},{2},{3,5}} => [4,2,5,1,3] => [2,4,5,1,3] => 0
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => [2,4,3,1,5] => 1
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [4,5,3,2,1] => 3
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [4,1,3,5,2] => 1
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [4,1,5,2,3] => 0
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [4,1,3,2,5] => 1
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [2,5,4,3,1] => 3
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [5,1,4,3,2] => 3
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,2,5,4,3] => 1
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,2,4,3,5] => 0
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [2,5,3,4,1] => 2
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [5,1,3,4,2] => 2
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,2,4,5,3] => 0
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,2,5,3,4] => 0
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4,5,6}} => [2,3,4,5,6,1] => [3,2,4,5,6,1] => 1
{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => [3,2,4,5,1,6] => 1
{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [3,2,4,6,5,1] => 2
{{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => [3,2,4,1,6,5] => 1
{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => [3,2,4,1,5,6] => 1
{{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => [3,2,5,4,6,1] => 2
{{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => [3,2,5,6,1,4] => 1
{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => [3,2,5,4,1,6] => 2
{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [3,2,6,5,4,1] => 4
{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => [3,2,1,5,6,4] => 1
{{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => [3,2,1,5,4,6] => 1
{{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => [3,2,6,4,5,1] => 3
{{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => [3,2,1,6,5,4] => 2
{{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => [3,2,1,4,6,5] => 1
{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => [3,2,1,4,5,6] => 1
{{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => [4,2,3,5,6,1] => 2
{{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => [4,2,6,5,1,3] => 3
{{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => [4,2,3,5,1,6] => 2
