Identifier
-
Mp00025:
Dyck paths
—to 132-avoiding permutation⟶
Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
St000575: Set partitions ⟶ ℤ
Values
[1,0,1,0] => [2,1] => {{1,2}} => {{1,2}} => 0
[1,1,0,0] => [1,2] => {{1},{2}} => {{1},{2}} => 1
[1,0,1,0,1,0] => [3,2,1] => {{1,3},{2}} => {{1,3},{2}} => 0
[1,0,1,1,0,0] => [2,3,1] => {{1,2,3}} => {{1,2,3}} => 0
[1,1,0,0,1,0] => [3,1,2] => {{1,2,3}} => {{1,2,3}} => 0
[1,1,0,1,0,0] => [2,1,3] => {{1,2},{3}} => {{1},{2,3}} => 0
[1,1,1,0,0,0] => [1,2,3] => {{1},{2},{3}} => {{1},{2},{3}} => 3
[1,0,1,0,1,0,1,0] => [4,3,2,1] => {{1,4},{2,3}} => {{1,4},{2,3}} => 0
[1,0,1,0,1,1,0,0] => [3,4,2,1] => {{1,2,3,4}} => {{1,2,3,4}} => 0
[1,0,1,1,0,0,1,0] => [4,2,3,1] => {{1,4},{2},{3}} => {{1,4},{2},{3}} => 1
[1,0,1,1,0,1,0,0] => [3,2,4,1] => {{1,3,4},{2}} => {{1,2,4},{3}} => 0
[1,0,1,1,1,0,0,0] => [2,3,4,1] => {{1,2,3,4}} => {{1,2,3,4}} => 0
[1,1,0,0,1,0,1,0] => [4,3,1,2] => {{1,2,3,4}} => {{1,2,3,4}} => 0
[1,1,0,0,1,1,0,0] => [3,4,1,2] => {{1,3},{2,4}} => {{1,3},{2,4}} => 0
[1,1,0,1,0,0,1,0] => [4,2,1,3] => {{1,3,4},{2}} => {{1,2,4},{3}} => 0
[1,1,0,1,0,1,0,0] => [3,2,1,4] => {{1,3},{2},{4}} => {{1},{2,4},{3}} => 1
[1,1,0,1,1,0,0,0] => [2,3,1,4] => {{1,2,3},{4}} => {{1},{2,3,4}} => 0
[1,1,1,0,0,0,1,0] => [4,1,2,3] => {{1,2,3,4}} => {{1,2,3,4}} => 0
[1,1,1,0,0,1,0,0] => [3,1,2,4] => {{1,2,3},{4}} => {{1},{2,3,4}} => 0
[1,1,1,0,1,0,0,0] => [2,1,3,4] => {{1,2},{3},{4}} => {{1},{2},{3,4}} => 1
[1,1,1,1,0,0,0,0] => [1,2,3,4] => {{1},{2},{3},{4}} => {{1},{2},{3},{4}} => 6
[1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => {{1,5},{2,4},{3}} => {{1,5},{2,4},{3}} => 0
[1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => {{1,2,4,5},{3}} => {{1,2,4,5},{3}} => 0
[1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => {{1,5},{2,3,4}} => {{1,5},{2,3,4}} => 0
[1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => {{1,2,3,4,5}} => {{1,2,3,4,5}} => 0
[1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => {{1,3,5},{2,4}} => {{1,3,5},{2,4}} => 0
[1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => {{1,5},{2,3,4}} => {{1,5},{2,3,4}} => 0
[1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => {{1,2,3,4,5}} => {{1,2,3,4,5}} => 0
[1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => {{1,5},{2,3},{4}} => {{1,5},{2},{3,4}} => 0
[1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => {{1,4,5},{2,3}} => {{1,2,5},{3,4}} => 0
[1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => {{1,2,3,4,5}} => {{1,2,3,4,5}} => 0
[1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => {{1,5},{2},{3},{4}} => {{1,5},{2},{3},{4}} => 3
[1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => {{1,4,5},{2},{3}} => {{1,2,5},{3},{4}} => 1
[1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => {{1,3,4,5},{2}} => {{1,2,3,5},{4}} => 0
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => {{1,2,3,4,5}} => {{1,2,3,4,5}} => 0
[1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => {{1,2,4,5},{3}} => {{1,2,4,5},{3}} => 0
[1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => {{1,4},{2,5},{3}} => {{1,4},{2,5},{3}} => 0
[1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => {{1,2,3,4,5}} => {{1,2,3,4,5}} => 0
[1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => {{1,4},{2,3,5}} => {{1,3,4},{2,5}} => 0
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => {{1,2,3,4,5}} => {{1,2,3,4,5}} => 0
[1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => {{1,2,3,4,5}} => {{1,2,3,4,5}} => 0
[1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => {{1,4},{2,3,5}} => {{1,3,4},{2,5}} => 0
[1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => {{1,4,5},{2,3}} => {{1,2,5},{3,4}} => 0
[1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => {{1,4},{2,3},{5}} => {{1},{2,5},{3,4}} => 0
[1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => {{1,2,3,4},{5}} => {{1},{2,3,4,5}} => 0
[1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => {{1,4,5},{2},{3}} => {{1,2,5},{3},{4}} => 1
[1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => {{1,4},{2},{3},{5}} => {{1},{2,5},{3},{4}} => 3
[1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => {{1,3,4},{2},{5}} => {{1},{2,3,5},{4}} => 1
[1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => {{1,2,3,4},{5}} => {{1},{2,3,4,5}} => 0
[1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => {{1,3,5},{2,4}} => {{1,3,5},{2,4}} => 0
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => {{1,2,3,4,5}} => {{1,2,3,4,5}} => 0
[1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => {{1,2,3,4,5}} => {{1,2,3,4,5}} => 0
[1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => {{1,2,3,4},{5}} => {{1},{2,3,4,5}} => 0
[1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => {{1,3},{2,4},{5}} => {{1},{2,4},{3,5}} => 0
[1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => {{1,3,4,5},{2}} => {{1,2,3,5},{4}} => 0
[1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => {{1,3,4},{2},{5}} => {{1},{2,3,5},{4}} => 1
[1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => {{1,3},{2},{4},{5}} => {{1},{2},{3,5},{4}} => 3
[1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => {{1,2,3},{4},{5}} => {{1},{2},{3,4,5}} => 1
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => {{1,2,3,4,5}} => {{1,2,3,4,5}} => 0
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => {{1,2,3,4},{5}} => {{1},{2,3,4,5}} => 0
[1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => {{1,2,3},{4},{5}} => {{1},{2},{3,4,5}} => 1
[1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => {{1,2},{3},{4},{5}} => {{1},{2},{3},{4,5}} => 3
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}} => {{1},{2},{3},{4},{5}} => 10
[1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => {{1,6},{2,5},{3,4}} => {{1,6},{2,5},{3,4}} => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => {{1,2,5,6},{3,4}} => {{1,2,5,6},{3,4}} => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => {{1,6},{2,3,4,5}} => {{1,6},{2,3,4,5}} => 0
[1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1] => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0
[1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => {{1,3,4,6},{2,5}} => {{1,3,4,6},{2,5}} => 0
[1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => {{1,6},{2,5},{3},{4}} => {{1,6},{2,5},{3},{4}} => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => {{1,2,5,6},{3},{4}} => {{1,2,5,6},{3},{4}} => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => {{1,6},{2,4,5},{3}} => {{1,6},{2,3,5},{4}} => 0
[1,0,1,0,1,1,0,1,0,1,0,0] => [5,4,3,6,2,1] => {{1,2,4,5,6},{3}} => {{1,2,3,5,6},{4}} => 0
[1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => {{1,4,6},{2,5},{3}} => {{1,3,6},{2,5},{4}} => 0
[1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => {{1,6},{2,3,4,5}} => {{1,6},{2,3,4,5}} => 0
[1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1] => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0
[1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1] => {{1,4,6},{2,3,5}} => {{1,3,6},{2,4,5}} => 0
[1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0
[1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => {{1,6},{2,3,4,5}} => {{1,6},{2,3,4,5}} => 0
[1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0
[1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => {{1,6},{2,4},{3,5}} => {{1,6},{2,4},{3,5}} => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1] => {{1,3,5,6},{2,4}} => {{1,2,4,6},{3,5}} => 0
[1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1] => {{1,6},{2,4,5},{3}} => {{1,6},{2,3,5},{4}} => 0
[1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1] => {{1,2,4,5,6},{3}} => {{1,2,3,5,6},{4}} => 0
[1,0,1,1,0,1,0,1,0,0,1,0] => [6,4,3,2,5,1] => {{1,6},{2,4},{3},{5}} => {{1,6},{2},{3,5},{4}} => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [5,4,3,2,6,1] => {{1,5,6},{2,4},{3}} => {{1,2,6},{3,5},{4}} => 0
[1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => {{1,2,4,5,6},{3}} => {{1,2,3,5,6},{4}} => 0
[1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => {{1,6},{2,3,4},{5}} => {{1,6},{2},{3,4,5}} => 0
[1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => {{1,5,6},{2,3,4}} => {{1,2,6},{3,4,5}} => 0
[1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0
[1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => {{1,3,5,6},{2,4}} => {{1,2,4,6},{3,5}} => 0
[1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => {{1,6},{2,3,4,5}} => {{1,6},{2,3,4,5}} => 0
[1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0
[1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1] => {{1,6},{2,3,4},{5}} => {{1,6},{2},{3,4,5}} => 0
[1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => {{1,5,6},{2,3,4}} => {{1,2,6},{3,4,5}} => 0
[1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0
[1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => {{1,6},{2,3},{4},{5}} => {{1,6},{2},{3},{4,5}} => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => {{1,5,6},{2,3},{4}} => {{1,2,6},{3},{4,5}} => 0
[1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => {{1,4,5,6},{2,3}} => {{1,2,3,6},{4,5}} => 0
[1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0
[1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => {{1,6},{2},{3},{4},{5}} => {{1,6},{2},{3},{4},{5}} => 6
>>> Load all 975 entries. <<<[1,0,1,1,1,1,0,0,0,1,0,0] => [5,2,3,4,6,1] => {{1,5,6},{2},{3},{4}} => {{1,2,6},{3},{4},{5}} => 3
[1,0,1,1,1,1,0,0,1,0,0,0] => [4,2,3,5,6,1] => {{1,4,5,6},{2},{3}} => {{1,2,3,6},{4},{5}} => 1
[1,0,1,1,1,1,0,1,0,0,0,0] => [3,2,4,5,6,1] => {{1,3,4,5,6},{2}} => {{1,2,3,4,6},{5}} => 0
[1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0
[1,1,0,0,1,0,1,0,1,0,1,0] => [6,5,4,3,1,2] => {{1,2,5,6},{3,4}} => {{1,2,5,6},{3,4}} => 0
[1,1,0,0,1,0,1,0,1,1,0,0] => [5,6,4,3,1,2] => {{1,5},{2,6},{3,4}} => {{1,5},{2,6},{3,4}} => 0
[1,1,0,0,1,0,1,1,0,0,1,0] => [6,4,5,3,1,2] => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0
[1,1,0,0,1,0,1,1,0,1,0,0] => [5,4,6,3,1,2] => {{1,5},{2,3,4,6}} => {{1,3,4,5},{2,6}} => 0
[1,1,0,0,1,0,1,1,1,0,0,0] => [4,5,6,3,1,2] => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0
[1,1,0,0,1,1,0,0,1,0,1,0] => [6,5,3,4,1,2] => {{1,2,5,6},{3},{4}} => {{1,2,5,6},{3},{4}} => 1
[1,1,0,0,1,1,0,0,1,1,0,0] => [5,6,3,4,1,2] => {{1,5},{2,6},{3},{4}} => {{1,5},{2,6},{3},{4}} => 1
[1,1,0,0,1,1,0,1,0,0,1,0] => [6,4,3,5,1,2] => {{1,2,4,5,6},{3}} => {{1,2,3,5,6},{4}} => 0
[1,1,0,0,1,1,0,1,0,1,0,0] => [5,4,3,6,1,2] => {{1,5},{2,4,6},{3}} => {{1,3,5},{2,6},{4}} => 0
[1,1,0,0,1,1,0,1,1,0,0,0] => [4,5,3,6,1,2] => {{1,2,4,5,6},{3}} => {{1,2,3,5,6},{4}} => 0
[1,1,0,0,1,1,1,0,0,0,1,0] => [6,3,4,5,1,2] => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0
[1,1,0,0,1,1,1,0,0,1,0,0] => [5,3,4,6,1,2] => {{1,5},{2,3,4,6}} => {{1,3,4,5},{2,6}} => 0
[1,1,0,0,1,1,1,0,1,0,0,0] => [4,3,5,6,1,2] => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0
[1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => {{1,3,5},{2,4,6}} => {{1,3,5},{2,4,6}} => 0
[1,1,0,1,0,0,1,0,1,0,1,0] => [6,5,4,2,1,3] => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0
[1,1,0,1,0,0,1,0,1,1,0,0] => [5,6,4,2,1,3] => {{1,5},{2,3,4,6}} => {{1,3,4,5},{2,6}} => 0
[1,1,0,1,0,0,1,1,0,0,1,0] => [6,4,5,2,1,3] => {{1,3,5,6},{2,4}} => {{1,2,4,6},{3,5}} => 0
[1,1,0,1,0,0,1,1,0,1,0,0] => [5,4,6,2,1,3] => {{1,5},{2,4},{3,6}} => {{1,4},{2,6},{3,5}} => 0
[1,1,0,1,0,0,1,1,1,0,0,0] => [4,5,6,2,1,3] => {{1,2,4,5},{3,6}} => {{1,4},{2,3,5,6}} => 0
[1,1,0,1,0,1,0,0,1,0,1,0] => [6,5,3,2,1,4] => {{1,2,4,5,6},{3}} => {{1,2,3,5,6},{4}} => 0
[1,1,0,1,0,1,0,0,1,1,0,0] => [5,6,3,2,1,4] => {{1,5},{2,4,6},{3}} => {{1,3,5},{2,6},{4}} => 0
[1,1,0,1,0,1,0,1,0,0,1,0] => [6,4,3,2,1,5] => {{1,5,6},{2,4},{3}} => {{1,2,6},{3,5},{4}} => 0
[1,1,0,1,0,1,0,1,0,1,0,0] => [5,4,3,2,1,6] => {{1,5},{2,4},{3},{6}} => {{1},{2,6},{3,5},{4}} => 1
[1,1,0,1,0,1,0,1,1,0,0,0] => [4,5,3,2,1,6] => {{1,2,4,5},{3},{6}} => {{1},{2,3,5,6},{4}} => 1
[1,1,0,1,0,1,1,0,0,0,1,0] => [6,3,4,2,1,5] => {{1,5,6},{2,3,4}} => {{1,2,6},{3,4,5}} => 0
[1,1,0,1,0,1,1,0,0,1,0,0] => [5,3,4,2,1,6] => {{1,5},{2,3,4},{6}} => {{1},{2,6},{3,4,5}} => 0
[1,1,0,1,0,1,1,0,1,0,0,0] => [4,3,5,2,1,6] => {{1,2,3,4,5},{6}} => {{1},{2,3,4,5,6}} => 0
[1,1,0,1,0,1,1,1,0,0,0,0] => [3,4,5,2,1,6] => {{1,3,5},{2,4},{6}} => {{1},{2,4,6},{3,5}} => 0
