Identifier
Values
{{1}} => [1] => [1] => [1,0] => 0
{{1,2}} => [2] => [1,1] => [1,0,1,0] => 1
{{1},{2}} => [1,1] => [2] => [1,1,0,0] => 0
{{1,2,3}} => [3] => [1,1,1] => [1,0,1,0,1,0] => 3
{{1,2},{3}} => [2,1] => [2,1] => [1,1,0,0,1,0] => 2
{{1,3},{2}} => [2,1] => [2,1] => [1,1,0,0,1,0] => 2
{{1},{2,3}} => [1,2] => [1,2] => [1,0,1,1,0,0] => 1
{{1},{2},{3}} => [1,1,1] => [3] => [1,1,1,0,0,0] => 0
{{1,2,3,4}} => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 6
{{1,2,3},{4}} => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0] => 5
{{1,2,4},{3}} => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0] => 5
{{1,2},{3,4}} => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0] => 4
{{1,2},{3},{4}} => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0] => 3
{{1,3,4},{2}} => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0] => 5
{{1,3},{2,4}} => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0] => 4
{{1,3},{2},{4}} => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0] => 3
{{1,4},{2,3}} => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0] => 4
{{1},{2,3,4}} => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0] => 3
{{1},{2,3},{4}} => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0] => 2
{{1,4},{2},{3}} => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0] => 3
{{1},{2,4},{3}} => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0] => 2
{{1},{2},{3,4}} => [1,1,2] => [1,3] => [1,0,1,1,1,0,0,0] => 1
{{1},{2},{3},{4}} => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0] => 0
{{1,2,3,4,5}} => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 10
{{1,2,3,4},{5}} => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 9
{{1,2,3,5},{4}} => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 9
{{1,2,3},{4,5}} => [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 8
{{1,2,3},{4},{5}} => [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 7
{{1,2,4,5},{3}} => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 9
{{1,2,4},{3,5}} => [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 8
{{1,2,4},{3},{5}} => [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 7
{{1,2,5},{3,4}} => [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 8
{{1,2},{3,4,5}} => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 7
{{1,2},{3,4},{5}} => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 6
{{1,2,5},{3},{4}} => [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 7
{{1,2},{3,5},{4}} => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 6
{{1,2},{3},{4,5}} => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 5
{{1,2},{3},{4},{5}} => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 4
{{1,3,4,5},{2}} => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 9
{{1,3,4},{2,5}} => [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 8
{{1,3,4},{2},{5}} => [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 7
{{1,3,5},{2,4}} => [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 8
{{1,3},{2,4,5}} => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 7
{{1,3},{2,4},{5}} => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 6
{{1,3,5},{2},{4}} => [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 7
{{1,3},{2,5},{4}} => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 6
{{1,3},{2},{4,5}} => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 5
{{1,3},{2},{4},{5}} => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 4
{{1,4,5},{2,3}} => [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 8
{{1,4},{2,3,5}} => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 7
{{1,4},{2,3},{5}} => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 6
{{1,5},{2,3,4}} => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 7
{{1},{2,3,4,5}} => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 6
{{1},{2,3,4},{5}} => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 5
{{1,5},{2,3},{4}} => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 6
{{1},{2,3,5},{4}} => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 5
{{1},{2,3},{4,5}} => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4
{{1},{2,3},{4},{5}} => [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 3
{{1,4,5},{2},{3}} => [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 7
{{1,4},{2,5},{3}} => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 6
{{1,4},{2},{3,5}} => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 5
{{1,4},{2},{3},{5}} => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 4
{{1,5},{2,4},{3}} => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 6
{{1},{2,4,5},{3}} => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 5
{{1},{2,4},{3,5}} => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4
{{1},{2,4},{3},{5}} => [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 3
{{1,5},{2},{3,4}} => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 5
{{1},{2,5},{3,4}} => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4
{{1},{2},{3,4,5}} => [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3
{{1},{2},{3,4},{5}} => [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2
{{1,5},{2},{3},{4}} => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 4
{{1},{2,5},{3},{4}} => [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 3
{{1},{2},{3,5},{4}} => [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2
{{1},{2},{3},{4,5}} => [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1
{{1},{2},{3},{4},{5}} => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0] => 0
{{1,2,3,4,5,6}} => [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 15
{{1,2,3,4,5},{6}} => [5,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 14
{{1,2,3,4,6},{5}} => [5,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 14
{{1,2,3,4},{5,6}} => [4,2] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 13
{{1,2,3,4},{5},{6}} => [4,1,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 12