{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => [4,2,5,6,3,1] => 2
{{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => [4,2,5,1,6,3] => 1
{{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => [4,2,5,1,3,6] => 1
{{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => [4,2,3,6,5,1] => 3
{{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => [4,2,6,1,5,3] => 2
{{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => [4,2,3,1,6,5] => 2
{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [4,2,3,1,5,6] => 2
{{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => [5,2,4,3,6,1] => 4
>>> Load all 325 entries. <<<
{{1,2,5},{3,4,6}} => [2,5,4,6,1,3] => [5,2,4,6,1,3] => 3
{{1,2,5},{3,4},{6}} => [2,5,4,3,1,6] => [5,2,4,3,1,6] => 4
{{1,2,6},{3,4,5}} => [2,6,4,5,3,1] => [6,2,4,5,3,1] => 6
{{1,2},{3,4,5,6}} => [2,1,4,5,6,3] => [2,1,5,4,6,3] => 1
{{1,2},{3,4,5},{6}} => [2,1,4,5,3,6] => [2,1,5,4,3,6] => 1
{{1,2,6},{3,4},{5}} => [2,6,4,3,5,1] => [6,2,4,3,5,1] => 5
{{1,2},{3,4,6},{5}} => [2,1,4,6,5,3] => [2,1,6,4,5,3] => 2
{{1,2},{3,4},{5,6}} => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 0
{{1,2},{3,4},{5},{6}} => [2,1,4,3,5,6] => [2,1,4,3,5,6] => 0
{{1,2,5,6},{3},{4}} => [2,5,3,4,6,1] => [5,2,3,4,6,1] => 3
{{1,2,5},{3,6},{4}} => [2,5,6,4,1,3] => [5,2,6,4,1,3] => 3
{{1,2,5},{3},{4,6}} => [2,5,3,6,1,4] => [5,2,3,6,1,4] => 2
{{1,2,5},{3},{4},{6}} => [2,5,3,4,1,6] => [5,2,3,4,1,6] => 3
{{1,2,6},{3,5},{4}} => [2,6,5,4,3,1] => [6,2,5,4,3,1] => 7
{{1,2},{3,5,6},{4}} => [2,1,5,4,6,3] => [2,1,4,5,6,3] => 0
{{1,2},{3,5},{4,6}} => [2,1,5,6,3,4] => [2,1,6,5,3,4] => 2
{{1,2},{3,5},{4},{6}} => [2,1,5,4,3,6] => [2,1,4,5,3,6] => 0
{{1,2,6},{3},{4,5}} => [2,6,3,5,4,1] => [6,2,3,5,4,1] => 5
{{1,2},{3,6},{4,5}} => [2,1,6,5,4,3] => [2,1,5,6,4,3] => 1
{{1,2},{3},{4,5,6}} => [2,1,3,5,6,4] => [2,1,5,3,6,4] => 0
{{1,2},{3},{4,5},{6}} => [2,1,3,5,4,6] => [2,1,5,3,4,6] => 0
{{1,2,6},{3},{4},{5}} => [2,6,3,4,5,1] => [6,2,3,4,5,1] => 4
{{1,2},{3,6},{4},{5}} => [2,1,6,4,5,3] => [2,1,4,6,5,3] => 1
{{1,2},{3},{4,6},{5}} => [2,1,3,6,5,4] => [2,1,6,3,5,4] => 1
{{1,2},{3},{4},{5,6}} => [2,1,3,4,6,5] => [2,1,3,4,6,5] => 0
{{1,2},{3},{4},{5},{6}} => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 0
{{1,3,4,5,6},{2}} => [3,2,4,5,6,1] => [2,3,4,5,6,1] => 0
{{1,3,4,5},{2,6}} => [3,6,4,5,1,2] => [6,3,4,5,1,2] => 6