[1,1,0,1,1,0,0,0,1,0,1,0] => [6,5,2,3,1,4] => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0
[1,1,0,1,1,0,0,0,1,1,0,0] => [5,6,2,3,1,4] => {{1,5},{2,3,4,6}} => {{1,3,4,5},{2,6}} => 0
[1,1,0,1,1,0,0,1,0,0,1,0] => [6,4,2,3,1,5] => {{1,5,6},{2,3,4}} => {{1,2,6},{3,4,5}} => 0
[1,1,0,1,1,0,0,1,0,1,0,0] => [5,4,2,3,1,6] => {{1,5},{2,3,4},{6}} => {{1},{2,6},{3,4,5}} => 0
[1,1,0,1,1,0,0,1,1,0,0,0] => [4,5,2,3,1,6] => {{1,2,3,4,5},{6}} => {{1},{2,3,4,5,6}} => 0
[1,1,0,1,1,0,1,0,0,0,1,0] => [6,3,2,4,1,5] => {{1,5,6},{2,3},{4}} => {{1,2,6},{3},{4,5}} => 0
[1,1,0,1,1,0,1,0,0,1,0,0] => [5,3,2,4,1,6] => {{1,5},{2,3},{4},{6}} => {{1},{2,6},{3},{4,5}} => 1
[1,1,0,1,1,0,1,0,1,0,0,0] => [4,3,2,5,1,6] => {{1,4,5},{2,3},{6}} => {{1},{2,3,6},{4,5}} => 0
[1,1,0,1,1,0,1,1,0,0,0,0] => [3,4,2,5,1,6] => {{1,2,3,4,5},{6}} => {{1},{2,3,4,5,6}} => 0
[1,1,0,1,1,1,0,0,0,0,1,0] => [6,2,3,4,1,5] => {{1,5,6},{2},{3},{4}} => {{1,2,6},{3},{4},{5}} => 3
[1,1,0,1,1,1,0,0,0,1,0,0] => [5,2,3,4,1,6] => {{1,5},{2},{3},{4},{6}} => {{1},{2,6},{3},{4},{5}} => 6
[1,1,0,1,1,1,0,0,1,0,0,0] => [4,2,3,5,1,6] => {{1,4,5},{2},{3},{6}} => {{1},{2,3,6},{4},{5}} => 3
[1,1,0,1,1,1,0,1,0,0,0,0] => [3,2,4,5,1,6] => {{1,3,4,5},{2},{6}} => {{1},{2,3,4,6},{5}} => 1
[1,1,0,1,1,1,1,0,0,0,0,0] => [2,3,4,5,1,6] => {{1,2,3,4,5},{6}} => {{1},{2,3,4,5,6}} => 0
[1,1,1,0,0,0,1,0,1,0,1,0] => [6,5,4,1,2,3] => {{1,3,4,6},{2,5}} => {{1,3,4,6},{2,5}} => 0
[1,1,1,0,0,0,1,0,1,1,0,0] => [5,6,4,1,2,3] => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0
[1,1,1,0,0,0,1,1,0,0,1,0] => [6,4,5,1,2,3] => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0
[1,1,1,0,0,0,1,1,0,1,0,0] => [5,4,6,1,2,3] => {{1,2,4,5},{3,6}} => {{1,4},{2,3,5,6}} => 0
[1,1,1,0,0,0,1,1,1,0,0,0] => [4,5,6,1,2,3] => {{1,4},{2,5},{3,6}} => {{1,4},{2,5},{3,6}} => 0
[1,1,1,0,0,1,0,0,1,0,1,0] => [6,5,3,1,2,4] => {{1,4,6},{2,5},{3}} => {{1,3,6},{2,5},{4}} => 0
[1,1,1,0,0,1,0,0,1,1,0,0] => [5,6,3,1,2,4] => {{1,2,4,5,6},{3}} => {{1,2,3,5,6},{4}} => 0
[1,1,1,0,0,1,0,1,0,0,1,0] => [6,4,3,1,2,5] => {{1,2,4,5,6},{3}} => {{1,2,3,5,6},{4}} => 0
[1,1,1,0,0,1,0,1,0,1,0,0] => [5,4,3,1,2,6] => {{1,2,4,5},{3},{6}} => {{1},{2,3,5,6},{4}} => 1
[1,1,1,0,0,1,0,1,1,0,0,0] => [4,5,3,1,2,6] => {{1,4},{2,5},{3},{6}} => {{1},{2,5},{3,6},{4}} => 1
[1,1,1,0,0,1,1,0,0,0,1,0] => [6,3,4,1,2,5] => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0
[1,1,1,0,0,1,1,0,0,1,0,0] => [5,3,4,1,2,6] => {{1,2,3,4,5},{6}} => {{1},{2,3,4,5,6}} => 0
[1,1,1,0,0,1,1,0,1,0,0,0] => [4,3,5,1,2,6] => {{1,4},{2,3,5},{6}} => {{1},{2,4,5},{3,6}} => 0
[1,1,1,0,0,1,1,1,0,0,0,0] => [3,4,5,1,2,6] => {{1,2,3,4,5},{6}} => {{1},{2,3,4,5,6}} => 0
[1,1,1,0,1,0,0,0,1,0,1,0] => [6,5,2,1,3,4] => {{1,4,6},{2,3,5}} => {{1,3,6},{2,4,5}} => 0
[1,1,1,0,1,0,0,0,1,1,0,0] => [5,6,2,1,3,4] => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0
[1,1,1,0,1,0,0,1,0,0,1,0] => [6,4,2,1,3,5] => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0
[1,1,1,0,1,0,0,1,0,1,0,0] => [5,4,2,1,3,6] => {{1,2,3,4,5},{6}} => {{1},{2,3,4,5,6}} => 0
[1,1,1,0,1,0,0,1,1,0,0,0] => [4,5,2,1,3,6] => {{1,4},{2,3,5},{6}} => {{1},{2,4,5},{3,6}} => 0
[1,1,1,0,1,0,1,0,0,0,1,0] => [6,3,2,1,4,5] => {{1,4,5,6},{2,3}} => {{1,2,3,6},{4,5}} => 0
[1,1,1,0,1,0,1,0,0,1,0,0] => [5,3,2,1,4,6] => {{1,4,5},{2,3},{6}} => {{1},{2,3,6},{4,5}} => 0
[1,1,1,0,1,0,1,0,1,0,0,0] => [4,3,2,1,5,6] => {{1,4},{2,3},{5},{6}} => {{1},{2},{3,6},{4,5}} => 1
[1,1,1,0,1,0,1,1,0,0,0,0] => [3,4,2,1,5,6] => {{1,2,3,4},{5},{6}} => {{1},{2},{3,4,5,6}} => 1
[1,1,1,0,1,1,0,0,0,0,1,0] => [6,2,3,1,4,5] => {{1,4,5,6},{2},{3}} => {{1,2,3,6},{4},{5}} => 1
[1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,3,1,4,6] => {{1,4,5},{2},{3},{6}} => {{1},{2,3,6},{4},{5}} => 3
[1,1,1,0,1,1,0,0,1,0,0,0] => [4,2,3,1,5,6] => {{1,4},{2},{3},{5},{6}} => {{1},{2},{3,6},{4},{5}} => 6
[1,1,1,0,1,1,0,1,0,0,0,0] => [3,2,4,1,5,6] => {{1,3,4},{2},{5},{6}} => {{1},{2},{3,4,6},{5}} => 3
[1,1,1,0,1,1,1,0,0,0,0,0] => [2,3,4,1,5,6] => {{1,2,3,4},{5},{6}} => {{1},{2},{3,4,5,6}} => 1
[1,1,1,1,0,0,0,0,1,0,1,0] => [6,5,1,2,3,4] => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0
[1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => {{1,3,5},{2,4,6}} => {{1,3,5},{2,4,6}} => 0
[1,1,1,1,0,0,0,1,0,0,1,0] => [6,4,1,2,3,5] => {{1,3,5,6},{2,4}} => {{1,2,4,6},{3,5}} => 0
[1,1,1,1,0,0,0,1,0,1,0,0] => [5,4,1,2,3,6] => {{1,3,5},{2,4},{6}} => {{1},{2,4,6},{3,5}} => 0
[1,1,1,1,0,0,0,1,1,0,0,0] => [4,5,1,2,3,6] => {{1,2,3,4,5},{6}} => {{1},{2,3,4,5,6}} => 0
[1,1,1,1,0,0,1,0,0,0,1,0] => [6,3,1,2,4,5] => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0
[1,1,1,1,0,0,1,0,0,1,0,0] => [5,3,1,2,4,6] => {{1,2,3,4,5},{6}} => {{1},{2,3,4,5,6}} => 0
[1,1,1,1,0,0,1,0,1,0,0,0] => [4,3,1,2,5,6] => {{1,2,3,4},{5},{6}} => {{1},{2},{3,4,5,6}} => 1
[1,1,1,1,0,0,1,1,0,0,0,0] => [3,4,1,2,5,6] => {{1,3},{2,4},{5},{6}} => {{1},{2},{3,5},{4,6}} => 1
[1,1,1,1,0,1,0,0,0,0,1,0] => [6,2,1,3,4,5] => {{1,3,4,5,6},{2}} => {{1,2,3,4,6},{5}} => 0
[1,1,1,1,0,1,0,0,0,1,0,0] => [5,2,1,3,4,6] => {{1,3,4,5},{2},{6}} => {{1},{2,3,4,6},{5}} => 1
[1,1,1,1,0,1,0,0,1,0,0,0] => [4,2,1,3,5,6] => {{1,3,4},{2},{5},{6}} => {{1},{2},{3,4,6},{5}} => 3
[1,1,1,1,0,1,0,1,0,0,0,0] => [3,2,1,4,5,6] => {{1,3},{2},{4},{5},{6}} => {{1},{2},{3},{4,6},{5}} => 6
[1,1,1,1,0,1,1,0,0,0,0,0] => [2,3,1,4,5,6] => {{1,2,3},{4},{5},{6}} => {{1},{2},{3},{4,5,6}} => 3
[1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0
[1,1,1,1,1,0,0,0,0,1,0,0] => [5,1,2,3,4,6] => {{1,2,3,4,5},{6}} => {{1},{2,3,4,5,6}} => 0
[1,1,1,1,1,0,0,0,1,0,0,0] => [4,1,2,3,5,6] => {{1,2,3,4},{5},{6}} => {{1},{2},{3,4,5,6}} => 1
[1,1,1,1,1,0,0,1,0,0,0,0] => [3,1,2,4,5,6] => {{1,2,3},{4},{5},{6}} => {{1},{2},{3},{4,5,6}} => 3
[1,1,1,1,1,0,1,0,0,0,0,0] => [2,1,3,4,5,6] => {{1,2},{3},{4},{5},{6}} => {{1},{2},{3},{4},{5,6}} => 6
[1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => {{1},{2},{3},{4},{5},{6}} => {{1},{2},{3},{4},{5},{6}} => 15
[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [7,6,5,4,3,2,1] => {{1,7},{2,6},{3,5},{4}} => {{1,7},{2,6},{3,5},{4}} => 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [6,7,5,4,3,2,1] => {{1,2,6,7},{3,5},{4}} => {{1,2,6,7},{3,5},{4}} => 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [7,5,6,4,3,2,1] => {{1,7},{2,3,5,6},{4}} => {{1,7},{2,3,5,6},{4}} => 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [6,5,7,4,3,2,1] => {{1,2,3,5,6,7},{4}} => {{1,2,3,5,6,7},{4}} => 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [5,6,7,4,3,2,1] => {{1,3,5,7},{2,6},{4}} => {{1,3,5,7},{2,6},{4}} => 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [7,6,4,5,3,2,1] => {{1,7},{2,6},{3,4,5}} => {{1,7},{2,6},{3,4,5}} => 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [6,7,4,5,3,2,1] => {{1,2,6,7},{3,4,5}} => {{1,2,6,7},{3,4,5}} => 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,0] => [7,5,4,6,3,2,1] => {{1,7},{2,3,4,5,6}} => {{1,7},{2,3,4,5,6}} => 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [6,5,4,7,3,2,1] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0] => [5,6,4,7,3,2,1] => {{1,3,4,5,7},{2,6}} => {{1,3,4,5,7},{2,6}} => 0
[1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [7,4,5,6,3,2,1] => {{1,7},{2,4,6},{3,5}} => {{1,7},{2,4,6},{3,5}} => 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [6,4,5,7,3,2,1] => {{1,2,4,6,7},{3,5}} => {{1,2,4,6,7},{3,5}} => 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [5,4,6,7,3,2,1] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [4,5,6,7,3,2,1] => {{1,4,7},{2,3,5,6}} => {{1,4,7},{2,3,5,6}} => 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [7,6,5,3,4,2,1] => {{1,7},{2,6},{3,4,5}} => {{1,7},{2,6},{3,4,5}} => 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [6,7,5,3,4,2,1] => {{1,2,6,7},{3,4,5}} => {{1,2,6,7},{3,4,5}} => 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [7,5,6,3,4,2,1] => {{1,7},{2,3,4,5,6}} => {{1,7},{2,3,4,5,6}} => 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,0] => [6,5,7,3,4,2,1] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [5,6,7,3,4,2,1] => {{1,3,4,5,7},{2,6}} => {{1,3,4,5,7},{2,6}} => 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0] => [7,6,4,3,5,2,1] => {{1,7},{2,6},{3,4},{5}} => {{1,7},{2,6},{3},{4,5}} => 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,0] => [6,7,4,3,5,2,1] => {{1,2,6,7},{3,4},{5}} => {{1,2,6,7},{3},{4,5}} => 0
[1,0,1,0,1,1,0,1,0,1,0,0,1,0] => [7,5,4,3,6,2,1] => {{1,7},{2,5,6},{3,4}} => {{1,7},{2,3,6},{4,5}} => 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [6,5,4,3,7,2,1] => {{1,2,5,6,7},{3,4}} => {{1,2,3,6,7},{4,5}} => 0
[1,0,1,0,1,1,0,1,0,1,1,0,0,0] => [5,6,4,3,7,2,1] => {{1,5,7},{2,6},{3,4}} => {{1,3,7},{2,6},{4,5}} => 0
[1,0,1,0,1,1,0,1,1,0,0,0,1,0] => [7,4,5,3,6,2,1] => {{1,7},{2,3,4,5,6}} => {{1,7},{2,3,4,5,6}} => 0
[1,0,1,0,1,1,0,1,1,0,0,1,0,0] => [6,4,5,3,7,2,1] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,0] => [5,4,6,3,7,2,1] => {{1,5,7},{2,3,4,6}} => {{1,3,7},{2,4,5,6}} => 0
[1,0,1,0,1,1,0,1,1,1,0,0,0,0] => [4,5,6,3,7,2,1] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [7,6,3,4,5,2,1] => {{1,7},{2,6},{3},{4},{5}} => {{1,7},{2,6},{3},{4},{5}} => 3
[1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [6,7,3,4,5,2,1] => {{1,2,6,7},{3},{4},{5}} => {{1,2,6,7},{3},{4},{5}} => 3
[1,0,1,0,1,1,1,0,0,1,0,0,1,0] => [7,5,3,4,6,2,1] => {{1,7},{2,5,6},{3},{4}} => {{1,7},{2,3,6},{4},{5}} => 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [6,5,3,4,7,2,1] => {{1,2,5,6,7},{3},{4}} => {{1,2,3,6,7},{4},{5}} => 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0] => [5,6,3,4,7,2,1] => {{1,5,7},{2,6},{3},{4}} => {{1,3,7},{2,6},{4},{5}} => 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0] => [7,4,3,5,6,2,1] => {{1,7},{2,4,5,6},{3}} => {{1,7},{2,3,4,6},{5}} => 0
[1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [6,4,3,5,7,2,1] => {{1,2,4,5,6,7},{3}} => {{1,2,3,4,6,7},{5}} => 0
[1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [5,4,3,6,7,2,1] => {{1,5,7},{2,4,6},{3}} => {{1,3,7},{2,4,6},{5}} => 0
[1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [4,5,3,6,7,2,1] => {{1,2,4,5,6,7},{3}} => {{1,2,3,4,6,7},{5}} => 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [7,3,4,5,6,2,1] => {{1,7},{2,3,4,5,6}} => {{1,7},{2,3,4,5,6}} => 0
[1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [6,3,4,5,7,2,1] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,0,1,0,1,1,1,1,0,0,1,0,0,0] => [5,3,4,6,7,2,1] => {{1,5,7},{2,3,4,6}} => {{1,3,7},{2,4,5,6}} => 0
[1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [4,3,5,6,7,2,1] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [3,4,5,6,7,2,1] => {{1,3,5,7},{2,4,6}} => {{1,3,5,7},{2,4,6}} => 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [7,6,5,4,2,3,1] => {{1,7},{2,3,5,6},{4}} => {{1,7},{2,3,5,6},{4}} => 0
[1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [6,7,5,4,2,3,1] => {{1,2,3,5,6,7},{4}} => {{1,2,3,5,6,7},{4}} => 0
[1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [7,5,6,4,2,3,1] => {{1,7},{2,5},{3,6},{4}} => {{1,7},{2,5},{3,6},{4}} => 0
[1,0,1,1,0,0,1,0,1,1,0,1,0,0] => [6,5,7,4,2,3,1] => {{1,3,6,7},{2,5},{4}} => {{1,2,5,7},{3,6},{4}} => 0
[1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [5,6,7,4,2,3,1] => {{1,2,3,5,6,7},{4}} => {{1,2,3,5,6,7},{4}} => 0
[1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [7,6,4,5,2,3,1] => {{1,7},{2,3,4,5,6}} => {{1,7},{2,3,4,5,6}} => 0
[1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [6,7,4,5,2,3,1] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,0,1,1,0,0,1,1,0,1,0,0,1,0] => [7,5,4,6,2,3,1] => {{1,7},{2,5},{3,4,6}} => {{1,7},{2,4,5},{3,6}} => 0
[1,0,1,1,0,0,1,1,0,1,0,1,0,0] => [6,5,4,7,2,3,1] => {{1,3,4,6,7},{2,5}} => {{1,2,4,5,7},{3,6}} => 0
[1,0,1,1,0,0,1,1,0,1,1,0,0,0] => [5,6,4,7,2,3,1] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [7,4,5,6,2,3,1] => {{1,7},{2,3,4,5,6}} => {{1,7},{2,3,4,5,6}} => 0
[1,0,1,1,0,0,1,1,1,0,0,1,0,0] => [6,4,5,7,2,3,1] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,0,1,1,0,0,1,1,1,0,1,0,0,0] => [5,4,6,7,2,3,1] => {{1,2,4,5,7},{3,6}} => {{1,3,4,6,7},{2,5}} => 0
[1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [4,5,6,7,2,3,1] => {{1,4,7},{2,5},{3,6}} => {{1,4,7},{2,5},{3,6}} => 0
[1,0,1,1,0,1,0,0,1,0,1,0,1,0] => [7,6,5,3,2,4,1] => {{1,7},{2,3,4,5,6}} => {{1,7},{2,3,4,5,6}} => 0