{{1,2,3,5,6},{4}} => [5,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 14
{{1,2,3,5},{4,6}} => [4,2] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 13
{{1,2,3,5},{4},{6}} => [4,1,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 12
{{1,2,3,6},{4,5}} => [4,2] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 13
{{1,2,3},{4,5,6}} => [3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 12
{{1,2,3},{4,5},{6}} => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 11
{{1,2,3,6},{4},{5}} => [4,1,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 12
{{1,2,3},{4,6},{5}} => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 11
{{1,2,3},{4},{5,6}} => [3,1,2] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 10
{{1,2,3},{4},{5},{6}} => [3,1,1,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 9
{{1,2,4,5,6},{3}} => [5,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 14
{{1,2,4,5},{3,6}} => [4,2] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 13
{{1,2,4,5},{3},{6}} => [4,1,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 12
{{1,2,4,6},{3,5}} => [4,2] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 13
{{1,2,4},{3,5,6}} => [3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 12
{{1,2,4},{3,5},{6}} => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 11
{{1,2,4,6},{3},{5}} => [4,1,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 12
{{1,2,4},{3,6},{5}} => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 11
{{1,2,4},{3},{5,6}} => [3,1,2] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 10
{{1,2,4},{3},{5},{6}} => [3,1,1,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 9
{{1,2,5,6},{3,4}} => [4,2] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 13
>>> Load all 278 entries. <<<
{{1,2,5},{3,4,6}} => [3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 12
{{1,2,5},{3,4},{6}} => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 11
{{1,2,6},{3,4,5}} => [3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 12
{{1,2},{3,4,5,6}} => [2,4] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 11
{{1,2},{3,4,5},{6}} => [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 10
{{1,2,6},{3,4},{5}} => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 11
{{1,2},{3,4,6},{5}} => [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 10
{{1,2},{3,4},{5,6}} => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 9
{{1,2},{3,4},{5},{6}} => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 8
{{1,2,5,6},{3},{4}} => [4,1,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 12
{{1,2,5},{3,6},{4}} => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 11
{{1,2,5},{3},{4,6}} => [3,1,2] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 10
{{1,2,5},{3},{4},{6}} => [3,1,1,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 9
{{1,2,6},{3,5},{4}} => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 11
{{1,2},{3,5,6},{4}} => [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 10
{{1,2},{3,5},{4,6}} => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 9
{{1,2},{3,5},{4},{6}} => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 8
{{1,2,6},{3},{4,5}} => [3,1,2] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 10
{{1,2},{3,6},{4,5}} => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 9
{{1,2},{3},{4,5,6}} => [2,1,3] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 8
{{1,2},{3},{4,5},{6}} => [2,1,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 7
{{1,2,6},{3},{4},{5}} => [3,1,1,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 9
{{1,2},{3,6},{4},{5}} => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 8
{{1,2},{3},{4,6},{5}} => [2,1,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 7
{{1,2},{3},{4},{5,6}} => [2,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 6
{{1,2},{3},{4},{5},{6}} => [2,1,1,1,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5
{{1,3,4,5,6},{2}} => [5,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 14
{{1,3,4,5},{2,6}} => [4,2] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 13
{{1,3,4,5},{2},{6}} => [4,1,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 12
{{1,3,4,6},{2,5}} => [4,2] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 13
{{1,3,4},{2,5,6}} => [3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 12
{{1,3,4},{2,5},{6}} => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 11
{{1,3,4,6},{2},{5}} => [4,1,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 12
{{1,3,4},{2,6},{5}} => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 11
{{1,3,4},{2},{5,6}} => [3,1,2] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 10
{{1,3,4},{2},{5},{6}} => [3,1,1,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 9
{{1,3,5,6},{2,4}} => [4,2] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 13
{{1,3,5},{2,4,6}} => [3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 12
{{1,3,5},{2,4},{6}} => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 11
{{1,3,6},{2,4,5}} => [3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 12
{{1,3},{2,4,5,6}} => [2,4] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 11
{{1,3},{2,4,5},{6}} => [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 10
{{1,3,6},{2,4},{5}} => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 11
{{1,3},{2,4,6},{5}} => [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 10
{{1,3},{2,4},{5,6}} => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 9
{{1,3},{2,4},{5},{6}} => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 8
{{1,3,5,6},{2},{4}} => [4,1,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 12
{{1,3,5},{2,6},{4}} => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 11
{{1,3,5},{2},{4,6}} => [3,1,2] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 10
{{1,3,5},{2},{4},{6}} => [3,1,1,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 9
{{1,3,6},{2,5},{4}} => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 11