{{1,3,4,5},{2},{6}} => [3,2,4,5,1,6] => [2,3,4,5,1,6] => 0
{{1,3,4,6},{2,5}} => [3,5,4,6,2,1] => [5,3,4,6,2,1] => 5
{{1,3,4},{2,5,6}} => [3,5,4,1,6,2] => [5,3,4,1,6,2] => 4
{{1,3,4},{2,5},{6}} => [3,5,4,1,2,6] => [5,3,4,1,2,6] => 4
{{1,3,4,6},{2},{5}} => [3,2,4,6,5,1] => [2,3,4,6,5,1] => 1
{{1,3,4},{2,6},{5}} => [3,6,4,1,5,2] => [6,3,4,1,5,2] => 5
{{1,3,4},{2},{5,6}} => [3,2,4,1,6,5] => [2,3,4,1,6,5] => 0
{{1,3,4},{2},{5},{6}} => [3,2,4,1,5,6] => [2,3,4,1,5,6] => 0
{{1,3,5,6},{2,4}} => [3,4,5,2,6,1] => [4,3,5,2,6,1] => 3
{{1,3,5},{2,4,6}} => [3,4,5,6,1,2] => [4,3,5,6,1,2] => 2
{{1,3,5},{2,4},{6}} => [3,4,5,2,1,6] => [4,3,5,2,1,6] => 3
{{1,3,6},{2,4,5}} => [3,4,6,5,2,1] => [4,3,6,5,2,1] => 5
{{1,3},{2,4,5,6}} => [3,4,1,5,6,2] => [4,3,1,5,6,2] => 2
{{1,3},{2,4,5},{6}} => [3,4,1,5,2,6] => [4,3,1,5,2,6] => 2
{{1,3,6},{2,4},{5}} => [3,4,6,2,5,1] => [4,3,6,2,5,1] => 4
{{1,3},{2,4,6},{5}} => [3,4,1,6,5,2] => [4,3,1,6,5,2] => 3
{{1,3},{2,4},{5,6}} => [3,4,1,2,6,5] => [4,3,1,2,6,5] => 2
{{1,3},{2,4},{5},{6}} => [3,4,1,2,5,6] => [4,3,1,2,5,6] => 2
{{1,3,5,6},{2},{4}} => [3,2,5,4,6,1] => [2,3,5,4,6,1] => 1
{{1,3,5},{2,6},{4}} => [3,6,5,4,1,2] => [6,3,5,4,1,2] => 7
{{1,3,5},{2},{4,6}} => [3,2,5,6,1,4] => [2,3,5,6,1,4] => 0
{{1,3,5},{2},{4},{6}} => [3,2,5,4,1,6] => [2,3,5,4,1,6] => 1
{{1,3,6},{2,5},{4}} => [3,5,6,4,2,1] => [5,3,6,4,2,1] => 5
{{1,3},{2,5,6},{4}} => [3,5,1,4,6,2] => [5,3,1,4,6,2] => 3
{{1,3},{2,5},{4,6}} => [3,5,1,6,2,4] => [5,3,1,6,2,4] => 2
{{1,3},{2,5},{4},{6}} => [3,5,1,4,2,6] => [5,3,1,4,2,6] => 3
{{1,3,6},{2},{4,5}} => [3,2,6,5,4,1] => [2,3,6,5,4,1] => 3
{{1,3},{2,6},{4,5}} => [3,6,1,5,4,2] => [6,3,1,5,4,2] => 5
{{1,3},{2},{4,5,6}} => [3,2,1,5,6,4] => [2,3,1,5,6,4] => 0
{{1,3},{2},{4,5},{6}} => [3,2,1,5,4,6] => [2,3,1,5,4,6] => 0
{{1,3,6},{2},{4},{5}} => [3,2,6,4,5,1] => [2,3,6,4,5,1] => 2
{{1,3},{2,6},{4},{5}} => [3,6,1,4,5,2] => [6,3,1,4,5,2] => 4
{{1,3},{2},{4,6},{5}} => [3,2,1,6,5,4] => [2,3,1,6,5,4] => 1
{{1,3},{2},{4},{5,6}} => [3,2,1,4,6,5] => [2,3,1,4,6,5] => 0
{{1,3},{2},{4},{5},{6}} => [3,2,1,4,5,6] => [2,3,1,4,5,6] => 0
{{1,4,5,6},{2,3}} => [4,3,2,5,6,1] => [3,4,2,5,6,1] => 1
{{1,4,5},{2,3,6}} => [4,3,6,5,1,2] => [3,4,6,5,1,2] => 2
{{1,4,5},{2,3},{6}} => [4,3,2,5,1,6] => [3,4,2,5,1,6] => 1
{{1,4,6},{2,3,5}} => [4,3,5,6,2,1] => [3,4,5,6,2,1] => 1
{{1,4},{2,3,5,6}} => [4,3,5,1,6,2] => [3,4,5,1,6,2] => 0