[1,0,1,1,0,1,0,0,1,0,1,1,0,0] => [6,7,5,3,2,4,1] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,0,1,1,0,1,0,0,1,1,0,0,1,0] => [7,5,6,3,2,4,1] => {{1,7},{2,5},{3,4,6}} => {{1,7},{2,4,5},{3,6}} => 0
[1,0,1,1,0,1,0,0,1,1,0,1,0,0] => [6,5,7,3,2,4,1] => {{1,3,4,6,7},{2,5}} => {{1,2,4,5,7},{3,6}} => 0
[1,0,1,1,0,1,0,0,1,1,1,0,0,0] => [5,6,7,3,2,4,1] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,0,1,1,0,1,0,1,0,0,1,0,1,0] => [7,6,4,3,2,5,1] => {{1,7},{2,5,6},{3,4}} => {{1,7},{2,3,6},{4,5}} => 0
[1,0,1,1,0,1,0,1,0,0,1,1,0,0] => [6,7,4,3,2,5,1] => {{1,2,5,6,7},{3,4}} => {{1,2,3,6,7},{4,5}} => 0
[1,0,1,1,0,1,0,1,0,1,0,0,1,0] => [7,5,4,3,2,6,1] => {{1,7},{2,5},{3,4},{6}} => {{1,7},{2},{3,6},{4,5}} => 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [6,5,4,3,2,7,1] => {{1,6,7},{2,5},{3,4}} => {{1,2,7},{3,6},{4,5}} => 0
[1,0,1,1,0,1,0,1,0,1,1,0,0,0] => [5,6,4,3,2,7,1] => {{1,2,5,6,7},{3,4}} => {{1,2,3,6,7},{4,5}} => 0
[1,0,1,1,0,1,0,1,1,0,0,0,1,0] => [7,4,5,3,2,6,1] => {{1,7},{2,3,4,5},{6}} => {{1,7},{2},{3,4,5,6}} => 0
[1,0,1,1,0,1,0,1,1,0,0,1,0,0] => [6,4,5,3,2,7,1] => {{1,6,7},{2,3,4,5}} => {{1,2,7},{3,4,5,6}} => 0
[1,0,1,1,0,1,0,1,1,0,1,0,0,0] => [5,4,6,3,2,7,1] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,0,1,1,0,1,0,1,1,1,0,0,0,0] => [4,5,6,3,2,7,1] => {{1,3,4,6,7},{2,5}} => {{1,2,4,5,7},{3,6}} => 0
[1,0,1,1,0,1,1,0,0,0,1,0,1,0] => [7,6,3,4,2,5,1] => {{1,7},{2,5,6},{3},{4}} => {{1,7},{2,3,6},{4},{5}} => 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0] => [6,7,3,4,2,5,1] => {{1,2,5,6,7},{3},{4}} => {{1,2,3,6,7},{4},{5}} => 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0] => [7,5,3,4,2,6,1] => {{1,7},{2,5},{3},{4},{6}} => {{1,7},{2},{3,6},{4},{5}} => 3
[1,0,1,1,0,1,1,0,0,1,0,1,0,0] => [6,5,3,4,2,7,1] => {{1,6,7},{2,5},{3},{4}} => {{1,2,7},{3,6},{4},{5}} => 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0] => [5,6,3,4,2,7,1] => {{1,2,5,6,7},{3},{4}} => {{1,2,3,6,7},{4},{5}} => 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0] => [7,4,3,5,2,6,1] => {{1,7},{2,4,5},{3},{6}} => {{1,7},{2},{3,4,6},{5}} => 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0] => [6,4,3,5,2,7,1] => {{1,6,7},{2,4,5},{3}} => {{1,2,7},{3,4,6},{5}} => 0
[1,0,1,1,0,1,1,0,1,0,1,0,0,0] => [5,4,3,6,2,7,1] => {{1,2,4,5,6,7},{3}} => {{1,2,3,4,6,7},{5}} => 0
[1,0,1,1,0,1,1,0,1,1,0,0,0,0] => [4,5,3,6,2,7,1] => {{1,4,6,7},{2,5},{3}} => {{1,2,4,7},{3,6},{5}} => 0
[1,0,1,1,0,1,1,1,0,0,0,0,1,0] => [7,3,4,5,2,6,1] => {{1,7},{2,3,4,5},{6}} => {{1,7},{2},{3,4,5,6}} => 0
[1,0,1,1,0,1,1,1,0,0,0,1,0,0] => [6,3,4,5,2,7,1] => {{1,6,7},{2,3,4,5}} => {{1,2,7},{3,4,5,6}} => 0
[1,0,1,1,0,1,1,1,0,0,1,0,0,0] => [5,3,4,6,2,7,1] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,0,1,1,0,1,1,1,0,1,0,0,0,0] => [4,3,5,6,2,7,1] => {{1,4,6,7},{2,3,5}} => {{1,2,4,7},{3,5,6}} => 0
[1,0,1,1,0,1,1,1,1,0,0,0,0,0] => [3,4,5,6,2,7,1] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [7,6,5,2,3,4,1] => {{1,7},{2,4,6},{3,5}} => {{1,7},{2,4,6},{3,5}} => 0
[1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [6,7,5,2,3,4,1] => {{1,2,4,6,7},{3,5}} => {{1,2,4,6,7},{3,5}} => 0
[1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [7,5,6,2,3,4,1] => {{1,7},{2,3,4,5,6}} => {{1,7},{2,3,4,5,6}} => 0
[1,0,1,1,1,0,0,0,1,1,0,1,0,0] => [6,5,7,2,3,4,1] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [5,6,7,2,3,4,1] => {{1,3,5,7},{2,4,6}} => {{1,3,5,7},{2,4,6}} => 0
[1,0,1,1,1,0,0,1,0,0,1,0,1,0] => [7,6,4,2,3,5,1] => {{1,7},{2,3,4,5,6}} => {{1,7},{2,3,4,5,6}} => 0
[1,0,1,1,1,0,0,1,0,0,1,1,0,0] => [6,7,4,2,3,5,1] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,0,1,1,1,0,0,1,0,1,0,0,1,0] => [7,5,4,2,3,6,1] => {{1,7},{2,3,4,5},{6}} => {{1,7},{2},{3,4,5,6}} => 0
[1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [6,5,4,2,3,7,1] => {{1,6,7},{2,3,4,5}} => {{1,2,7},{3,4,5,6}} => 0
[1,0,1,1,1,0,0,1,0,1,1,0,0,0] => [5,6,4,2,3,7,1] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,0,1,1,1,0,0,1,1,0,0,0,1,0] => [7,4,5,2,3,6,1] => {{1,7},{2,4},{3,5},{6}} => {{1,7},{2},{3,5},{4,6}} => 0
[1,0,1,1,1,0,0,1,1,0,0,1,0,0] => [6,4,5,2,3,7,1] => {{1,6,7},{2,4},{3,5}} => {{1,2,7},{3,5},{4,6}} => 0
[1,0,1,1,1,0,0,1,1,0,1,0,0,0] => [5,4,6,2,3,7,1] => {{1,3,5,6,7},{2,4}} => {{1,2,3,5,7},{4,6}} => 0
[1,0,1,1,1,0,0,1,1,1,0,0,0,0] => [4,5,6,2,3,7,1] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,0,1,1,1,0,1,0,0,0,1,0,1,0] => [7,6,3,2,4,5,1] => {{1,7},{2,4,5,6},{3}} => {{1,7},{2,3,4,6},{5}} => 0
[1,0,1,1,1,0,1,0,0,0,1,1,0,0] => [6,7,3,2,4,5,1] => {{1,2,4,5,6,7},{3}} => {{1,2,3,4,6,7},{5}} => 0
[1,0,1,1,1,0,1,0,0,1,0,0,1,0] => [7,5,3,2,4,6,1] => {{1,7},{2,4,5},{3},{6}} => {{1,7},{2},{3,4,6},{5}} => 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [6,5,3,2,4,7,1] => {{1,6,7},{2,4,5},{3}} => {{1,2,7},{3,4,6},{5}} => 0
[1,0,1,1,1,0,1,0,0,1,1,0,0,0] => [5,6,3,2,4,7,1] => {{1,2,4,5,6,7},{3}} => {{1,2,3,4,6,7},{5}} => 0
[1,0,1,1,1,0,1,0,1,0,0,0,1,0] => [7,4,3,2,5,6,1] => {{1,7},{2,4},{3},{5},{6}} => {{1,7},{2},{3},{4,6},{5}} => 3
[1,0,1,1,1,0,1,0,1,0,0,1,0,0] => [6,4,3,2,5,7,1] => {{1,6,7},{2,4},{3},{5}} => {{1,2,7},{3},{4,6},{5}} => 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [5,4,3,2,6,7,1] => {{1,5,6,7},{2,4},{3}} => {{1,2,3,7},{4,6},{5}} => 0
[1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [4,5,3,2,6,7,1] => {{1,2,4,5,6,7},{3}} => {{1,2,3,4,6,7},{5}} => 0
[1,0,1,1,1,0,1,1,0,0,0,0,1,0] => [7,3,4,2,5,6,1] => {{1,7},{2,3,4},{5},{6}} => {{1,7},{2},{3},{4,5,6}} => 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [6,3,4,2,5,7,1] => {{1,6,7},{2,3,4},{5}} => {{1,2,7},{3},{4,5,6}} => 0
[1,0,1,1,1,0,1,1,0,0,1,0,0,0] => [5,3,4,2,6,7,1] => {{1,5,6,7},{2,3,4}} => {{1,2,3,7},{4,5,6}} => 0
[1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [4,3,5,2,6,7,1] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [3,4,5,2,6,7,1] => {{1,3,5,6,7},{2,4}} => {{1,2,3,5,7},{4,6}} => 0
[1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [7,6,2,3,4,5,1] => {{1,7},{2,3,4,5,6}} => {{1,7},{2,3,4,5,6}} => 0
[1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [6,7,2,3,4,5,1] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,0,1,1,1,1,0,0,0,1,0,0,1,0] => [7,5,2,3,4,6,1] => {{1,7},{2,3,4,5},{6}} => {{1,7},{2},{3,4,5,6}} => 0
[1,0,1,1,1,1,0,0,0,1,0,1,0,0] => [6,5,2,3,4,7,1] => {{1,6,7},{2,3,4,5}} => {{1,2,7},{3,4,5,6}} => 0
[1,0,1,1,1,1,0,0,0,1,1,0,0,0] => [5,6,2,3,4,7,1] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,0,1,1,1,1,0,0,1,0,0,0,1,0] => [7,4,2,3,5,6,1] => {{1,7},{2,3,4},{5},{6}} => {{1,7},{2},{3},{4,5,6}} => 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0] => [6,4,2,3,5,7,1] => {{1,6,7},{2,3,4},{5}} => {{1,2,7},{3},{4,5,6}} => 0
[1,0,1,1,1,1,0,0,1,0,1,0,0,0] => [5,4,2,3,6,7,1] => {{1,5,6,7},{2,3,4}} => {{1,2,3,7},{4,5,6}} => 0
[1,0,1,1,1,1,0,0,1,1,0,0,0,0] => [4,5,2,3,6,7,1] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,0,1,1,1,1,0,1,0,0,0,0,1,0] => [7,3,2,4,5,6,1] => {{1,7},{2,3},{4},{5},{6}} => {{1,7},{2},{3},{4},{5,6}} => 3
[1,0,1,1,1,1,0,1,0,0,0,1,0,0] => [6,3,2,4,5,7,1] => {{1,6,7},{2,3},{4},{5}} => {{1,2,7},{3},{4},{5,6}} => 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0] => [5,3,2,4,6,7,1] => {{1,5,6,7},{2,3},{4}} => {{1,2,3,7},{4},{5,6}} => 0
[1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [4,3,2,5,6,7,1] => {{1,4,5,6,7},{2,3}} => {{1,2,3,4,7},{5,6}} => 0
[1,0,1,1,1,1,0,1,1,0,0,0,0,0] => [3,4,2,5,6,7,1] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [7,2,3,4,5,6,1] => {{1,7},{2},{3},{4},{5},{6}} => {{1,7},{2},{3},{4},{5},{6}} => 10
[1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [6,2,3,4,5,7,1] => {{1,6,7},{2},{3},{4},{5}} => {{1,2,7},{3},{4},{5},{6}} => 6
[1,0,1,1,1,1,1,0,0,0,1,0,0,0] => [5,2,3,4,6,7,1] => {{1,5,6,7},{2},{3},{4}} => {{1,2,3,7},{4},{5},{6}} => 3
[1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [4,2,3,5,6,7,1] => {{1,4,5,6,7},{2},{3}} => {{1,2,3,4,7},{5},{6}} => 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [3,2,4,5,6,7,1] => {{1,3,4,5,6,7},{2}} => {{1,2,3,4,5,7},{6}} => 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [7,6,5,4,3,1,2] => {{1,2,6,7},{3,5},{4}} => {{1,2,6,7},{3,5},{4}} => 0
[1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [6,7,5,4,3,1,2] => {{1,6},{2,7},{3,5},{4}} => {{1,6},{2,7},{3,5},{4}} => 0
[1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [7,5,6,4,3,1,2] => {{1,2,3,5,6,7},{4}} => {{1,2,3,5,6,7},{4}} => 0
[1,1,0,0,1,0,1,0,1,1,0,1,0,0] => [6,5,7,4,3,1,2] => {{1,6},{2,3,5,7},{4}} => {{1,3,5,6},{2,7},{4}} => 0
[1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [5,6,7,4,3,1,2] => {{1,2,3,5,6,7},{4}} => {{1,2,3,5,6,7},{4}} => 0
[1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [7,6,4,5,3,1,2] => {{1,2,6,7},{3,4,5}} => {{1,2,6,7},{3,4,5}} => 0
[1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [6,7,4,5,3,1,2] => {{1,6},{2,7},{3,4,5}} => {{1,6},{2,7},{3,4,5}} => 0
[1,1,0,0,1,0,1,1,0,1,0,0,1,0] => [7,5,4,6,3,1,2] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,0,0,1,0,1,1,0,1,0,1,0,0] => [6,5,4,7,3,1,2] => {{1,6},{2,3,4,5,7}} => {{1,3,4,5,6},{2,7}} => 0
[1,1,0,0,1,0,1,1,0,1,1,0,0,0] => [5,6,4,7,3,1,2] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [7,4,5,6,3,1,2] => {{1,2,4,6,7},{3,5}} => {{1,2,4,6,7},{3,5}} => 0
[1,1,0,0,1,0,1,1,1,0,0,1,0,0] => [6,4,5,7,3,1,2] => {{1,6},{2,4,7},{3,5}} => {{1,4,6},{2,7},{3,5}} => 0
[1,1,0,0,1,0,1,1,1,0,1,0,0,0] => [5,4,6,7,3,1,2] => {{1,3,5,6},{2,4,7}} => {{1,4,6},{2,3,5,7}} => 0
[1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [4,5,6,7,3,1,2] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [7,6,5,3,4,1,2] => {{1,2,6,7},{3,4,5}} => {{1,2,6,7},{3,4,5}} => 0
[1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [6,7,5,3,4,1,2] => {{1,6},{2,7},{3,4,5}} => {{1,6},{2,7},{3,4,5}} => 0
[1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [7,5,6,3,4,1,2] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,0,0,1,1,0,0,1,1,0,1,0,0] => [6,5,7,3,4,1,2] => {{1,6},{2,3,4,5,7}} => {{1,3,4,5,6},{2,7}} => 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [5,6,7,3,4,1,2] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,0,0,1,1,0,1,0,0,1,0,1,0] => [7,6,4,3,5,1,2] => {{1,2,6,7},{3,4},{5}} => {{1,2,6,7},{3},{4,5}} => 0
[1,1,0,0,1,1,0,1,0,0,1,1,0,0] => [6,7,4,3,5,1,2] => {{1,6},{2,7},{3,4},{5}} => {{1,6},{2,7},{3},{4,5}} => 0
[1,1,0,0,1,1,0,1,0,1,0,0,1,0] => [7,5,4,3,6,1,2] => {{1,2,5,6,7},{3,4}} => {{1,2,3,6,7},{4,5}} => 0
[1,1,0,0,1,1,0,1,0,1,0,1,0,0] => [6,5,4,3,7,1,2] => {{1,6},{2,5,7},{3,4}} => {{1,3,6},{2,7},{4,5}} => 0
[1,1,0,0,1,1,0,1,0,1,1,0,0,0] => [5,6,4,3,7,1,2] => {{1,2,5,6,7},{3,4}} => {{1,2,3,6,7},{4,5}} => 0
[1,1,0,0,1,1,0,1,1,0,0,0,1,0] => [7,4,5,3,6,1,2] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,0,0,1,1,0,1,1,0,0,1,0,0] => [6,4,5,3,7,1,2] => {{1,6},{2,3,4,5,7}} => {{1,3,4,5,6},{2,7}} => 0
[1,1,0,0,1,1,0,1,1,0,1,0,0,0] => [5,4,6,3,7,1,2] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,0,0,1,1,0,1,1,1,0,0,0,0] => [4,5,6,3,7,1,2] => {{1,3,4,6},{2,5,7}} => {{1,3,6},{2,4,5,7}} => 0
[1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [7,6,3,4,5,1,2] => {{1,2,6,7},{3},{4},{5}} => {{1,2,6,7},{3},{4},{5}} => 3
[1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [6,7,3,4,5,1,2] => {{1,6},{2,7},{3},{4},{5}} => {{1,6},{2,7},{3},{4},{5}} => 3
[1,1,0,0,1,1,1,0,0,1,0,0,1,0] => [7,5,3,4,6,1,2] => {{1,2,5,6,7},{3},{4}} => {{1,2,3,6,7},{4},{5}} => 1
[1,1,0,0,1,1,1,0,0,1,0,1,0,0] => [6,5,3,4,7,1,2] => {{1,6},{2,5,7},{3},{4}} => {{1,3,6},{2,7},{4},{5}} => 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,0] => [5,6,3,4,7,1,2] => {{1,2,5,6,7},{3},{4}} => {{1,2,3,6,7},{4},{5}} => 1
[1,1,0,0,1,1,1,0,1,0,0,0,1,0] => [7,4,3,5,6,1,2] => {{1,2,4,5,6,7},{3}} => {{1,2,3,4,6,7},{5}} => 0
[1,1,0,0,1,1,1,0,1,0,0,1,0,0] => [6,4,3,5,7,1,2] => {{1,6},{2,4,5,7},{3}} => {{1,3,4,6},{2,7},{5}} => 0
[1,1,0,0,1,1,1,0,1,0,1,0,0,0] => [5,4,3,6,7,1,2] => {{1,2,4,5,6,7},{3}} => {{1,2,3,4,6,7},{5}} => 0
[1,1,0,0,1,1,1,0,1,1,0,0,0,0] => [4,5,3,6,7,1,2] => {{1,4,6},{2,5,7},{3}} => {{1,3,6},{2,4,7},{5}} => 0
[1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [7,3,4,5,6,1,2] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,0,0,1,1,1,1,0,0,0,1,0,0] => [6,3,4,5,7,1,2] => {{1,6},{2,3,4,5,7}} => {{1,3,4,5,6},{2,7}} => 0
[1,1,0,0,1,1,1,1,0,0,1,0,0,0] => [5,3,4,6,7,1,2] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,0,0,1,1,1,1,0,1,0,0,0,0] => [4,3,5,6,7,1,2] => {{1,4,6},{2,3,5,7}} => {{1,3,5,6},{2,4,7}} => 0