{{1,3},{2,5,6},{4}} => [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 10
{{1,3},{2,5},{4,6}} => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 9
{{1,3},{2,5},{4},{6}} => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 8
{{1,3,6},{2},{4,5}} => [3,1,2] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 10
{{1,3},{2,6},{4,5}} => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 9
{{1,3},{2},{4,5,6}} => [2,1,3] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 8
{{1,3},{2},{4,5},{6}} => [2,1,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 7
{{1,3,6},{2},{4},{5}} => [3,1,1,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 9
{{1,3},{2,6},{4},{5}} => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 8
{{1,3},{2},{4,6},{5}} => [2,1,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 7
{{1,3},{2},{4},{5,6}} => [2,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 6
{{1,3},{2},{4},{5},{6}} => [2,1,1,1,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5
{{1,4,5,6},{2,3}} => [4,2] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 13
{{1,4,5},{2,3,6}} => [3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 12
{{1,4,5},{2,3},{6}} => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 11
{{1,4,6},{2,3,5}} => [3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 12
{{1,4},{2,3,5,6}} => [2,4] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 11
{{1,4},{2,3,5},{6}} => [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 10
{{1,4,6},{2,3},{5}} => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 11
{{1,4},{2,3,6},{5}} => [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 10
{{1,4},{2,3},{5,6}} => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 9
{{1,4},{2,3},{5},{6}} => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 8
{{1,5,6},{2,3,4}} => [3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 12
{{1,5},{2,3,4,6}} => [2,4] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 11
{{1,5},{2,3,4},{6}} => [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 10
{{1,6},{2,3,4,5}} => [2,4] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 11
{{1},{2,3,4,5,6}} => [1,5] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 10
{{1},{2,3,4,5},{6}} => [1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 9
{{1,6},{2,3,4},{5}} => [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 10
{{1},{2,3,4,6},{5}} => [1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 9
{{1},{2,3,4},{5,6}} => [1,3,2] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 8
{{1},{2,3,4},{5},{6}} => [1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 7
{{1,5,6},{2,3},{4}} => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 11
{{1,5},{2,3,6},{4}} => [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 10
{{1,5},{2,3},{4,6}} => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 9
{{1,5},{2,3},{4},{6}} => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 8
{{1,6},{2,3,5},{4}} => [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 10
{{1},{2,3,5,6},{4}} => [1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 9
{{1},{2,3,5},{4,6}} => [1,3,2] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 8
{{1},{2,3,5},{4},{6}} => [1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 7
{{1,6},{2,3},{4,5}} => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 9
{{1},{2,3,6},{4,5}} => [1,3,2] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 8
{{1},{2,3},{4,5,6}} => [1,2,3] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 7
{{1},{2,3},{4,5},{6}} => [1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 6
{{1,6},{2,3},{4},{5}} => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 8
{{1},{2,3,6},{4},{5}} => [1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 7
{{1},{2,3},{4,6},{5}} => [1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 6
{{1},{2,3},{4},{5,6}} => [1,2,1,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
{{1},{2,3},{4},{5},{6}} => [1,2,1,1,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4
{{1,4,5,6},{2},{3}} => [4,1,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 12
{{1,4,5},{2,6},{3}} => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 11
{{1,4,5},{2},{3,6}} => [3,1,2] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 10
{{1,4,5},{2},{3},{6}} => [3,1,1,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 9
{{1,4,6},{2,5},{3}} => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 11
{{1,4},{2,5,6},{3}} => [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 10
{{1,4},{2,5},{3,6}} => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 9
{{1,4},{2,5},{3},{6}} => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 8
{{1,4,6},{2},{3,5}} => [3,1,2] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 10
{{1,4},{2,6},{3,5}} => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 9
{{1,4},{2},{3,5,6}} => [2,1,3] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 8
{{1,4},{2},{3,5},{6}} => [2,1,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 7
{{1,4,6},{2},{3},{5}} => [3,1,1,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 9
{{1,4},{2,6},{3},{5}} => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 8
{{1,4},{2},{3,6},{5}} => [2,1,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 7
{{1,4},{2},{3},{5,6}} => [2,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 6
{{1,4},{2},{3},{5},{6}} => [2,1,1,1,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5
{{1,5,6},{2,4},{3}} => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 11
{{1,5},{2,4,6},{3}} => [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 10
{{1,5},{2,4},{3,6}} => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 9
{{1,5},{2,4},{3},{6}} => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 8
{{1,6},{2,4,5},{3}} => [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 10
{{1},{2,4,5,6},{3}} => [1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 9