{{1,4},{2,3,5},{6}} => [4,3,5,1,2,6] => [3,4,5,1,2,6] => 0
{{1,4,6},{2,3},{5}} => [4,3,2,6,5,1] => [3,4,2,6,5,1] => 2
{{1,4},{2,3,6},{5}} => [4,3,6,1,5,2] => [3,4,6,1,5,2] => 1
{{1,4},{2,3},{5,6}} => [4,3,2,1,6,5] => [3,4,2,1,6,5] => 1
{{1,4},{2,3},{5},{6}} => [4,3,2,1,5,6] => [3,4,2,1,5,6] => 1
{{1,5,6},{2,3,4}} => [5,3,4,2,6,1] => [3,5,4,2,6,1] => 3
{{1,5},{2,3,4,6}} => [5,3,4,6,1,2] => [3,5,4,6,1,2] => 2
{{1,5},{2,3,4},{6}} => [5,3,4,2,1,6] => [3,5,4,2,1,6] => 3
{{1,6},{2,3,4,5}} => [6,3,4,5,2,1] => [3,6,4,5,2,1] => 5
{{1},{2,3,4,5,6}} => [1,3,4,5,6,2] => [3,1,4,5,6,2] => 0
{{1},{2,3,4,5},{6}} => [1,3,4,5,2,6] => [3,1,4,5,2,6] => 0
{{1,6},{2,3,4},{5}} => [6,3,4,2,5,1] => [3,6,4,2,5,1] => 4
{{1},{2,3,4,6},{5}} => [1,3,4,6,5,2] => [3,1,4,6,5,2] => 1
{{1},{2,3,4},{5,6}} => [1,3,4,2,6,5] => [3,1,4,2,6,5] => 0
{{1},{2,3,4},{5},{6}} => [1,3,4,2,5,6] => [3,1,4,2,5,6] => 0
{{1,5,6},{2,3},{4}} => [5,3,2,4,6,1] => [3,5,2,4,6,1] => 2
{{1,5},{2,3,6},{4}} => [5,3,6,4,1,2] => [3,5,6,4,1,2] => 2
{{1,5},{2,3},{4,6}} => [5,3,2,6,1,4] => [3,5,2,6,1,4] => 1
{{1,5},{2,3},{4},{6}} => [5,3,2,4,1,6] => [3,5,2,4,1,6] => 2
{{1,6},{2,3,5},{4}} => [6,3,5,4,2,1] => [3,6,5,4,2,1] => 6
{{1},{2,3,5,6},{4}} => [1,3,5,4,6,2] => [3,1,5,4,6,2] => 1
{{1},{2,3,5},{4,6}} => [1,3,5,6,2,4] => [3,1,5,6,2,4] => 0
{{1},{2,3,5},{4},{6}} => [1,3,5,4,2,6] => [3,1,5,4,2,6] => 1
{{1,6},{2,3},{4,5}} => [6,3,2,5,4,1] => [3,6,2,5,4,1] => 4
{{1},{2,3,6},{4,5}} => [1,3,6,5,4,2] => [3,1,6,5,4,2] => 3
{{1},{2,3},{4,5,6}} => [1,3,2,5,6,4] => [3,1,2,5,6,4] => 0
{{1},{2,3},{4,5},{6}} => [1,3,2,5,4,6] => [3,1,2,5,4,6] => 0
{{1,6},{2,3},{4},{5}} => [6,3,2,4,5,1] => [3,6,2,4,5,1] => 3
{{1},{2,3,6},{4},{5}} => [1,3,6,4,5,2] => [3,1,6,4,5,2] => 2
{{1},{2,3},{4,6},{5}} => [1,3,2,6,5,4] => [3,1,2,6,5,4] => 1
{{1},{2,3},{4},{5,6}} => [1,3,2,4,6,5] => [3,1,2,4,6,5] => 0
{{1},{2,3},{4},{5},{6}} => [1,3,2,4,5,6] => [3,1,2,4,5,6] => 0
{{1,4,5,6},{2},{3}} => [4,2,3,5,6,1] => [2,4,3,5,6,1] => 1
{{1,4,5},{2,6},{3}} => [4,6,3,5,1,2] => [6,4,3,5,1,2] => 7
{{1,4,5},{2},{3,6}} => [4,2,6,5,1,3] => [2,4,6,5,1,3] => 2
{{1,4,5},{2},{3},{6}} => [4,2,3,5,1,6] => [2,4,3,5,1,6] => 1
{{1,4,6},{2,5},{3}} => [4,5,3,6,2,1] => [5,4,3,6,2,1] => 6
{{1,4},{2,5,6},{3}} => [4,5,3,1,6,2] => [5,4,3,1,6,2] => 5
{{1,4},{2,5},{3,6}} => [4,5,6,1,2,3] => [5,4,6,1,2,3] => 3