[1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [3,4,5,6,7,1,2] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,0,1,0,0,1,0,1,0,1,0,1,0] => [7,6,5,4,2,1,3] => {{1,2,3,5,6,7},{4}} => {{1,2,3,5,6,7},{4}} => 0
[1,1,0,1,0,0,1,0,1,0,1,1,0,0] => [6,7,5,4,2,1,3] => {{1,6},{2,3,5,7},{4}} => {{1,3,5,6},{2,7},{4}} => 0
[1,1,0,1,0,0,1,0,1,1,0,0,1,0] => [7,5,6,4,2,1,3] => {{1,3,6,7},{2,5},{4}} => {{1,2,5,7},{3,6},{4}} => 0
[1,1,0,1,0,0,1,0,1,1,0,1,0,0] => [6,5,7,4,2,1,3] => {{1,6},{2,5},{3,7},{4}} => {{1,5},{2,7},{3,6},{4}} => 0
[1,1,0,1,0,0,1,0,1,1,1,0,0,0] => [5,6,7,4,2,1,3] => {{1,2,5,6},{3,7},{4}} => {{1,5},{2,3,6,7},{4}} => 0
[1,1,0,1,0,0,1,1,0,0,1,0,1,0] => [7,6,4,5,2,1,3] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,0,1,0,0,1,1,0,0,1,1,0,0] => [6,7,4,5,2,1,3] => {{1,6},{2,3,4,5,7}} => {{1,3,4,5,6},{2,7}} => 0
[1,1,0,1,0,0,1,1,0,1,0,0,1,0] => [7,5,4,6,2,1,3] => {{1,3,4,6,7},{2,5}} => {{1,2,4,5,7},{3,6}} => 0
[1,1,0,1,0,0,1,1,0,1,0,1,0,0] => [6,5,4,7,2,1,3] => {{1,6},{2,5},{3,4,7}} => {{1,4,5},{2,7},{3,6}} => 0
[1,1,0,1,0,0,1,1,0,1,1,0,0,0] => [5,6,4,7,2,1,3] => {{1,2,5,6},{3,4,7}} => {{1,4,5},{2,3,6,7}} => 0
[1,1,0,1,0,0,1,1,1,0,0,0,1,0] => [7,4,5,6,2,1,3] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,0,1,0,0,1,1,1,0,0,1,0,0] => [6,4,5,7,2,1,3] => {{1,6},{2,3,4,5,7}} => {{1,3,4,5,6},{2,7}} => 0
[1,1,0,1,0,0,1,1,1,0,1,0,0,0] => [5,4,6,7,2,1,3] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,0,1,0,0,1,1,1,1,0,0,0,0] => [4,5,6,7,2,1,3] => {{1,3,4,6,7},{2,5}} => {{1,2,4,5,7},{3,6}} => 0
[1,1,0,1,0,1,0,0,1,0,1,0,1,0] => [7,6,5,3,2,1,4] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,0,1,0,1,0,0,1,0,1,1,0,0] => [6,7,5,3,2,1,4] => {{1,6},{2,3,4,5,7}} => {{1,3,4,5,6},{2,7}} => 0
[1,1,0,1,0,1,0,0,1,1,0,0,1,0] => [7,5,6,3,2,1,4] => {{1,3,4,6,7},{2,5}} => {{1,2,4,5,7},{3,6}} => 0
[1,1,0,1,0,1,0,0,1,1,0,1,0,0] => [6,5,7,3,2,1,4] => {{1,6},{2,5},{3,4,7}} => {{1,4,5},{2,7},{3,6}} => 0
[1,1,0,1,0,1,0,0,1,1,1,0,0,0] => [5,6,7,3,2,1,4] => {{1,2,5,6},{3,4,7}} => {{1,4,5},{2,3,6,7}} => 0
[1,1,0,1,0,1,0,1,0,0,1,0,1,0] => [7,6,4,3,2,1,5] => {{1,2,5,6,7},{3,4}} => {{1,2,3,6,7},{4,5}} => 0
[1,1,0,1,0,1,0,1,0,0,1,1,0,0] => [6,7,4,3,2,1,5] => {{1,6},{2,5,7},{3,4}} => {{1,3,6},{2,7},{4,5}} => 0
[1,1,0,1,0,1,0,1,0,1,0,0,1,0] => [7,5,4,3,2,1,6] => {{1,6,7},{2,5},{3,4}} => {{1,2,7},{3,6},{4,5}} => 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [6,5,4,3,2,1,7] => {{1,6},{2,5},{3,4},{7}} => {{1},{2,7},{3,6},{4,5}} => 0
[1,1,0,1,0,1,0,1,0,1,1,0,0,0] => [5,6,4,3,2,1,7] => {{1,2,5,6},{3,4},{7}} => {{1},{2,3,6,7},{4,5}} => 0
[1,1,0,1,0,1,0,1,1,0,0,0,1,0] => [7,4,5,3,2,1,6] => {{1,6,7},{2,3,4,5}} => {{1,2,7},{3,4,5,6}} => 0
[1,1,0,1,0,1,0,1,1,0,0,1,0,0] => [6,4,5,3,2,1,7] => {{1,6},{2,3,4,5},{7}} => {{1},{2,7},{3,4,5,6}} => 0
[1,1,0,1,0,1,0,1,1,0,1,0,0,0] => [5,4,6,3,2,1,7] => {{1,2,3,4,5,6},{7}} => {{1},{2,3,4,5,6,7}} => 0
[1,1,0,1,0,1,0,1,1,1,0,0,0,0] => [4,5,6,3,2,1,7] => {{1,3,4,6},{2,5},{7}} => {{1},{2,4,5,7},{3,6}} => 0
[1,1,0,1,0,1,1,0,0,0,1,0,1,0] => [7,6,3,4,2,1,5] => {{1,2,5,6,7},{3},{4}} => {{1,2,3,6,7},{4},{5}} => 1
[1,1,0,1,0,1,1,0,0,0,1,1,0,0] => [6,7,3,4,2,1,5] => {{1,6},{2,5,7},{3},{4}} => {{1,3,6},{2,7},{4},{5}} => 1
[1,1,0,1,0,1,1,0,0,1,0,0,1,0] => [7,5,3,4,2,1,6] => {{1,6,7},{2,5},{3},{4}} => {{1,2,7},{3,6},{4},{5}} => 1
[1,1,0,1,0,1,1,0,0,1,0,1,0,0] => [6,5,3,4,2,1,7] => {{1,6},{2,5},{3},{4},{7}} => {{1},{2,7},{3,6},{4},{5}} => 3
[1,1,0,1,0,1,1,0,0,1,1,0,0,0] => [5,6,3,4,2,1,7] => {{1,2,5,6},{3},{4},{7}} => {{1},{2,3,6,7},{4},{5}} => 3
[1,1,0,1,0,1,1,0,1,0,0,0,1,0] => [7,4,3,5,2,1,6] => {{1,6,7},{2,4,5},{3}} => {{1,2,7},{3,4,6},{5}} => 0
[1,1,0,1,0,1,1,0,1,0,0,1,0,0] => [6,4,3,5,2,1,7] => {{1,6},{2,4,5},{3},{7}} => {{1},{2,7},{3,4,6},{5}} => 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0] => [5,4,3,6,2,1,7] => {{1,2,4,5,6},{3},{7}} => {{1},{2,3,4,6,7},{5}} => 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,0] => [4,5,3,6,2,1,7] => {{1,4,6},{2,5},{3},{7}} => {{1},{2,4,7},{3,6},{5}} => 1
[1,1,0,1,0,1,1,1,0,0,0,0,1,0] => [7,3,4,5,2,1,6] => {{1,6,7},{2,3,4,5}} => {{1,2,7},{3,4,5,6}} => 0
[1,1,0,1,0,1,1,1,0,0,0,1,0,0] => [6,3,4,5,2,1,7] => {{1,6},{2,3,4,5},{7}} => {{1},{2,7},{3,4,5,6}} => 0
[1,1,0,1,0,1,1,1,0,0,1,0,0,0] => [5,3,4,6,2,1,7] => {{1,2,3,4,5,6},{7}} => {{1},{2,3,4,5,6,7}} => 0
[1,1,0,1,0,1,1,1,0,1,0,0,0,0] => [4,3,5,6,2,1,7] => {{1,4,6},{2,3,5},{7}} => {{1},{2,4,7},{3,5,6}} => 0
[1,1,0,1,0,1,1,1,1,0,0,0,0,0] => [3,4,5,6,2,1,7] => {{1,2,3,4,5,6},{7}} => {{1},{2,3,4,5,6,7}} => 0
[1,1,0,1,1,0,0,0,1,0,1,0,1,0] => [7,6,5,2,3,1,4] => {{1,2,4,6,7},{3,5}} => {{1,2,4,6,7},{3,5}} => 0
[1,1,0,1,1,0,0,0,1,0,1,1,0,0] => [6,7,5,2,3,1,4] => {{1,6},{2,4,7},{3,5}} => {{1,4,6},{2,7},{3,5}} => 0
[1,1,0,1,1,0,0,0,1,1,0,0,1,0] => [7,5,6,2,3,1,4] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,0,1,1,0,0,0,1,1,0,1,0,0] => [6,5,7,2,3,1,4] => {{1,6},{2,3,4,5,7}} => {{1,3,4,5,6},{2,7}} => 0
[1,1,0,1,1,0,0,0,1,1,1,0,0,0] => [5,6,7,2,3,1,4] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,0,1,1,0,0,1,0,0,1,0,1,0] => [7,6,4,2,3,1,5] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,0,1,1,0,0,1,0,0,1,1,0,0] => [6,7,4,2,3,1,5] => {{1,6},{2,3,4,5,7}} => {{1,3,4,5,6},{2,7}} => 0
[1,1,0,1,1,0,0,1,0,1,0,0,1,0] => [7,5,4,2,3,1,6] => {{1,6,7},{2,3,4,5}} => {{1,2,7},{3,4,5,6}} => 0
[1,1,0,1,1,0,0,1,0,1,0,1,0,0] => [6,5,4,2,3,1,7] => {{1,6},{2,3,4,5},{7}} => {{1},{2,7},{3,4,5,6}} => 0
[1,1,0,1,1,0,0,1,0,1,1,0,0,0] => [5,6,4,2,3,1,7] => {{1,2,3,4,5,6},{7}} => {{1},{2,3,4,5,6,7}} => 0
[1,1,0,1,1,0,0,1,1,0,0,0,1,0] => [7,4,5,2,3,1,6] => {{1,6,7},{2,4},{3,5}} => {{1,2,7},{3,5},{4,6}} => 0
[1,1,0,1,1,0,0,1,1,0,0,1,0,0] => [6,4,5,2,3,1,7] => {{1,6},{2,4},{3,5},{7}} => {{1},{2,7},{3,5},{4,6}} => 0
[1,1,0,1,1,0,0,1,1,0,1,0,0,0] => [5,4,6,2,3,1,7] => {{1,3,5,6},{2,4},{7}} => {{1},{2,3,5,7},{4,6}} => 0
[1,1,0,1,1,0,0,1,1,1,0,0,0,0] => [4,5,6,2,3,1,7] => {{1,2,3,4,5,6},{7}} => {{1},{2,3,4,5,6,7}} => 0
[1,1,0,1,1,0,1,0,0,0,1,0,1,0] => [7,6,3,2,4,1,5] => {{1,2,4,5,6,7},{3}} => {{1,2,3,4,6,7},{5}} => 0
[1,1,0,1,1,0,1,0,0,0,1,1,0,0] => [6,7,3,2,4,1,5] => {{1,6},{2,4,5,7},{3}} => {{1,3,4,6},{2,7},{5}} => 0
[1,1,0,1,1,0,1,0,0,1,0,0,1,0] => [7,5,3,2,4,1,6] => {{1,6,7},{2,4,5},{3}} => {{1,2,7},{3,4,6},{5}} => 0
[1,1,0,1,1,0,1,0,0,1,0,1,0,0] => [6,5,3,2,4,1,7] => {{1,6},{2,4,5},{3},{7}} => {{1},{2,7},{3,4,6},{5}} => 1
[1,1,0,1,1,0,1,0,0,1,1,0,0,0] => [5,6,3,2,4,1,7] => {{1,2,4,5,6},{3},{7}} => {{1},{2,3,4,6,7},{5}} => 1
[1,1,0,1,1,0,1,0,1,0,0,0,1,0] => [7,4,3,2,5,1,6] => {{1,6,7},{2,4},{3},{5}} => {{1,2,7},{3},{4,6},{5}} => 1
[1,1,0,1,1,0,1,0,1,0,0,1,0,0] => [6,4,3,2,5,1,7] => {{1,6},{2,4},{3},{5},{7}} => {{1},{2,7},{3},{4,6},{5}} => 3
[1,1,0,1,1,0,1,0,1,0,1,0,0,0] => [5,4,3,2,6,1,7] => {{1,5,6},{2,4},{3},{7}} => {{1},{2,3,7},{4,6},{5}} => 1
[1,1,0,1,1,0,1,0,1,1,0,0,0,0] => [4,5,3,2,6,1,7] => {{1,2,4,5,6},{3},{7}} => {{1},{2,3,4,6,7},{5}} => 1
[1,1,0,1,1,0,1,1,0,0,0,0,1,0] => [7,3,4,2,5,1,6] => {{1,6,7},{2,3,4},{5}} => {{1,2,7},{3},{4,5,6}} => 0
[1,1,0,1,1,0,1,1,0,0,0,1,0,0] => [6,3,4,2,5,1,7] => {{1,6},{2,3,4},{5},{7}} => {{1},{2,7},{3},{4,5,6}} => 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,0] => [5,3,4,2,6,1,7] => {{1,5,6},{2,3,4},{7}} => {{1},{2,3,7},{4,5,6}} => 0
[1,1,0,1,1,0,1,1,0,1,0,0,0,0] => [4,3,5,2,6,1,7] => {{1,2,3,4,5,6},{7}} => {{1},{2,3,4,5,6,7}} => 0
[1,1,0,1,1,0,1,1,1,0,0,0,0,0] => [3,4,5,2,6,1,7] => {{1,3,5,6},{2,4},{7}} => {{1},{2,3,5,7},{4,6}} => 0
[1,1,0,1,1,1,0,0,0,0,1,0,1,0] => [7,6,2,3,4,1,5] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,0,1,1,1,0,0,0,0,1,1,0,0] => [6,7,2,3,4,1,5] => {{1,6},{2,3,4,5,7}} => {{1,3,4,5,6},{2,7}} => 0
[1,1,0,1,1,1,0,0,0,1,0,0,1,0] => [7,5,2,3,4,1,6] => {{1,6,7},{2,3,4,5}} => {{1,2,7},{3,4,5,6}} => 0
[1,1,0,1,1,1,0,0,0,1,0,1,0,0] => [6,5,2,3,4,1,7] => {{1,6},{2,3,4,5},{7}} => {{1},{2,7},{3,4,5,6}} => 0
[1,1,0,1,1,1,0,0,0,1,1,0,0,0] => [5,6,2,3,4,1,7] => {{1,2,3,4,5,6},{7}} => {{1},{2,3,4,5,6,7}} => 0
[1,1,0,1,1,1,0,0,1,0,0,0,1,0] => [7,4,2,3,5,1,6] => {{1,6,7},{2,3,4},{5}} => {{1,2,7},{3},{4,5,6}} => 0
[1,1,0,1,1,1,0,0,1,0,0,1,0,0] => [6,4,2,3,5,1,7] => {{1,6},{2,3,4},{5},{7}} => {{1},{2,7},{3},{4,5,6}} => 1
[1,1,0,1,1,1,0,0,1,0,1,0,0,0] => [5,4,2,3,6,1,7] => {{1,5,6},{2,3,4},{7}} => {{1},{2,3,7},{4,5,6}} => 0
[1,1,0,1,1,1,0,0,1,1,0,0,0,0] => [4,5,2,3,6,1,7] => {{1,2,3,4,5,6},{7}} => {{1},{2,3,4,5,6,7}} => 0
[1,1,0,1,1,1,0,1,0,0,0,0,1,0] => [7,3,2,4,5,1,6] => {{1,6,7},{2,3},{4},{5}} => {{1,2,7},{3},{4},{5,6}} => 1
[1,1,0,1,1,1,0,1,0,0,0,1,0,0] => [6,3,2,4,5,1,7] => {{1,6},{2,3},{4},{5},{7}} => {{1},{2,7},{3},{4},{5,6}} => 3
[1,1,0,1,1,1,0,1,0,0,1,0,0,0] => [5,3,2,4,6,1,7] => {{1,5,6},{2,3},{4},{7}} => {{1},{2,3,7},{4},{5,6}} => 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0] => [4,3,2,5,6,1,7] => {{1,4,5,6},{2,3},{7}} => {{1},{2,3,4,7},{5,6}} => 0
[1,1,0,1,1,1,0,1,1,0,0,0,0,0] => [3,4,2,5,6,1,7] => {{1,2,3,4,5,6},{7}} => {{1},{2,3,4,5,6,7}} => 0
[1,1,0,1,1,1,1,0,0,0,0,0,1,0] => [7,2,3,4,5,1,6] => {{1,6,7},{2},{3},{4},{5}} => {{1,2,7},{3},{4},{5},{6}} => 6
[1,1,0,1,1,1,1,0,0,0,0,1,0,0] => [6,2,3,4,5,1,7] => {{1,6},{2},{3},{4},{5},{7}} => {{1},{2,7},{3},{4},{5},{6}} => 10
[1,1,0,1,1,1,1,0,0,0,1,0,0,0] => [5,2,3,4,6,1,7] => {{1,5,6},{2},{3},{4},{7}} => {{1},{2,3,7},{4},{5},{6}} => 6
[1,1,0,1,1,1,1,0,0,1,0,0,0,0] => [4,2,3,5,6,1,7] => {{1,4,5,6},{2},{3},{7}} => {{1},{2,3,4,7},{5},{6}} => 3
[1,1,0,1,1,1,1,0,1,0,0,0,0,0] => [3,2,4,5,6,1,7] => {{1,3,4,5,6},{2},{7}} => {{1},{2,3,4,5,7},{6}} => 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,1,7] => {{1,2,3,4,5,6},{7}} => {{1},{2,3,4,5,6,7}} => 0
[1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [7,6,5,4,1,2,3] => {{1,3,5,7},{2,6},{4}} => {{1,3,5,7},{2,6},{4}} => 0
[1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [6,7,5,4,1,2,3] => {{1,2,3,5,6,7},{4}} => {{1,2,3,5,6,7},{4}} => 0
[1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [7,5,6,4,1,2,3] => {{1,2,3,5,6,7},{4}} => {{1,2,3,5,6,7},{4}} => 0
[1,1,1,0,0,0,1,0,1,1,0,1,0,0] => [6,5,7,4,1,2,3] => {{1,2,5,6},{3,7},{4}} => {{1,5},{2,3,6,7},{4}} => 0
[1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [5,6,7,4,1,2,3] => {{1,5},{2,6},{3,7},{4}} => {{1,5},{2,6},{3,7},{4}} => 0
[1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [7,6,4,5,1,2,3] => {{1,3,4,5,7},{2,6}} => {{1,3,4,5,7},{2,6}} => 0
[1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [6,7,4,5,1,2,3] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,1,0,0,0,1,1,0,1,0,0,1,0] => [7,5,4,6,1,2,3] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,1,0,0,0,1,1,0,1,0,1,0,0] => [6,5,4,7,1,2,3] => {{1,2,5,6},{3,4,7}} => {{1,4,5},{2,3,6,7}} => 0
[1,1,1,0,0,0,1,1,0,1,1,0,0,0] => [5,6,4,7,1,2,3] => {{1,5},{2,6},{3,4,7}} => {{1,4,5},{2,6},{3,7}} => 0
[1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [7,4,5,6,1,2,3] => {{1,3,5,7},{2,4,6}} => {{1,3,5,7},{2,4,6}} => 0
[1,1,1,0,0,0,1,1,1,0,0,1,0,0] => [6,4,5,7,1,2,3] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,1,0,0,0,1,1,1,0,1,0,0,0] => [5,4,6,7,1,2,3] => {{1,5},{2,3,4,6,7}} => {{1,2,4,5,6},{3,7}} => 0
[1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [4,5,6,7,1,2,3] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,1,0,0,1,0,0,1,0,1,0,1,0] => [7,6,5,3,1,2,4] => {{1,3,4,5,7},{2,6}} => {{1,3,4,5,7},{2,6}} => 0
[1,1,1,0,0,1,0,0,1,0,1,1,0,0] => [6,7,5,3,1,2,4] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,1,0,0,1,0,0,1,1,0,0,1,0] => [7,5,6,3,1,2,4] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,1,0,0,1,0,0,1,1,0,1,0,0] => [6,5,7,3,1,2,4] => {{1,2,5,6},{3,4,7}} => {{1,4,5},{2,3,6,7}} => 0
[1,1,1,0,0,1,0,0,1,1,1,0,0,0] => [5,6,7,3,1,2,4] => {{1,5},{2,6},{3,4,7}} => {{1,4,5},{2,6},{3,7}} => 0
[1,1,1,0,0,1,0,1,0,0,1,0,1,0] => [7,6,4,3,1,2,5] => {{1,5,7},{2,6},{3,4}} => {{1,3,7},{2,6},{4,5}} => 0
[1,1,1,0,0,1,0,1,0,0,1,1,0,0] => [6,7,4,3,1,2,5] => {{1,2,5,6,7},{3,4}} => {{1,2,3,6,7},{4,5}} => 0
[1,1,1,0,0,1,0,1,0,1,0,0,1,0] => [7,5,4,3,1,2,6] => {{1,2,5,6,7},{3,4}} => {{1,2,3,6,7},{4,5}} => 0
[1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [6,5,4,3,1,2,7] => {{1,2,5,6},{3,4},{7}} => {{1},{2,3,6,7},{4,5}} => 0
[1,1,1,0,0,1,0,1,0,1,1,0,0,0] => [5,6,4,3,1,2,7] => {{1,5},{2,6},{3,4},{7}} => {{1},{2,6},{3,7},{4,5}} => 0