{{1},{2,4,5},{3,6}} => [1,3,2] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 8
{{1},{2,4,5},{3},{6}} => [1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 7
{{1,6},{2,4},{3,5}} => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 9
{{1},{2,4,6},{3,5}} => [1,3,2] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 8
{{1},{2,4},{3,5,6}} => [1,2,3] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 7
{{1},{2,4},{3,5},{6}} => [1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 6
{{1,6},{2,4},{3},{5}} => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 8
{{1},{2,4,6},{3},{5}} => [1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 7
{{1},{2,4},{3,6},{5}} => [1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 6
{{1},{2,4},{3},{5,6}} => [1,2,1,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
{{1},{2,4},{3},{5},{6}} => [1,2,1,1,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4
{{1,5,6},{2},{3,4}} => [3,1,2] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 10
{{1,5},{2,6},{3,4}} => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 9
{{1,5},{2},{3,4,6}} => [2,1,3] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 8
{{1,5},{2},{3,4},{6}} => [2,1,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 7
{{1,6},{2,5},{3,4}} => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 9
{{1},{2,5,6},{3,4}} => [1,3,2] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 8
{{1},{2,5},{3,4,6}} => [1,2,3] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 7
{{1},{2,5},{3,4},{6}} => [1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 6
{{1,6},{2},{3,4,5}} => [2,1,3] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 8
{{1},{2,6},{3,4,5}} => [1,2,3] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 7
{{1},{2},{3,4,5,6}} => [1,1,4] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 6
{{1},{2},{3,4,5},{6}} => [1,1,3,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 5
{{1,6},{2},{3,4},{5}} => [2,1,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 7
{{1},{2,6},{3,4},{5}} => [1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 6
{{1},{2},{3,4,6},{5}} => [1,1,3,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 5
{{1},{2},{3,4},{5,6}} => [1,1,2,2] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 4
{{1},{2},{3,4},{5},{6}} => [1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
{{1,5,6},{2},{3},{4}} => [3,1,1,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 9
{{1,5},{2,6},{3},{4}} => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 8
{{1,5},{2},{3,6},{4}} => [2,1,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 7
{{1,5},{2},{3},{4,6}} => [2,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 6
{{1,5},{2},{3},{4},{6}} => [2,1,1,1,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5
{{1,6},{2,5},{3},{4}} => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 8
{{1},{2,5,6},{3},{4}} => [1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 7
{{1},{2,5},{3,6},{4}} => [1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 6
{{1},{2,5},{3},{4,6}} => [1,2,1,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
{{1},{2,5},{3},{4},{6}} => [1,2,1,1,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4
{{1,6},{2},{3,5},{4}} => [2,1,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 7
{{1},{2,6},{3,5},{4}} => [1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 6
{{1},{2},{3,5,6},{4}} => [1,1,3,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 5
{{1},{2},{3,5},{4,6}} => [1,1,2,2] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 4
{{1},{2},{3,5},{4},{6}} => [1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
{{1,6},{2},{3},{4,5}} => [2,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 6
{{1},{2,6},{3},{4,5}} => [1,2,1,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
{{1},{2},{3,6},{4,5}} => [1,1,2,2] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 4
{{1},{2},{3},{4,5,6}} => [1,1,1,3] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 3
{{1},{2},{3},{4,5},{6}} => [1,1,1,2,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2
{{1,6},{2},{3},{4},{5}} => [2,1,1,1,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5
{{1},{2,6},{3},{4},{5}} => [1,2,1,1,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4
{{1},{2},{3,6},{4},{5}} => [1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
{{1},{2},{3},{4,6},{5}} => [1,1,1,2,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2
{{1},{2},{3},{4},{5,6}} => [1,1,1,1,2] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
{{1},{2},{3},{4},{5},{6}} => [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
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Description
The bounce statistic of a Dyck path.
The bounce path $D'$ of a Dyck path $D$ is the Dyck path obtained from $D$ by starting at the end point $(2n,0)$, traveling north-west until hitting $D$, then bouncing back south-west to the $x$-axis, and repeating this procedure until finally reaching the point $(0,0)$.
The points where $D'$ touches the $x$-axis are called bounce points, and a bounce path is uniquely determined by its bounce points.
This statistic is given by the sum of all $i$ for which the bounce path $D'$ of $D$ touches the $x$-axis at $(2i,0)$.
In particular, the bounce statistics of $D$ and $D'$ coincide.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
conjugate
Description
The conjugate of a composition.
The conjugate of a composition $C$ is defined as the complement (Mp00039complement) of the reversal (Mp00038reverse) of $C$.
Equivalently, the ribbon shape corresponding to the conjugate of $C$ is the conjugate of the ribbon shape of $C$.
Map
to composition
Description
The integer composition of block sizes of a set partition.
For a set partition of $\{1,2,\dots,n\}$, this is the integer composition of $n$ obtained by sorting the blocks by their minimal element and then taking the block sizes.