{{1,4},{2,5},{3},{6}} => [4,5,3,1,2,6] => [5,4,3,1,2,6] => 5
{{1,4,6},{2},{3,5}} => [4,2,5,6,3,1] => [2,4,5,6,3,1] => 1
{{1,4},{2,6},{3,5}} => [4,6,5,1,3,2] => [6,4,5,1,3,2] => 7
{{1,4},{2},{3,5,6}} => [4,2,5,1,6,3] => [2,4,5,1,6,3] => 0
{{1,4},{2},{3,5},{6}} => [4,2,5,1,3,6] => [2,4,5,1,3,6] => 0
{{1,4,6},{2},{3},{5}} => [4,2,3,6,5,1] => [2,4,3,6,5,1] => 2
{{1,4},{2,6},{3},{5}} => [4,6,3,1,5,2] => [6,4,3,1,5,2] => 6
{{1,4},{2},{3,6},{5}} => [4,2,6,1,5,3] => [2,4,6,1,5,3] => 1
{{1,4},{2},{3},{5,6}} => [4,2,3,1,6,5] => [2,4,3,1,6,5] => 1
{{1,4},{2},{3},{5},{6}} => [4,2,3,1,5,6] => [2,4,3,1,5,6] => 1
{{1,5,6},{2,4},{3}} => [5,4,3,2,6,1] => [4,5,3,2,6,1] => 3
{{1,5},{2,4,6},{3}} => [5,4,3,6,1,2] => [4,5,3,6,1,2] => 2
{{1,5},{2,4},{3,6}} => [5,4,6,2,1,3] => [4,5,6,2,1,3] => 1
{{1,5},{2,4},{3},{6}} => [5,4,3,2,1,6] => [4,5,3,2,1,6] => 3
{{1,6},{2,4,5},{3}} => [6,4,3,5,2,1] => [4,6,3,5,2,1] => 5
{{1},{2,4,5,6},{3}} => [1,4,3,5,6,2] => [4,1,3,5,6,2] => 1
{{1},{2,4,5},{3,6}} => [1,4,6,5,2,3] => [4,1,6,5,2,3] => 2
{{1},{2,4,5},{3},{6}} => [1,4,3,5,2,6] => [4,1,3,5,2,6] => 1
{{1,6},{2,4},{3,5}} => [6,4,5,2,3,1] => [4,6,5,2,3,1] => 5
{{1},{2,4,6},{3,5}} => [1,4,5,6,3,2] => [4,1,5,6,3,2] => 1
{{1},{2,4},{3,5,6}} => [1,4,5,2,6,3] => [4,1,5,2,6,3] => 0
{{1},{2,4},{3,5},{6}} => [1,4,5,2,3,6] => [4,1,5,2,3,6] => 0
{{1,6},{2,4},{3},{5}} => [6,4,3,2,5,1] => [4,6,3,2,5,1] => 4
{{1},{2,4,6},{3},{5}} => [1,4,3,6,5,2] => [4,1,3,6,5,2] => 2
{{1},{2,4},{3,6},{5}} => [1,4,6,2,5,3] => [4,1,6,2,5,3] => 1
{{1},{2,4},{3},{5,6}} => [1,4,3,2,6,5] => [4,1,3,2,6,5] => 1
{{1},{2,4},{3},{5},{6}} => [1,4,3,2,5,6] => [4,1,3,2,5,6] => 1
{{1,5,6},{2},{3,4}} => [5,2,4,3,6,1] => [2,5,4,3,6,1] => 3
{{1,5},{2,6},{3,4}} => [5,6,4,3,1,2] => [6,5,4,3,1,2] => 9
{{1,5},{2},{3,4,6}} => [5,2,4,6,1,3] => [2,5,4,6,1,3] => 2
{{1,5},{2},{3,4},{6}} => [5,2,4,3,1,6] => [2,5,4,3,1,6] => 3
{{1,6},{2,5},{3,4}} => [6,5,4,3,2,1] => [5,6,4,3,2,1] => 6
{{1},{2,5,6},{3,4}} => [1,5,4,3,6,2] => [5,1,4,3,6,2] => 3
{{1},{2,5},{3,4,6}} => [1,5,4,6,2,3] => [5,1,4,6,2,3] => 2
{{1},{2,5},{3,4},{6}} => [1,5,4,3,2,6] => [5,1,4,3,2,6] => 3
{{1,6},{2},{3,4,5}} => [6,2,4,5,3,1] => [2,6,4,5,3,1] => 5
{{1},{2,6},{3,4,5}} => [1,6,4,5,3,2] => [6,1,4,5,3,2] => 5
{{1},{2},{3,4,5,6}} => [1,2,4,5,6,3] => [1,2,5,4,6,3] => 1
{{1},{2},{3,4,5},{6}} => [1,2,4,5,3,6] => [1,2,5,4,3,6] => 1