[1,1,1,0,0,1,0,1,1,0,0,0,1,0] => [7,4,5,3,1,2,6] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,1,0,0,1,0,1,1,0,0,1,0,0] => [6,4,5,3,1,2,7] => {{1,2,3,4,5,6},{7}} => {{1},{2,3,4,5,6,7}} => 0
[1,1,1,0,0,1,0,1,1,0,1,0,0,0] => [5,4,6,3,1,2,7] => {{1,5},{2,3,4,6},{7}} => {{1},{2,4,5,6},{3,7}} => 0
[1,1,1,0,0,1,0,1,1,1,0,0,0,0] => [4,5,6,3,1,2,7] => {{1,2,3,4,5,6},{7}} => {{1},{2,3,4,5,6,7}} => 0
[1,1,1,0,0,1,1,0,0,0,1,0,1,0] => [7,6,3,4,1,2,5] => {{1,5,7},{2,6},{3},{4}} => {{1,3,7},{2,6},{4},{5}} => 1
[1,1,1,0,0,1,1,0,0,0,1,1,0,0] => [6,7,3,4,1,2,5] => {{1,2,5,6,7},{3},{4}} => {{1,2,3,6,7},{4},{5}} => 1
[1,1,1,0,0,1,1,0,0,1,0,0,1,0] => [7,5,3,4,1,2,6] => {{1,2,5,6,7},{3},{4}} => {{1,2,3,6,7},{4},{5}} => 1
[1,1,1,0,0,1,1,0,0,1,0,1,0,0] => [6,5,3,4,1,2,7] => {{1,2,5,6},{3},{4},{7}} => {{1},{2,3,6,7},{4},{5}} => 3
[1,1,1,0,0,1,1,0,0,1,1,0,0,0] => [5,6,3,4,1,2,7] => {{1,5},{2,6},{3},{4},{7}} => {{1},{2,6},{3,7},{4},{5}} => 3
[1,1,1,0,0,1,1,0,1,0,0,0,1,0] => [7,4,3,5,1,2,6] => {{1,2,4,5,6,7},{3}} => {{1,2,3,4,6,7},{5}} => 0
[1,1,1,0,0,1,1,0,1,0,0,1,0,0] => [6,4,3,5,1,2,7] => {{1,2,4,5,6},{3},{7}} => {{1},{2,3,4,6,7},{5}} => 1
[1,1,1,0,0,1,1,0,1,0,1,0,0,0] => [5,4,3,6,1,2,7] => {{1,5},{2,4,6},{3},{7}} => {{1},{2,4,6},{3,7},{5}} => 1
[1,1,1,0,0,1,1,0,1,1,0,0,0,0] => [4,5,3,6,1,2,7] => {{1,2,4,5,6},{3},{7}} => {{1},{2,3,4,6,7},{5}} => 1
[1,1,1,0,0,1,1,1,0,0,0,0,1,0] => [7,3,4,5,1,2,6] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,1,0,0,1,1,1,0,0,0,1,0,0] => [6,3,4,5,1,2,7] => {{1,2,3,4,5,6},{7}} => {{1},{2,3,4,5,6,7}} => 0
[1,1,1,0,0,1,1,1,0,0,1,0,0,0] => [5,3,4,6,1,2,7] => {{1,5},{2,3,4,6},{7}} => {{1},{2,4,5,6},{3,7}} => 0
[1,1,1,0,0,1,1,1,0,1,0,0,0,0] => [4,3,5,6,1,2,7] => {{1,2,3,4,5,6},{7}} => {{1},{2,3,4,5,6,7}} => 0
[1,1,1,0,0,1,1,1,1,0,0,0,0,0] => [3,4,5,6,1,2,7] => {{1,3,5},{2,4,6},{7}} => {{1},{2,4,6},{3,5,7}} => 0
[1,1,1,0,1,0,0,0,1,0,1,0,1,0] => [7,6,5,2,1,3,4] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,1,0,1,0,0,0,1,0,1,1,0,0] => [6,7,5,2,1,3,4] => {{1,3,5,6},{2,4,7}} => {{1,4,6},{2,3,5,7}} => 0
[1,1,1,0,1,0,0,0,1,1,0,0,1,0] => [7,5,6,2,1,3,4] => {{1,2,4,5,7},{3,6}} => {{1,3,4,6,7},{2,5}} => 0
[1,1,1,0,1,0,0,0,1,1,0,1,0,0] => [6,5,7,2,1,3,4] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,1,0,1,0,0,0,1,1,1,0,0,0] => [5,6,7,2,1,3,4] => {{1,5},{2,3,4,6,7}} => {{1,2,4,5,6},{3,7}} => 0
[1,1,1,0,1,0,0,1,0,0,1,0,1,0] => [7,6,4,2,1,3,5] => {{1,5,7},{2,3,4,6}} => {{1,3,7},{2,4,5,6}} => 0
[1,1,1,0,1,0,0,1,0,0,1,1,0,0] => [6,7,4,2,1,3,5] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,1,0,1,0,0,1,0,1,0,0,1,0] => [7,5,4,2,1,3,6] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,1,0,1,0,0,1,0,1,0,1,0,0] => [6,5,4,2,1,3,7] => {{1,2,3,4,5,6},{7}} => {{1},{2,3,4,5,6,7}} => 0
[1,1,1,0,1,0,0,1,0,1,1,0,0,0] => [5,6,4,2,1,3,7] => {{1,5},{2,3,4,6},{7}} => {{1},{2,4,5,6},{3,7}} => 0
[1,1,1,0,1,0,0,1,1,0,0,0,1,0] => [7,4,5,2,1,3,6] => {{1,3,5,6,7},{2,4}} => {{1,2,3,5,7},{4,6}} => 0
[1,1,1,0,1,0,0,1,1,0,0,1,0,0] => [6,4,5,2,1,3,7] => {{1,3,5,6},{2,4},{7}} => {{1},{2,3,5,7},{4,6}} => 0
[1,1,1,0,1,0,0,1,1,0,1,0,0,0] => [5,4,6,2,1,3,7] => {{1,5},{2,4},{3,6},{7}} => {{1},{2,5},{3,7},{4,6}} => 0
[1,1,1,0,1,0,0,1,1,1,0,0,0,0] => [4,5,6,2,1,3,7] => {{1,2,4,5},{3,6},{7}} => {{1},{2,5},{3,4,6,7}} => 0
[1,1,1,0,1,0,1,0,0,0,1,0,1,0] => [7,6,3,2,1,4,5] => {{1,5,7},{2,4,6},{3}} => {{1,3,7},{2,4,6},{5}} => 0
[1,1,1,0,1,0,1,0,0,0,1,1,0,0] => [6,7,3,2,1,4,5] => {{1,2,4,5,6,7},{3}} => {{1,2,3,4,6,7},{5}} => 0
[1,1,1,0,1,0,1,0,0,1,0,0,1,0] => [7,5,3,2,1,4,6] => {{1,2,4,5,6,7},{3}} => {{1,2,3,4,6,7},{5}} => 0
[1,1,1,0,1,0,1,0,0,1,0,1,0,0] => [6,5,3,2,1,4,7] => {{1,2,4,5,6},{3},{7}} => {{1},{2,3,4,6,7},{5}} => 1
[1,1,1,0,1,0,1,0,0,1,1,0,0,0] => [5,6,3,2,1,4,7] => {{1,5},{2,4,6},{3},{7}} => {{1},{2,4,6},{3,7},{5}} => 1
[1,1,1,0,1,0,1,0,1,0,0,0,1,0] => [7,4,3,2,1,5,6] => {{1,5,6,7},{2,4},{3}} => {{1,2,3,7},{4,6},{5}} => 0
[1,1,1,0,1,0,1,0,1,0,0,1,0,0] => [6,4,3,2,1,5,7] => {{1,5,6},{2,4},{3},{7}} => {{1},{2,3,7},{4,6},{5}} => 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [5,4,3,2,1,6,7] => {{1,5},{2,4},{3},{6},{7}} => {{1},{2},{3,7},{4,6},{5}} => 3
[1,1,1,0,1,0,1,0,1,1,0,0,0,0] => [4,5,3,2,1,6,7] => {{1,2,4,5},{3},{6},{7}} => {{1},{2},{3,4,6,7},{5}} => 3
[1,1,1,0,1,0,1,1,0,0,0,0,1,0] => [7,3,4,2,1,5,6] => {{1,5,6,7},{2,3,4}} => {{1,2,3,7},{4,5,6}} => 0
[1,1,1,0,1,0,1,1,0,0,0,1,0,0] => [6,3,4,2,1,5,7] => {{1,5,6},{2,3,4},{7}} => {{1},{2,3,7},{4,5,6}} => 0
[1,1,1,0,1,0,1,1,0,0,1,0,0,0] => [5,3,4,2,1,6,7] => {{1,5},{2,3,4},{6},{7}} => {{1},{2},{3,7},{4,5,6}} => 1
[1,1,1,0,1,0,1,1,0,1,0,0,0,0] => [4,3,5,2,1,6,7] => {{1,2,3,4,5},{6},{7}} => {{1},{2},{3,4,5,6,7}} => 1
[1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [3,4,5,2,1,6,7] => {{1,3,5},{2,4},{6},{7}} => {{1},{2},{3,5,7},{4,6}} => 1
[1,1,1,0,1,1,0,0,0,0,1,0,1,0] => [7,6,2,3,1,4,5] => {{1,5,7},{2,3,4,6}} => {{1,3,7},{2,4,5,6}} => 0
[1,1,1,0,1,1,0,0,0,0,1,1,0,0] => [6,7,2,3,1,4,5] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,1,0,1,1,0,0,0,1,0,0,1,0] => [7,5,2,3,1,4,6] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,1,0,1,1,0,0,0,1,0,1,0,0] => [6,5,2,3,1,4,7] => {{1,2,3,4,5,6},{7}} => {{1},{2,3,4,5,6,7}} => 0
[1,1,1,0,1,1,0,0,0,1,1,0,0,0] => [5,6,2,3,1,4,7] => {{1,5},{2,3,4,6},{7}} => {{1},{2,4,5,6},{3,7}} => 0
[1,1,1,0,1,1,0,0,1,0,0,0,1,0] => [7,4,2,3,1,5,6] => {{1,5,6,7},{2,3,4}} => {{1,2,3,7},{4,5,6}} => 0
[1,1,1,0,1,1,0,0,1,0,0,1,0,0] => [6,4,2,3,1,5,7] => {{1,5,6},{2,3,4},{7}} => {{1},{2,3,7},{4,5,6}} => 0
[1,1,1,0,1,1,0,0,1,0,1,0,0,0] => [5,4,2,3,1,6,7] => {{1,5},{2,3,4},{6},{7}} => {{1},{2},{3,7},{4,5,6}} => 1
[1,1,1,0,1,1,0,0,1,1,0,0,0,0] => [4,5,2,3,1,6,7] => {{1,2,3,4,5},{6},{7}} => {{1},{2},{3,4,5,6,7}} => 1
[1,1,1,0,1,1,0,1,0,0,0,0,1,0] => [7,3,2,4,1,5,6] => {{1,5,6,7},{2,3},{4}} => {{1,2,3,7},{4},{5,6}} => 0
[1,1,1,0,1,1,0,1,0,0,0,1,0,0] => [6,3,2,4,1,5,7] => {{1,5,6},{2,3},{4},{7}} => {{1},{2,3,7},{4},{5,6}} => 1
[1,1,1,0,1,1,0,1,0,0,1,0,0,0] => [5,3,2,4,1,6,7] => {{1,5},{2,3},{4},{6},{7}} => {{1},{2},{3,7},{4},{5,6}} => 3
[1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [4,3,2,5,1,6,7] => {{1,4,5},{2,3},{6},{7}} => {{1},{2},{3,4,7},{5,6}} => 1
[1,1,1,0,1,1,0,1,1,0,0,0,0,0] => [3,4,2,5,1,6,7] => {{1,2,3,4,5},{6},{7}} => {{1},{2},{3,4,5,6,7}} => 1
[1,1,1,0,1,1,1,0,0,0,0,0,1,0] => [7,2,3,4,1,5,6] => {{1,5,6,7},{2},{3},{4}} => {{1,2,3,7},{4},{5},{6}} => 3
[1,1,1,0,1,1,1,0,0,0,0,1,0,0] => [6,2,3,4,1,5,7] => {{1,5,6},{2},{3},{4},{7}} => {{1},{2,3,7},{4},{5},{6}} => 6
[1,1,1,0,1,1,1,0,0,0,1,0,0,0] => [5,2,3,4,1,6,7] => {{1,5},{2},{3},{4},{6},{7}} => {{1},{2},{3,7},{4},{5},{6}} => 10
[1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [4,2,3,5,1,6,7] => {{1,4,5},{2},{3},{6},{7}} => {{1},{2},{3,4,7},{5},{6}} => 6
[1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [3,2,4,5,1,6,7] => {{1,3,4,5},{2},{6},{7}} => {{1},{2},{3,4,5,7},{6}} => 3
[1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,1,6,7] => {{1,2,3,4,5},{6},{7}} => {{1},{2},{3,4,5,6,7}} => 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [7,6,5,1,2,3,4] => {{1,4,7},{2,3,5,6}} => {{1,4,7},{2,3,5,6}} => 0
[1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [6,7,5,1,2,3,4] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [7,5,6,1,2,3,4] => {{1,4,7},{2,5},{3,6}} => {{1,4,7},{2,5},{3,6}} => 0
[1,1,1,1,0,0,0,0,1,1,0,1,0,0] => [6,5,7,1,2,3,4] => {{1,3,4,6,7},{2,5}} => {{1,2,4,5,7},{3,6}} => 0
[1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [5,6,7,1,2,3,4] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,1,1,0,0,0,1,0,0,1,0,1,0] => [7,6,4,1,2,3,5] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,1,1,0,0,0,1,0,0,1,1,0,0] => [6,7,4,1,2,3,5] => {{1,3,4,6},{2,5,7}} => {{1,3,6},{2,4,5,7}} => 0
[1,1,1,1,0,0,0,1,0,1,0,0,1,0] => [7,5,4,1,2,3,6] => {{1,3,4,6,7},{2,5}} => {{1,2,4,5,7},{3,6}} => 0
[1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [6,5,4,1,2,3,7] => {{1,3,4,6},{2,5},{7}} => {{1},{2,4,5,7},{3,6}} => 0
[1,1,1,1,0,0,0,1,0,1,1,0,0,0] => [5,6,4,1,2,3,7] => {{1,2,3,4,5,6},{7}} => {{1},{2,3,4,5,6,7}} => 0
[1,1,1,1,0,0,0,1,1,0,0,0,1,0] => [7,4,5,1,2,3,6] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,1,1,0,0,0,1,1,0,0,1,0,0] => [6,4,5,1,2,3,7] => {{1,2,3,4,5,6},{7}} => {{1},{2,3,4,5,6,7}} => 0
[1,1,1,1,0,0,0,1,1,0,1,0,0,0] => [5,4,6,1,2,3,7] => {{1,2,4,5},{3,6},{7}} => {{1},{2,5},{3,4,6,7}} => 0
[1,1,1,1,0,0,0,1,1,1,0,0,0,0] => [4,5,6,1,2,3,7] => {{1,4},{2,5},{3,6},{7}} => {{1},{2,5},{3,6},{4,7}} => 0
[1,1,1,1,0,0,1,0,0,0,1,0,1,0] => [7,6,3,1,2,4,5] => {{1,2,4,5,6,7},{3}} => {{1,2,3,4,6,7},{5}} => 0
[1,1,1,1,0,0,1,0,0,0,1,1,0,0] => [6,7,3,1,2,4,5] => {{1,4,6},{2,5,7},{3}} => {{1,3,6},{2,4,7},{5}} => 0
[1,1,1,1,0,0,1,0,0,1,0,0,1,0] => [7,5,3,1,2,4,6] => {{1,4,6,7},{2,5},{3}} => {{1,2,4,7},{3,6},{5}} => 0
[1,1,1,1,0,0,1,0,0,1,0,1,0,0] => [6,5,3,1,2,4,7] => {{1,4,6},{2,5},{3},{7}} => {{1},{2,4,7},{3,6},{5}} => 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,0] => [5,6,3,1,2,4,7] => {{1,2,4,5,6},{3},{7}} => {{1},{2,3,4,6,7},{5}} => 1
[1,1,1,1,0,0,1,0,1,0,0,0,1,0] => [7,4,3,1,2,5,6] => {{1,2,4,5,6,7},{3}} => {{1,2,3,4,6,7},{5}} => 0
[1,1,1,1,0,0,1,0,1,0,0,1,0,0] => [6,4,3,1,2,5,7] => {{1,2,4,5,6},{3},{7}} => {{1},{2,3,4,6,7},{5}} => 1
[1,1,1,1,0,0,1,0,1,0,1,0,0,0] => [5,4,3,1,2,6,7] => {{1,2,4,5},{3},{6},{7}} => {{1},{2},{3,4,6,7},{5}} => 3
[1,1,1,1,0,0,1,0,1,1,0,0,0,0] => [4,5,3,1,2,6,7] => {{1,4},{2,5},{3},{6},{7}} => {{1},{2},{3,6},{4,7},{5}} => 3
[1,1,1,1,0,0,1,1,0,0,0,0,1,0] => [7,3,4,1,2,5,6] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,1,1,0,0,1,1,0,0,0,1,0,0] => [6,3,4,1,2,5,7] => {{1,2,3,4,5,6},{7}} => {{1},{2,3,4,5,6,7}} => 0
[1,1,1,1,0,0,1,1,0,0,1,0,0,0] => [5,3,4,1,2,6,7] => {{1,2,3,4,5},{6},{7}} => {{1},{2},{3,4,5,6,7}} => 1
[1,1,1,1,0,0,1,1,0,1,0,0,0,0] => [4,3,5,1,2,6,7] => {{1,4},{2,3,5},{6},{7}} => {{1},{2},{3,5,6},{4,7}} => 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0] => [3,4,5,1,2,6,7] => {{1,2,3,4,5},{6},{7}} => {{1},{2},{3,4,5,6,7}} => 1
[1,1,1,1,0,1,0,0,0,0,1,0,1,0] => [7,6,2,1,3,4,5] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,1,1,0,1,0,0,0,0,1,1,0,0] => [6,7,2,1,3,4,5] => {{1,4,6},{2,3,5,7}} => {{1,3,5,6},{2,4,7}} => 0
[1,1,1,1,0,1,0,0,0,1,0,0,1,0] => [7,5,2,1,3,4,6] => {{1,4,6,7},{2,3,5}} => {{1,2,4,7},{3,5,6}} => 0
[1,1,1,1,0,1,0,0,0,1,0,1,0,0] => [6,5,2,1,3,4,7] => {{1,4,6},{2,3,5},{7}} => {{1},{2,4,7},{3,5,6}} => 0
[1,1,1,1,0,1,0,0,0,1,1,0,0,0] => [5,6,2,1,3,4,7] => {{1,2,3,4,5,6},{7}} => {{1},{2,3,4,5,6,7}} => 0
[1,1,1,1,0,1,0,0,1,0,0,0,1,0] => [7,4,2,1,3,5,6] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,1,1,0,1,0,0,1,0,0,1,0,0] => [6,4,2,1,3,5,7] => {{1,2,3,4,5,6},{7}} => {{1},{2,3,4,5,6,7}} => 0
[1,1,1,1,0,1,0,0,1,0,1,0,0,0] => [5,4,2,1,3,6,7] => {{1,2,3,4,5},{6},{7}} => {{1},{2},{3,4,5,6,7}} => 1
[1,1,1,1,0,1,0,0,1,1,0,0,0,0] => [4,5,2,1,3,6,7] => {{1,4},{2,3,5},{6},{7}} => {{1},{2},{3,5,6},{4,7}} => 1
[1,1,1,1,0,1,0,1,0,0,0,0,1,0] => [7,3,2,1,4,5,6] => {{1,4,5,6,7},{2,3}} => {{1,2,3,4,7},{5,6}} => 0
[1,1,1,1,0,1,0,1,0,0,0,1,0,0] => [6,3,2,1,4,5,7] => {{1,4,5,6},{2,3},{7}} => {{1},{2,3,4,7},{5,6}} => 0
[1,1,1,1,0,1,0,1,0,0,1,0,0,0] => [5,3,2,1,4,6,7] => {{1,4,5},{2,3},{6},{7}} => {{1},{2},{3,4,7},{5,6}} => 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [4,3,2,1,5,6,7] => {{1,4},{2,3},{5},{6},{7}} => {{1},{2},{3},{4,7},{5,6}} => 3
[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [3,4,2,1,5,6,7] => {{1,2,3,4},{5},{6},{7}} => {{1},{2},{3},{4,5,6,7}} => 3
[1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [7,2,3,1,4,5,6] => {{1,4,5,6,7},{2},{3}} => {{1,2,3,4,7},{5},{6}} => 1
[1,1,1,1,0,1,1,0,0,0,0,1,0,0] => [6,2,3,1,4,5,7] => {{1,4,5,6},{2},{3},{7}} => {{1},{2,3,4,7},{5},{6}} => 3
[1,1,1,1,0,1,1,0,0,0,1,0,0,0] => [5,2,3,1,4,6,7] => {{1,4,5},{2},{3},{6},{7}} => {{1},{2},{3,4,7},{5},{6}} => 6
[1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [4,2,3,1,5,6,7] => {{1,4},{2},{3},{5},{6},{7}} => {{1},{2},{3},{4,7},{5},{6}} => 10
[1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [3,2,4,1,5,6,7] => {{1,3,4},{2},{5},{6},{7}} => {{1},{2},{3},{4,5,7},{6}} => 6
[1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [2,3,4,1,5,6,7] => {{1,2,3,4},{5},{6},{7}} => {{1},{2},{3},{4,5,6,7}} => 3
[1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [7,6,1,2,3,4,5] => {{1,3,5,7},{2,4,6}} => {{1,3,5,7},{2,4,6}} => 0
[1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [6,7,1,2,3,4,5] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,1,1,1,0,0,0,0,1,0,0,1,0] => [7,5,1,2,3,4,6] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,1,1,1,0,0,0,0,1,0,1,0,0] => [6,5,1,2,3,4,7] => {{1,2,3,4,5,6},{7}} => {{1},{2,3,4,5,6,7}} => 0
[1,1,1,1,1,0,0,0,0,1,1,0,0,0] => [5,6,1,2,3,4,7] => {{1,3,5},{2,4,6},{7}} => {{1},{2,4,6},{3,5,7}} => 0
[1,1,1,1,1,0,0,0,1,0,0,0,1,0] => [7,4,1,2,3,5,6] => {{1,3,5,6,7},{2,4}} => {{1,2,3,5,7},{4,6}} => 0
[1,1,1,1,1,0,0,0,1,0,0,1,0,0] => [6,4,1,2,3,5,7] => {{1,3,5,6},{2,4},{7}} => {{1},{2,3,5,7},{4,6}} => 0
[1,1,1,1,1,0,0,0,1,0,1,0,0,0] => [5,4,1,2,3,6,7] => {{1,3,5},{2,4},{6},{7}} => {{1},{2},{3,5,7},{4,6}} => 1
[1,1,1,1,1,0,0,0,1,1,0,0,0,0] => [4,5,1,2,3,6,7] => {{1,2,3,4,5},{6},{7}} => {{1},{2},{3,4,5,6,7}} => 1
[1,1,1,1,1,0,0,1,0,0,0,0,1,0] => [7,3,1,2,4,5,6] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,1,1,1,0,0,1,0,0,0,1,0,0] => [6,3,1,2,4,5,7] => {{1,2,3,4,5,6},{7}} => {{1},{2,3,4,5,6,7}} => 0
[1,1,1,1,1,0,0,1,0,0,1,0,0,0] => [5,3,1,2,4,6,7] => {{1,2,3,4,5},{6},{7}} => {{1},{2},{3,4,5,6,7}} => 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [4,3,1,2,5,6,7] => {{1,2,3,4},{5},{6},{7}} => {{1},{2},{3},{4,5,6,7}} => 3
[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [3,4,1,2,5,6,7] => {{1,3},{2,4},{5},{6},{7}} => {{1},{2},{3},{4,6},{5,7}} => 3
[1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [7,2,1,3,4,5,6] => {{1,3,4,5,6,7},{2}} => {{1,2,3,4,5,7},{6}} => 0
[1,1,1,1,1,0,1,0,0,0,0,1,0,0] => [6,2,1,3,4,5,7] => {{1,3,4,5,6},{2},{7}} => {{1},{2,3,4,5,7},{6}} => 1
[1,1,1,1,1,0,1,0,0,0,1,0,0,0] => [5,2,1,3,4,6,7] => {{1,3,4,5},{2},{6},{7}} => {{1},{2},{3,4,5,7},{6}} => 3
[1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [4,2,1,3,5,6,7] => {{1,3,4},{2},{5},{6},{7}} => {{1},{2},{3},{4,5,7},{6}} => 6
[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [3,2,1,4,5,6,7] => {{1,3},{2},{4},{5},{6},{7}} => {{1},{2},{3},{4},{5,7},{6}} => 10
[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [2,3,1,4,5,6,7] => {{1,2,3},{4},{5},{6},{7}} => {{1},{2},{3},{4},{5,6,7}} => 6
[1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [7,1,2,3,4,5,6] => {{1,2,3,4,5,6,7}} => {{1,2,3,4,5,6,7}} => 0
[1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [6,1,2,3,4,5,7] => {{1,2,3,4,5,6},{7}} => {{1},{2,3,4,5,6,7}} => 0
[1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [5,1,2,3,4,6,7] => {{1,2,3,4,5},{6},{7}} => {{1},{2},{3,4,5,6,7}} => 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [4,1,2,3,5,6,7] => {{1,2,3,4},{5},{6},{7}} => {{1},{2},{3},{4,5,6,7}} => 3
[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [3,1,2,4,5,6,7] => {{1,2,3},{4},{5},{6},{7}} => {{1},{2},{3},{4},{5,6,7}} => 6
[1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [2,1,3,4,5,6,7] => {{1,2},{3},{4},{5},{6},{7}} => {{1},{2},{3},{4},{5},{6,7}} => 10
[1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7] => {{1},{2},{3},{4},{5},{6},{7}} => {{1},{2},{3},{4},{5},{6},{7}} => 21
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [7,6,8,5,4,3,2,1] => {{1,2,3,6,7,8},{4,5}} => {{1,2,3,6,7,8},{4,5}} => 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0] => [8,6,5,7,4,3,2,1] => {{1,8},{2,3,4,5,6,7}} => {{1,8},{2,3,4,5,6,7}} => 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [7,6,5,8,4,3,2,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0] => [6,7,5,8,4,3,2,1] => {{1,3,4,5,6,8},{2,7}} => {{1,3,4,5,6,8},{2,7}} => 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [7,5,6,8,4,3,2,1] => {{1,2,4,5,7,8},{3,6}} => {{1,2,4,5,7,8},{3,6}} => 0
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [6,5,7,8,4,3,2,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [7,6,5,4,8,3,2,1] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0] => [6,5,7,4,8,3,2,1] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0] => [8,6,4,5,7,3,2,1] => {{1,8},{2,3,4,5,6,7}} => {{1,8},{2,3,4,5,6,7}} => 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [7,6,4,5,8,3,2,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0] => [6,7,4,5,8,3,2,1] => {{1,3,4,5,6,8},{2,7}} => {{1,3,4,5,6,8},{2,7}} => 0
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [6,5,4,7,8,3,2,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [8,4,5,6,7,3,2,1] => {{1,8},{2,3,4,5,6,7}} => {{1,8},{2,3,4,5,6,7}} => 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [7,4,5,6,8,3,2,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [4,5,6,7,8,3,2,1] => {{1,2,4,5,7,8},{3,6}} => {{1,2,4,5,7,8},{3,6}} => 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [8,6,7,5,3,4,2,1] => {{1,8},{2,3,4,5,6,7}} => {{1,8},{2,3,4,5,6,7}} => 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0] => [7,6,8,5,3,4,2,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0] => [7,6,5,8,3,4,2,1] => {{1,2,4,6,7,8},{3,5}} => {{1,2,3,5,7,8},{4,6}} => 0
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [8,5,6,7,3,4,2,1] => {{1,8},{2,3,4,5,6,7}} => {{1,8},{2,3,4,5,6,7}} => 0
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [5,6,7,8,3,4,2,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0] => [7,6,8,4,3,5,2,1] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0] => [7,5,6,4,3,8,2,1] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0] => [5,6,7,4,3,8,2,1] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0] => [7,5,4,6,3,8,2,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0] => [7,4,5,6,3,8,2,1] => {{1,2,4,6,7,8},{3,5}} => {{1,2,3,5,7,8},{4,6}} => 0
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0] => [4,5,6,7,3,8,2,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [8,6,7,3,4,5,2,1] => {{1,8},{2,3,4,5,6,7}} => {{1,8},{2,3,4,5,6,7}} => 0
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0] => [7,6,8,3,4,5,2,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [6,7,8,3,4,5,2,1] => {{1,3,4,5,6,8},{2,7}} => {{1,3,4,5,6,8},{2,7}} => 0
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0] => [7,5,4,3,6,8,2,1] => {{1,2,5,6,7,8},{3,4}} => {{1,2,3,4,7,8},{5,6}} => 0
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [5,4,3,6,7,8,2,1] => {{1,2,4,5,6,7,8},{3}} => {{1,2,3,4,5,7,8},{6}} => 0
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [8,3,4,5,6,7,2,1] => {{1,8},{2,3,4,5,6,7}} => {{1,8},{2,3,4,5,6,7}} => 0
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [7,3,4,5,6,8,2,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [5,3,4,6,7,8,2,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [3,4,5,6,7,8,2,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [7,8,6,5,4,2,3,1] => {{1,2,3,6,7,8},{4,5}} => {{1,2,3,6,7,8},{4,5}} => 0
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [8,7,5,6,4,2,3,1] => {{1,8},{2,3,4,5,6,7}} => {{1,8},{2,3,4,5,6,7}} => 0
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0] => [7,6,5,8,4,2,3,1] => {{1,3,4,5,7,8},{2,6}} => {{1,2,4,5,6,8},{3,7}} => 0
[1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0] => [6,7,5,8,4,2,3,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0] => [7,5,6,8,4,2,3,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0] => [7,5,6,4,8,2,3,1] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [8,7,4,5,6,2,3,1] => {{1,8},{2,3,4,5,6,7}} => {{1,8},{2,3,4,5,6,7}} => 0
[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [7,8,4,5,6,2,3,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0] => [6,7,4,5,8,2,3,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [4,5,6,7,8,2,3,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0] => [8,7,6,5,3,2,4,1] => {{1,8},{2,3,4,5,6,7}} => {{1,8},{2,3,4,5,6,7}} => 0
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0] => [7,8,6,5,3,2,4,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0] => [7,6,8,5,3,2,4,1] => {{1,3,4,5,7,8},{2,6}} => {{1,2,4,5,6,8},{3,7}} => 0
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0] => [6,7,8,5,3,2,4,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,0,1,0,0,1,1,0,1,1,0,0,0] => [6,7,5,8,3,2,4,1] => {{1,2,4,6,7,8},{3,5}} => {{1,2,3,5,7,8},{4,6}} => 0
[1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0] => [5,6,7,8,3,2,4,1] => {{1,3,4,5,7,8},{2,6}} => {{1,2,4,5,6,8},{3,7}} => 0
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0] => [7,8,6,4,3,2,5,1] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,0,1,1,0,1,0,1,0,0,1,1,1,0,0,0] => [6,7,8,4,3,2,5,1] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0] => [6,5,7,4,3,2,8,1] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0] => [6,5,4,7,3,2,8,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0] => [6,4,5,7,3,2,8,1] => {{1,2,4,6,7,8},{3,5}} => {{1,2,3,5,7,8},{4,6}} => 0
[1,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0] => [8,7,6,3,4,2,5,1] => {{1,8},{2,3,4,5,6,7}} => {{1,8},{2,3,4,5,6,7}} => 0
[1,0,1,1,0,1,1,0,0,0,1,0,1,1,0,0] => [7,8,6,3,4,2,5,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0] => [7,6,8,3,4,2,5,1] => {{1,3,4,5,7,8},{2,6}} => {{1,2,4,5,6,8},{3,7}} => 0
[1,0,1,1,0,1,1,0,0,0,1,1,1,0,0,0] => [6,7,8,3,4,2,5,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0] => [6,5,7,3,4,2,8,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0] => [6,5,4,3,7,2,8,1] => {{1,2,5,6,7,8},{3,4}} => {{1,2,3,4,7,8},{5,6}} => 0
[1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0] => [6,4,5,3,7,2,8,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0] => [4,5,6,3,7,2,8,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0] => [6,4,3,5,7,2,8,1] => {{1,2,4,5,6,7,8},{3}} => {{1,2,3,4,5,7,8},{6}} => 0
[1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0] => [4,5,3,6,7,2,8,1] => {{1,2,4,5,6,7,8},{3}} => {{1,2,3,4,5,7,8},{6}} => 0
[1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0] => [6,3,4,5,7,2,8,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0] => [4,3,5,6,7,2,8,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [8,6,7,5,2,3,4,1] => {{1,8},{2,3,4,5,6,7}} => {{1,8},{2,3,4,5,6,7}} => 0
[1,0,1,1,1,0,0,0,1,0,1,1,0,1,0,0] => [7,6,8,5,2,3,4,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [8,7,5,6,2,3,4,1] => {{1,8},{2,3,4,5,6,7}} => {{1,8},{2,3,4,5,6,7}} => 0
[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [7,8,5,6,2,3,4,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,1,0,0,0,1,1,0,1,1,0,0,0] => [6,7,5,8,2,3,4,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,1,0,0,0,1,1,1,0,1,0,0,0] => [6,5,7,8,2,3,4,1] => {{1,3,4,6,7,8},{2,5}} => {{1,2,3,5,6,8},{4,7}} => 0
[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [5,6,7,8,2,3,4,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,1,0,0,1,0,0,1,1,0,1,0,0] => [7,6,8,4,2,3,5,1] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,0,1,1,1,0,0,1,0,1,0,1,1,0,0,0] => [6,7,5,4,2,3,8,1] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,0] => [5,6,7,4,2,3,8,1] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0] => [6,7,4,5,2,3,8,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0] => [6,5,4,7,2,3,8,1] => {{1,3,4,6,7,8},{2,5}} => {{1,2,3,5,6,8},{4,7}} => 0
[1,0,1,1,1,0,0,1,1,0,1,1,0,0,0,0] => [5,6,4,7,2,3,8,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0] => [5,4,6,7,2,3,8,1] => {{1,2,4,5,7,8},{3,6}} => {{1,2,4,5,7,8},{3,6}} => 0
[1,0,1,1,1,0,1,0,0,0,1,1,0,0,1,0] => [8,6,7,3,2,4,5,1] => {{1,8},{2,3,4,5,6,7}} => {{1,8},{2,3,4,5,6,7}} => 0
[1,0,1,1,1,0,1,0,0,1,0,0,1,0,1,0] => [8,7,5,3,2,4,6,1] => {{1,8},{2,3,4,5,6,7}} => {{1,8},{2,3,4,5,6,7}} => 0
[1,0,1,1,1,0,1,0,0,1,0,1,1,0,0,0] => [6,7,5,3,2,4,8,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0] => [6,5,7,3,2,4,8,1] => {{1,3,4,6,7,8},{2,5}} => {{1,2,3,5,6,8},{4,7}} => 0
[1,0,1,1,1,0,1,0,1,0,0,0,1,1,0,0] => [7,8,4,3,2,5,6,1] => {{1,2,5,6,7,8},{3,4}} => {{1,2,3,4,7,8},{5,6}} => 0
[1,0,1,1,1,0,1,0,1,0,0,1,1,0,0,0] => [6,7,4,3,2,5,8,1] => {{1,2,5,6,7,8},{3,4}} => {{1,2,3,4,7,8},{5,6}} => 0
[1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0] => [5,4,6,3,2,7,8,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [4,5,6,3,2,7,8,1] => {{1,3,4,6,7,8},{2,5}} => {{1,2,3,5,6,8},{4,7}} => 0
[1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [5,4,3,6,2,7,8,1] => {{1,2,4,5,6,7,8},{3}} => {{1,2,3,4,5,7,8},{6}} => 0
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [3,4,5,6,2,7,8,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [8,7,6,2,3,4,5,1] => {{1,8},{2,3,4,5,6,7}} => {{1,8},{2,3,4,5,6,7}} => 0