{{1,6},{2},{3,4},{5}} => [6,2,4,3,5,1] => [2,6,4,3,5,1] => 4
{{1},{2,6},{3,4},{5}} => [1,6,4,3,5,2] => [6,1,4,3,5,2] => 4
{{1},{2},{3,4,6},{5}} => [1,2,4,6,5,3] => [1,2,6,4,5,3] => 2
{{1},{2},{3,4},{5,6}} => [1,2,4,3,6,5] => [1,2,4,3,6,5] => 0
{{1},{2},{3,4},{5},{6}} => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 0
{{1,5,6},{2},{3},{4}} => [5,2,3,4,6,1] => [2,5,3,4,6,1] => 2
{{1,5},{2,6},{3},{4}} => [5,6,3,4,1,2] => [6,5,3,4,1,2] => 8
{{1,5},{2},{3,6},{4}} => [5,2,6,4,1,3] => [2,5,6,4,1,3] => 2
{{1,5},{2},{3},{4,6}} => [5,2,3,6,1,4] => [2,5,3,6,1,4] => 1
{{1,5},{2},{3},{4},{6}} => [5,2,3,4,1,6] => [2,5,3,4,1,6] => 2
{{1,6},{2,5},{3},{4}} => [6,5,3,4,2,1] => [5,6,3,4,2,1] => 5
{{1},{2,5,6},{3},{4}} => [1,5,3,4,6,2] => [5,1,3,4,6,2] => 2
{{1},{2,5},{3,6},{4}} => [1,5,6,4,2,3] => [5,1,6,4,2,3] => 2
{{1},{2,5},{3},{4,6}} => [1,5,3,6,2,4] => [5,1,3,6,2,4] => 1
{{1},{2,5},{3},{4},{6}} => [1,5,3,4,2,6] => [5,1,3,4,2,6] => 2
{{1,6},{2},{3,5},{4}} => [6,2,5,4,3,1] => [2,6,5,4,3,1] => 6
{{1},{2,6},{3,5},{4}} => [1,6,5,4,3,2] => [6,1,5,4,3,2] => 6
{{1},{2},{3,5,6},{4}} => [1,2,5,4,6,3] => [1,2,4,5,6,3] => 0
{{1},{2},{3,5},{4,6}} => [1,2,5,6,3,4] => [1,2,6,5,3,4] => 2
{{1},{2},{3,5},{4},{6}} => [1,2,5,4,3,6] => [1,2,4,5,3,6] => 0
{{1,6},{2},{3},{4,5}} => [6,2,3,5,4,1] => [2,6,3,5,4,1] => 4
{{1},{2,6},{3},{4,5}} => [1,6,3,5,4,2] => [6,1,3,5,4,2] => 4
{{1},{2},{3,6},{4,5}} => [1,2,6,5,4,3] => [1,2,5,6,4,3] => 1
{{1},{2},{3},{4,5,6}} => [1,2,3,5,6,4] => [1,2,5,3,6,4] => 0
{{1},{2},{3},{4,5},{6}} => [1,2,3,5,4,6] => [1,2,5,3,4,6] => 0
{{1,6},{2},{3},{4},{5}} => [6,2,3,4,5,1] => [2,6,3,4,5,1] => 3
{{1},{2,6},{3},{4},{5}} => [1,6,3,4,5,2] => [6,1,3,4,5,2] => 3
{{1},{2},{3,6},{4},{5}} => [1,2,6,4,5,3] => [1,2,4,6,5,3] => 1
{{1},{2},{3},{4,6},{5}} => [1,2,3,6,5,4] => [1,2,6,3,5,4] => 1
{{1},{2},{3},{4},{5,6}} => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 0
{{1},{2},{3},{4},{5},{6}} => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
{{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => [3,2,4,5,6,7,1] => 1
{{1,2,3},{4},{5},{6},{7}} => [2,3,1,4,5,6,7] => [3,2,1,4,5,6,7] => 1
{{1,2,4,5,6,7},{3}} => [2,4,3,5,6,7,1] => [4,2,3,5,6,7,1] => 2
{{1,2,4},{3,5},{6},{7}} => [2,4,5,1,3,6,7] => [4,2,5,1,3,6,7] => 1
{{1,2,6},{3,4,5},{7}} => [2,6,4,5,3,1,7] => [6,2,4,5,3,1,7] => 6
{{1,2,5,6,7},{3},{4}} => [2,5,3,4,6,7,1] => [5,2,3,4,6,7,1] => 3