[1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [7,8,6,2,3,4,5,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [6,7,8,2,3,4,5,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,1,1,0,0,1,0,0,0,1,1,0,0] => [7,8,4,2,3,5,6,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0] => [5,6,4,2,3,7,8,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0] => [5,4,6,2,3,7,8,1] => {{1,3,5,6,7,8},{2,4}} => {{1,2,3,4,6,8},{5,7}} => 0
[1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0] => [4,5,6,2,3,7,8,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,1,1,0,1,0,0,0,0,1,1,0,0] => [7,8,3,2,4,5,6,1] => {{1,2,4,5,6,7,8},{3}} => {{1,2,3,4,5,7,8},{6}} => 0
[1,0,1,1,1,1,0,1,0,0,0,1,1,0,0,0] => [6,7,3,2,4,5,8,1] => {{1,2,4,5,6,7,8},{3}} => {{1,2,3,4,5,7,8},{6}} => 0
[1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [4,5,3,2,6,7,8,1] => {{1,2,4,5,6,7,8},{3}} => {{1,2,3,4,5,7,8},{6}} => 0
[1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [4,3,5,2,6,7,8,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [3,4,5,2,6,7,8,1] => {{1,3,5,6,7,8},{2,4}} => {{1,2,3,4,6,8},{5,7}} => 0
[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [8,7,2,3,4,5,6,1] => {{1,8},{2,3,4,5,6,7}} => {{1,8},{2,3,4,5,6,7}} => 0
[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [7,8,2,3,4,5,6,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0] => [6,7,2,3,4,5,8,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [4,5,2,3,6,7,8,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [4,3,2,5,6,7,8,1] => {{1,4,5,6,7,8},{2,3}} => {{1,2,3,4,5,8},{6,7}} => 0
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [3,4,2,5,6,7,8,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [3,2,4,5,6,7,8,1] => {{1,3,4,5,6,7,8},{2}} => {{1,2,3,4,5,6,8},{7}} => 0
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0] => [8,6,7,5,4,3,1,2] => {{1,2,3,6,7,8},{4,5}} => {{1,2,3,6,7,8},{4,5}} => 0
[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0] => [6,7,8,5,4,3,1,2] => {{1,2,3,6,7,8},{4,5}} => {{1,2,3,6,7,8},{4,5}} => 0
[1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0] => [8,6,5,7,4,3,1,2] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0] => [7,6,5,8,4,3,1,2] => {{1,7},{2,3,4,5,6,8}} => {{1,3,4,5,6,7},{2,8}} => 0
[1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0] => [6,7,5,8,4,3,1,2] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0] => [5,6,7,8,4,3,1,2] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,0,1,0,1,1,1,0,0,1,0,0,1,0] => [8,6,4,5,7,3,1,2] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,0] => [7,6,4,5,8,3,1,2] => {{1,7},{2,3,4,5,6,8}} => {{1,3,4,5,6,7},{2,8}} => 0
[1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0] => [6,7,4,5,8,3,1,2] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,0,1,0,1,1,1,0,1,1,0,0,0,0] => [5,6,4,7,8,3,1,2] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,0,1,0,1,1,1,1,0,1,0,0,0,0] => [5,4,6,7,8,3,1,2] => {{1,2,4,5,7,8},{3,6}} => {{1,2,4,5,7,8},{3,6}} => 0
[1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0] => [7,6,8,5,3,4,1,2] => {{1,7},{2,3,4,5,6,8}} => {{1,3,4,5,6,7},{2,8}} => 0
[1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0] => [6,7,8,5,3,4,1,2] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0] => [8,6,5,7,3,4,1,2] => {{1,2,4,6,7,8},{3,5}} => {{1,2,3,5,7,8},{4,6}} => 0
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0] => [6,7,5,8,3,4,1,2] => {{1,2,4,6,7,8},{3,5}} => {{1,2,3,5,7,8},{4,6}} => 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0] => [8,5,6,7,3,4,1,2] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0] => [7,5,6,8,3,4,1,2] => {{1,7},{2,3,4,5,6,8}} => {{1,3,4,5,6,7},{2,8}} => 0
[1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0] => [6,5,7,8,3,4,1,2] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,0,1,1,0,1,0,0,1,1,1,0,0,0] => [6,7,8,4,3,5,1,2] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,1,0,0,1,1,0,1,1,0,1,0,0,0,1,0] => [8,5,4,6,3,7,1,2] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,0,1,1,0,1,1,0,1,0,0,1,0,0] => [7,5,4,6,3,8,1,2] => {{1,7},{2,3,4,5,6,8}} => {{1,3,4,5,6,7},{2,8}} => 0
[1,1,0,0,1,1,0,1,1,0,1,0,1,0,0,0] => [6,5,4,7,3,8,1,2] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0] => [8,6,7,3,4,5,1,2] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,0,1,1,1,0,1,0,1,0,0,0,1,0] => [8,5,4,3,6,7,1,2] => {{1,2,5,6,7,8},{3,4}} => {{1,2,3,4,7,8},{5,6}} => 0
[1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0] => [6,5,4,3,7,8,1,2] => {{1,2,5,6,7,8},{3,4}} => {{1,2,3,4,7,8},{5,6}} => 0
[1,1,0,0,1,1,1,0,1,1,0,0,0,1,0,0] => [7,4,5,3,6,8,1,2] => {{1,7},{2,3,4,5,6,8}} => {{1,3,4,5,6,7},{2,8}} => 0
[1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0] => [4,5,6,3,7,8,1,2] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,0,1,1,1,1,0,1,0,0,0,0,1,0] => [8,4,3,5,6,7,1,2] => {{1,2,4,5,6,7,8},{3}} => {{1,2,3,4,5,7,8},{6}} => 0
[1,1,0,0,1,1,1,1,0,1,0,0,1,0,0,0] => [6,4,3,5,7,8,1,2] => {{1,2,4,5,6,7,8},{3}} => {{1,2,3,4,5,7,8},{6}} => 0
[1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0] => [4,5,3,6,7,8,1,2] => {{1,2,4,5,6,7,8},{3}} => {{1,2,3,4,5,7,8},{6}} => 0
[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0] => [8,3,4,5,6,7,1,2] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0] => [7,3,4,5,6,8,1,2] => {{1,7},{2,3,4,5,6,8}} => {{1,3,4,5,6,7},{2,8}} => 0
[1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0] => [6,3,4,5,7,8,1,2] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0] => [4,3,5,6,7,8,1,2] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0] => [8,7,6,5,4,2,1,3] => {{1,2,3,6,7,8},{4,5}} => {{1,2,3,6,7,8},{4,5}} => 0
[1,1,0,1,0,0,1,0,1,1,0,0,1,0,1,0] => [8,7,5,6,4,2,1,3] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,1,0,0,1,0,1,1,0,0,1,1,0,0] => [7,8,5,6,4,2,1,3] => {{1,7},{2,3,4,5,6,8}} => {{1,3,4,5,6,7},{2,8}} => 0
[1,1,0,1,0,0,1,0,1,1,1,0,0,1,0,0] => [7,5,6,8,4,2,1,3] => {{1,7},{2,3,4,5,6,8}} => {{1,3,4,5,6,7},{2,8}} => 0
[1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,0] => [6,5,7,8,4,2,1,3] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,1,0,0,1,0,1,1,1,1,0,0,0,0] => [5,6,7,8,4,2,1,3] => {{1,3,4,5,7,8},{2,6}} => {{1,2,4,5,6,8},{3,7}} => 0
[1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0] => [6,5,7,4,8,2,1,3] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,1,0,1,0,0,1,1,1,1,0,1,0,0,0,0] => [5,4,6,7,8,2,1,3] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0] => [8,7,6,5,3,2,1,4] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0] => [7,8,6,5,3,2,1,4] => {{1,7},{2,3,4,5,6,8}} => {{1,3,4,5,6,7},{2,8}} => 0
[1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0] => [8,6,7,5,3,2,1,4] => {{1,3,4,5,7,8},{2,6}} => {{1,2,4,5,6,8},{3,7}} => 0
[1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0] => [8,7,5,6,3,2,1,4] => {{1,2,4,6,7,8},{3,5}} => {{1,2,3,5,7,8},{4,6}} => 0
[1,1,0,1,0,1,0,0,1,1,1,0,1,0,0,0] => [6,5,7,8,3,2,1,4] => {{1,2,3,5,6,7},{4,8}} => {{1,5},{2,3,4,6,7,8}} => 0
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0] => [8,7,6,4,3,2,1,5] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0] => [6,5,7,4,3,2,1,8] => {{1,2,3,5,6,7},{4},{8}} => {{1},{2,3,4,6,7,8},{5}} => 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0] => [6,5,4,7,3,2,1,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0] => [5,4,6,7,3,2,1,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0] => [8,7,6,3,4,2,1,5] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,1,0,1,1,0,0,0,1,0,1,1,0,0] => [7,8,6,3,4,2,1,5] => {{1,7},{2,3,4,5,6,8}} => {{1,3,4,5,6,7},{2,8}} => 0
[1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0] => [6,5,7,3,4,2,1,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0] => [8,7,6,5,2,3,1,4] => {{1,2,4,5,7,8},{3,6}} => {{1,2,4,5,7,8},{3,6}} => 0
[1,1,0,1,1,0,0,0,1,0,1,1,0,0,1,0] => [8,6,7,5,2,3,1,4] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0] => [7,6,8,5,2,3,1,4] => {{1,7},{2,3,4,5,6,8}} => {{1,3,4,5,6,7},{2,8}} => 0
[1,1,0,1,1,0,0,0,1,0,1,1,1,0,0,0] => [6,7,8,5,2,3,1,4] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,1,1,0,0,0,1,1,0,0,1,0,1,0] => [8,7,5,6,2,3,1,4] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0] => [7,8,5,6,2,3,1,4] => {{1,7},{2,3,4,5,6,8}} => {{1,3,4,5,6,7},{2,8}} => 0
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0] => [6,7,5,8,2,3,1,4] => {{1,2,3,5,6,7},{4,8}} => {{1,5},{2,3,4,6,7,8}} => 0
[1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0] => [5,6,7,8,2,3,1,4] => {{1,2,3,5,6,7},{4,8}} => {{1,5},{2,3,4,6,7,8}} => 0
[1,1,0,1,1,0,0,1,0,0,1,1,1,0,0,0] => [6,7,8,4,2,3,1,5] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0] => [6,7,5,4,2,3,1,8] => {{1,2,3,5,6,7},{4},{8}} => {{1},{2,3,4,6,7,8},{5}} => 1
[1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0] => [5,6,7,4,2,3,1,8] => {{1,2,3,5,6,7},{4},{8}} => {{1},{2,3,4,6,7,8},{5}} => 1
[1,1,0,1,1,0,0,1,1,0,0,0,1,0,1,0] => [8,7,4,5,2,3,1,6] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0] => [7,8,4,5,2,3,1,6] => {{1,7},{2,3,4,5,6,8}} => {{1,3,4,5,6,7},{2,8}} => 0
[1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0] => [6,7,4,5,2,3,1,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0] => [5,6,4,7,2,3,1,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,0,1,1,0,1,0,0,0,1,1,0,0,1,0] => [8,6,7,3,2,4,1,5] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,1,1,0,1,0,0,0,1,1,1,0,0,0] => [6,7,8,3,2,4,1,5] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0] => [8,7,5,3,2,4,1,6] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0] => [5,6,7,3,2,4,1,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0] => [8,7,4,3,2,5,1,6] => {{1,2,5,6,7,8},{3,4}} => {{1,2,3,4,7,8},{5,6}} => 0
[1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0] => [5,4,6,3,2,7,1,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0] => [5,4,3,6,2,7,1,8] => {{1,2,4,5,6,7},{3},{8}} => {{1},{2,3,4,5,7,8},{6}} => 1
[1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0] => [5,3,4,6,2,7,1,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0] => [3,4,5,6,2,7,1,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0] => [8,7,6,2,3,4,1,5] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,1,1,1,0,0,0,1,1,0,1,0,0,0] => [6,5,7,2,3,4,1,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,0,1,1,1,0,0,1,0,0,1,1,0,0,0] => [6,7,4,2,3,5,1,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,0,1,1,1,0,1,0,0,0,1,1,0,0,0] => [6,7,3,2,4,5,1,8] => {{1,2,4,5,6,7},{3},{8}} => {{1},{2,3,4,5,7,8},{6}} => 1
[1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0] => [8,7,2,3,4,5,1,6] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,0,1,1,1,1,0,0,0,0,0,1,1,0,0] => [7,8,2,3,4,5,1,6] => {{1,7},{2,3,4,5,6,8}} => {{1,3,4,5,6,7},{2,8}} => 0
[1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0] => [6,7,2,3,4,5,1,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0] => [4,5,2,3,6,7,1,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0] => [3,4,2,5,6,7,1,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0] => [3,2,4,5,6,7,1,8] => {{1,3,4,5,6,7},{2},{8}} => {{1},{2,3,4,5,6,8},{7}} => 1
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,1,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0] => [7,8,6,5,4,1,2,3] => {{1,2,3,6,7,8},{4,5}} => {{1,2,3,6,7,8},{4,5}} => 0
[1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0] => [7,8,5,6,4,1,2,3] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,0,0,1,0,1,1,0,1,0,0,1,0] => [8,6,5,7,4,1,2,3] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0] => [7,5,6,8,4,1,2,3] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,0,0,1,0,1,1,1,0,1,0,0,0] => [6,5,7,8,4,1,2,3] => {{1,6},{2,3,4,5,7,8}} => {{1,2,4,5,6,7},{3,8}} => 0
[1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0] => [5,6,7,8,4,1,2,3] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0] => [7,8,5,4,6,1,2,3] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0] => [8,6,5,4,7,1,2,3] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0] => [5,6,7,4,8,1,2,3] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0] => [8,7,4,5,6,1,2,3] => {{1,3,4,5,6,8},{2,7}} => {{1,3,4,5,6,8},{2,7}} => 0
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0] => [7,8,4,5,6,1,2,3] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,0,0,1,1,1,0,0,1,0,0,1,0] => [8,6,4,5,7,1,2,3] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,0,0,1,1,1,0,1,0,0,1,0,0] => [7,5,4,6,8,1,2,3] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,0,0,1,1,1,0,1,0,1,0,0,0] => [6,5,4,7,8,1,2,3] => {{1,6},{2,3,4,5,7,8}} => {{1,2,4,5,6,7},{3,8}} => 0