{{1,2,5},{3,6},{4,7}} => [2,5,6,7,1,3,4] => [5,2,6,7,1,3,4] => 1
{{1,2,6},{3,5},{4},{7}} => [2,6,5,4,3,1,7] => [6,2,5,4,3,1,7] => 7
{{1,2,6,7},{3},{4},{5}} => [2,6,3,4,5,7,1] => [6,2,3,4,5,7,1] => 4
{{1,2},{3},{4},{5},{6,7}} => [2,1,3,4,5,7,6] => [2,1,3,4,7,5,6] => 0
{{1,2},{3},{4},{5},{6},{7}} => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => 0
{{1,3,4,5,6,7},{2}} => [3,2,4,5,6,7,1] => [2,3,4,5,6,7,1] => 0
{{1,3,4,5,6},{2},{7}} => [3,2,4,5,6,1,7] => [2,3,4,5,6,1,7] => 0
{{1,3,4,5},{2},{6,7}} => [3,2,4,5,1,7,6] => [2,3,4,5,1,7,6] => 0
{{1,3,4,5},{2},{6},{7}} => [3,2,4,5,1,6,7] => [2,3,4,5,1,6,7] => 0
{{1,3,4,6},{2},{5,7}} => [3,2,4,6,7,1,5] => [2,3,4,6,7,1,5] => 0
{{1,3,4},{2},{5},{6,7}} => [3,2,4,1,5,7,6] => [2,3,4,1,5,7,6] => 0
{{1,3,4},{2},{5},{6},{7}} => [3,2,4,1,5,6,7] => [2,3,4,1,5,6,7] => 0
{{1,3,5,6,7},{2,4}} => [3,4,5,2,6,7,1] => [4,3,5,2,6,7,1] => 3
{{1,3,6},{2,4,5},{7}} => [3,4,6,5,2,1,7] => [4,3,6,5,2,1,7] => 5
{{1,3},{2},{4},{5},{6},{7}} => [3,2,1,4,5,6,7] => [2,3,1,4,5,6,7] => 0
{{1,4,5,6,7},{2,3}} => [4,3,2,5,6,7,1] => [3,4,2,5,6,7,1] => 1
{{1,4,6,7},{2,3,5}} => [4,3,5,6,2,7,1] => [3,4,5,6,2,7,1] => 1
{{1,4,6},{2,3,5,7}} => [4,3,5,6,7,1,2] => [3,4,5,6,7,1,2] => 0
{{1,6},{2,3,5},{4},{7}} => [6,3,5,4,2,1,7] => [3,6,5,4,2,1,7] => 6
{{1},{2,3},{4},{5},{6,7}} => [1,3,2,4,5,7,6] => [3,1,2,4,5,7,6] => 0
{{1},{2,3},{4},{5},{6},{7}} => [1,3,2,4,5,6,7] => [3,1,2,4,5,6,7] => 0
{{1,4,6},{2,5},{3},{7}} => [4,5,3,6,2,1,7] => [5,4,3,6,2,1,7] => 6
{{1,6},{2,5},{3,4},{7}} => [6,5,4,3,2,1,7] => [5,6,4,3,2,1,7] => 6
{{1,6},{2,7},{3,5},{4}} => [6,7,5,4,3,1,2] => [7,6,5,4,3,1,2] => 14
{{1,6},{2},{3,5},{4},{7}} => [6,2,5,4,3,1,7] => [2,6,5,4,3,1,7] => 6
{{1},{2},{3},{4},{5,6},{7}} => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => 0
{{1},{2},{3},{4},{5},{6},{7}} => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
{{1,2},{3,4},{5,6},{7,8}} => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 0
{{1,7},{2,8},{3,5},{4,6}} => [7,8,5,6,3,4,1,2] => [8,7,5,6,3,4,1,2] => 18
{{1,7},{2,8},{3,6},{4,5}} => [7,8,6,5,4,3,1,2] => [8,7,6,5,4,3,1,2] => 20
{{1},{2},{3},{4},{5},{6},{7},{8}} => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 0
{{1},{2},{3,4},{5,6},{7},{8}} => [1,2,4,3,6,5,7,8] => [1,2,4,3,6,5,7,8] => 0
{{1,2},{3},{4},{5},{6},{7,8}} => [2,1,3,4,5,6,8,7] => [2,1,3,4,5,6,8,7] => 0
{{1,2},{3,4},{5,6,7,8}} => [2,1,4,3,6,7,8,5] => [2,1,4,3,7,6,8,5] => 1
{{1,2},{3,4,5,6},{7,8}} => [2,1,4,5,6,3,8,7] => [2,1,5,4,6,3,8,7] => 1
{{1,2,3,4},{5,6},{7,8}} => [2,3,4,1,6,5,8,7] => [3,2,4,1,6,5,8,7] => 1
{{1,7},{2,6},{3,5},{4},{8}} => [7,6,5,4,3,2,1,8] => [6,7,5,4,3,2,1,8] => 10
{{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => [3,2,4,5,6,7,8,1] => 1
{{1,4},{2},{3,6},{5,7,8}} => [4,2,6,1,7,3,8,5] => [2,4,6,1,7,3,8,5] => 0
{{1,2},{3,5,7},{4,6,8}} => [2,1,5,6,7,8,3,4] => [2,1,6,5,7,8,3,4] => 2
{{1,2,4},{3,6},{5,7,8}} => [2,4,6,1,7,3,8,5] => [4,2,6,1,7,3,8,5] => 1
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Description
The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map.
  • The Simion-Schmidt map takes a permutation and turns each occurrence of [3,2,1] into an occurrence of [3,1,2], thus reducing the number of inversions of the permutation. This statistic records the difference in length of the permutation and its image.
  • It is the number of pairs of positions for the pattern letters 2 and 1 in occurrences of 321 in a permutation. Thus, for a permutation $\pi$ this is the number of pairs $(j,k)$ such that there exists an index $i$ satisfying $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. See also St000119The number of occurrences of the pattern 321 in a permutation. and St000371The number of mid points of decreasing subsequences of length 3 in a permutation..
  • Apparently, this statistic can be described as the number of occurrences of the mesh pattern ([3,2,1], {(0,3),(0,2)}). Equivalent mesh patterns are ([3,2,1], {(0,2),(1,2)}), ([3,2,1], {(0,3),(1,3)}) and ([3,2,1], {(1,2),(1,3)}).
Map
Alexandersson Kebede
Description
Sends a permutation to a permutation and it preserves the set of right-to-left minima.
Take a permutation $\pi$ of length $n$. The mapping looks for a smallest odd integer $i\in[n-1]$ such that swapping the entries $\pi(i)$ and $\pi(i+1)$ preserves the set of right-to-left minima. Otherwise, $\pi$ will be a fixed element of the mapping. Note that the map changes the sign of all non-fixed elements.
There are exactly $\binom{\lfloor n/2 \rfloor}{k-\lceil n/2 \rceil}$ elements in $S_n$ fixed under this map, with exactly $k$ right-to-left minima, see Lemma 35 in [1].
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.