[1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0] => [5,6,4,7,8,1,2,3] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0] => [8,4,5,6,7,1,2,3] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0] => [4,5,6,7,8,1,2,3] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0] => [8,7,6,5,3,1,2,4] => {{1,3,4,5,6,8},{2,7}} => {{1,3,4,5,6,8},{2,7}} => 0
[1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0] => [7,8,6,5,3,1,2,4] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,0,1,0,0,1,0,1,1,0,0,1,0] => [8,6,7,5,3,1,2,4] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0] => [7,8,5,6,3,1,2,4] => {{1,2,4,6,7,8},{3,5}} => {{1,2,3,5,7,8},{4,6}} => 0
[1,1,1,0,0,1,0,0,1,1,0,1,0,0,1,0] => [8,6,5,7,3,1,2,4] => {{1,2,4,6,7,8},{3,5}} => {{1,2,3,5,7,8},{4,6}} => 0
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0] => [7,5,6,8,3,1,2,4] => {{1,2,3,5,6,7},{4,8}} => {{1,5},{2,3,4,6,7,8}} => 0
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0] => [5,6,7,8,3,1,2,4] => {{1,2,3,5,6,7},{4,8}} => {{1,5},{2,3,4,6,7,8}} => 0
[1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0] => [7,5,6,4,3,1,2,8] => {{1,2,3,5,6,7},{4},{8}} => {{1},{2,3,4,6,7,8},{5}} => 1
[1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0] => [4,5,6,7,3,1,2,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,1,0,0,1,1,0,0,0,1,0,1,0,1,0] => [8,7,6,3,4,1,2,5] => {{1,3,4,5,6,8},{2,7}} => {{1,3,4,5,6,8},{2,7}} => 0
[1,1,1,0,0,1,1,0,0,0,1,0,1,1,0,0] => [7,8,6,3,4,1,2,5] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,0,1,1,0,0,0,1,1,0,0,1,0] => [8,6,7,3,4,1,2,5] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0] => [8,5,6,3,4,1,2,7] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,0,1,1,0,1,0,1,0,0,0,1,0] => [8,5,4,3,6,1,2,7] => {{1,2,5,6,7,8},{3,4}} => {{1,2,3,4,7,8},{5,6}} => 0
[1,1,1,0,0,1,1,0,1,1,0,0,0,0,1,0] => [8,4,5,3,6,1,2,7] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,0,1,1,1,0,1,0,0,0,0,1,0] => [8,4,3,5,6,1,2,7] => {{1,2,4,5,6,7,8},{3}} => {{1,2,3,4,5,7,8},{6}} => 0
[1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0] => [8,3,4,5,6,1,2,7] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0] => [3,4,5,6,7,1,2,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0] => [8,7,6,5,2,1,3,4] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,1,0,0,0,1,0,1,1,0,1,0,0] => [7,6,8,5,2,1,3,4] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,1,0,0,0,1,0,1,1,1,0,0,0] => [6,7,8,5,2,1,3,4] => {{1,6},{2,3,4,5,7,8}} => {{1,2,4,5,6,7},{3,8}} => 0
[1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0] => [7,8,5,6,2,1,3,4] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,1,0,0,0,1,1,0,1,0,0,1,0] => [8,6,5,7,2,1,3,4] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,1,0,0,0,1,1,0,1,0,1,0,0] => [7,6,5,8,2,1,3,4] => {{1,2,3,5,6,7},{4,8}} => {{1,5},{2,3,4,6,7,8}} => 0
[1,1,1,0,1,0,0,0,1,1,1,0,0,0,1,0] => [8,5,6,7,2,1,3,4] => {{1,3,4,6,7,8},{2,5}} => {{1,2,3,5,6,8},{4,7}} => 0
[1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0] => [8,7,6,4,2,1,3,5] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0] => [7,8,5,4,2,1,3,6] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0] => [8,6,5,4,2,1,3,7] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0] => [7,6,5,4,2,1,3,8] => {{1,2,3,5,6,7},{4},{8}} => {{1},{2,3,4,6,7,8},{5}} => 1
[1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0] => [7,6,4,5,2,1,3,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,1,0,1,0,0,1,1,1,0,0,0,0,1,0] => [8,4,5,6,2,1,3,7] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,1,0,0,1,1,1,0,1,0,0,0,0] => [5,4,6,7,2,1,3,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0] => [8,7,6,3,2,1,4,5] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,1,0,1,0,0,0,1,1,0,1,0,0] => [7,6,8,3,2,1,4,5] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0] => [6,7,8,3,2,1,4,5] => {{1,6},{2,3,4,5,7,8}} => {{1,2,4,5,6,7},{3,8}} => 0
[1,1,1,0,1,0,1,0,0,1,0,0,1,1,0,0] => [7,8,5,3,2,1,4,6] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0] => [8,6,5,3,2,1,4,7] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0] => [7,6,5,3,2,1,4,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,1,0,1,0,1,0,0,1,1,0,0,0,1,0] => [8,5,6,3,2,1,4,7] => {{1,3,4,6,7,8},{2,5}} => {{1,2,3,5,6,8},{4,7}} => 0
[1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0] => [7,8,4,3,2,1,5,6] => {{1,2,5,6,7,8},{3,4}} => {{1,2,3,4,7,8},{5,6}} => 0
[1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0] => [8,6,4,3,2,1,5,7] => {{1,2,5,6,7,8},{3,4}} => {{1,2,3,4,7,8},{5,6}} => 0
[1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0] => [5,4,6,3,2,1,7,8] => {{1,2,3,4,5,6},{7},{8}} => {{1},{2},{3,4,5,6,7,8}} => 1
[1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0] => [3,4,5,6,2,1,7,8] => {{1,2,3,4,5,6},{7},{8}} => {{1},{2},{3,4,5,6,7,8}} => 1
[1,1,1,0,1,1,0,0,0,0,1,0,1,1,0,0] => [7,8,6,2,3,1,4,5] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,1,1,0,0,0,0,1,1,0,0,1,0] => [8,6,7,2,3,1,4,5] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,1,1,0,0,0,1,0,1,0,0,1,0] => [8,6,5,2,3,1,4,7] => {{1,2,4,6,7,8},{3,5}} => {{1,2,3,5,7,8},{4,6}} => 0
[1,1,1,0,1,1,0,0,0,1,1,0,0,0,1,0] => [8,5,6,2,3,1,4,7] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,1,1,0,0,0,1,1,0,0,1,0,0] => [7,5,6,2,3,1,4,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,1,0,1,1,0,0,0,1,1,1,0,0,0,0] => [5,6,7,2,3,1,4,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0] => [7,8,4,2,3,1,5,6] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,0] => [8,6,4,2,3,1,5,7] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,1,1,0,0,1,0,0,1,0,1,0,0] => [7,6,4,2,3,1,5,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,1,0,1,1,0,1,0,0,0,0,1,1,0,0] => [7,8,3,2,4,1,5,6] => {{1,2,4,5,6,7,8},{3}} => {{1,2,3,4,5,7,8},{6}} => 0
[1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0] => [8,6,3,2,4,1,5,7] => {{1,2,4,5,6,7,8},{3}} => {{1,2,3,4,5,7,8},{6}} => 0
[1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0] => [7,6,3,2,4,1,5,8] => {{1,2,4,5,6,7},{3},{8}} => {{1},{2,3,4,5,7,8},{6}} => 1
[1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0] => [4,3,5,2,6,1,7,8] => {{1,2,3,4,5,6},{7},{8}} => {{1},{2},{3,4,5,6,7,8}} => 1
[1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0] => [7,8,2,3,4,1,5,6] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,1,1,1,0,0,0,0,1,0,0,1,0] => [8,6,2,3,4,1,5,7] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0] => [7,6,2,3,4,1,5,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0] => [5,6,2,3,4,1,7,8] => {{1,2,3,4,5,6},{7},{8}} => {{1},{2},{3,4,5,6,7,8}} => 1
[1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0] => [3,4,2,5,6,1,7,8] => {{1,2,3,4,5,6},{7},{8}} => {{1},{2},{3,4,5,6,7,8}} => 1
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,1,7,8] => {{1,2,3,4,5,6},{7},{8}} => {{1},{2},{3,4,5,6,7,8}} => 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0] => [7,8,6,5,1,2,3,4] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0] => [7,6,8,5,1,2,3,4] => {{1,3,4,5,7,8},{2,6}} => {{1,2,4,5,6,8},{3,7}} => 0
[1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0] => [6,7,8,5,1,2,3,4] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0] => [8,7,5,6,1,2,3,4] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,0,0,0,0,1,1,0,1,0,0,1,0] => [8,6,5,7,1,2,3,4] => {{1,3,4,5,7,8},{2,6}} => {{1,2,4,5,6,8},{3,7}} => 0
[1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0] => [6,7,5,8,1,2,3,4] => {{1,2,3,5,6,7},{4,8}} => {{1,5},{2,3,4,6,7,8}} => 0
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0] => [8,5,6,7,1,2,3,4] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0] => [7,5,6,8,1,2,3,4] => {{1,2,3,5,6,7},{4,8}} => {{1,5},{2,3,4,6,7,8}} => 0
[1,1,1,1,0,0,0,1,0,0,1,0,1,1,0,0] => [7,8,6,4,1,2,3,5] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,1,1,1,0,0,0,1,0,0,1,1,1,0,0,0] => [6,7,8,4,1,2,3,5] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0] => [8,7,5,4,1,2,3,6] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,1,1,1,0,0,0,1,0,1,1,0,0,0,1,0] => [8,5,6,4,1,2,3,7] => {{1,2,3,5,6,7,8},{4}} => {{1,2,3,4,6,7,8},{5}} => 0
[1,1,1,1,0,0,0,1,1,0,0,0,1,0,1,0] => [8,7,4,5,1,2,3,6] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,0,0,0,1,1,0,0,1,0,0,1,0] => [8,6,4,5,1,2,3,7] => {{1,3,4,5,7,8},{2,6}} => {{1,2,4,5,6,8},{3,7}} => 0
[1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0] => [4,5,6,7,1,2,3,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,1,1,0,0,1,0,0,0,1,0,1,1,0,0] => [7,8,6,3,1,2,4,5] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,0,0,1,0,0,0,1,1,0,1,0,0] => [7,6,8,3,1,2,4,5] => {{1,3,4,5,7,8},{2,6}} => {{1,2,4,5,6,8},{3,7}} => 0
[1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0] => [6,7,8,3,1,2,4,5] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0] => [8,7,5,3,1,2,4,6] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,0,0,1,0,0,1,1,0,0,0,1,0] => [8,5,6,3,1,2,4,7] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,0,0,1,0,1,1,0,0,1,0,0,0] => [6,4,5,3,1,2,7,8] => {{1,2,3,4,5,6},{7},{8}} => {{1},{2},{3,4,5,6,7,8}} => 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0,1,0] => [8,3,4,5,1,2,6,7] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0] => [4,3,5,6,1,2,7,8] => {{1,2,3,4,5,6},{7},{8}} => {{1},{2},{3,4,5,6,7,8}} => 1
[1,1,1,1,0,1,0,0,0,0,1,0,1,1,0,0] => [7,8,6,2,1,3,4,5] => {{1,2,4,5,7,8},{3,6}} => {{1,2,4,5,7,8},{3,6}} => 0
[1,1,1,1,0,1,0,0,0,1,0,0,1,1,0,0] => [7,8,5,2,1,3,4,6] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0] => [8,6,5,2,1,3,4,7] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0] => [7,6,5,2,1,3,4,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,1,1,0,1,0,0,0,1,1,0,0,0,1,0] => [8,5,6,2,1,3,4,7] => {{1,2,4,5,7,8},{3,6}} => {{1,2,4,5,7,8},{3,6}} => 0
[1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0] => [8,7,4,2,1,3,5,6] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0] => [8,5,4,2,1,3,6,7] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0] => [6,5,4,2,1,3,7,8] => {{1,2,3,4,5,6},{7},{8}} => {{1},{2},{3,4,5,6,7,8}} => 1
[1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0] => [8,7,3,2,1,4,5,6] => {{1,2,4,5,6,7,8},{3}} => {{1,2,3,4,5,7,8},{6}} => 0
[1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0] => [6,7,3,2,1,4,5,8] => {{1,2,4,5,6,7},{3},{8}} => {{1},{2,3,4,5,7,8},{6}} => 1
[1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0] => [8,5,3,2,1,4,6,7] => {{1,2,4,5,6,7,8},{3}} => {{1,2,3,4,5,7,8},{6}} => 0
[1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0] => [7,5,3,2,1,4,6,8] => {{1,2,4,5,6,7},{3},{8}} => {{1},{2,3,4,5,7,8},{6}} => 1
[1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0] => [8,7,2,3,1,4,5,6] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0] => [6,7,2,3,1,4,5,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0] => [7,5,2,3,1,4,6,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0] => [8,7,6,1,2,3,4,5] => {{1,2,4,5,7,8},{3,6}} => {{1,2,4,5,7,8},{3,6}} => 0
[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0] => [8,6,7,1,2,3,4,5] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0] => [6,7,8,1,2,3,4,5] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0] => [8,7,5,1,2,3,4,6] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0] => [6,7,5,1,2,3,4,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0] => [5,6,7,1,2,3,4,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0] => [7,8,4,1,2,3,5,6] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0] => [8,6,4,1,2,3,5,7] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0] => [8,5,4,1,2,3,6,7] => {{1,3,4,6,7,8},{2,5}} => {{1,2,3,5,6,8},{4,7}} => 0
[1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0] => [8,4,5,1,2,3,6,7] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0] => [7,4,5,1,2,3,6,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0] => [7,8,3,1,2,4,5,6] => {{1,2,4,5,6,7,8},{3}} => {{1,2,3,4,5,7,8},{6}} => 0
[1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0] => [7,4,3,1,2,5,6,8] => {{1,2,4,5,6,7},{3},{8}} => {{1},{2,3,4,5,7,8},{6}} => 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0] => [8,3,4,1,2,5,6,7] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0] => [7,3,4,1,2,5,6,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0] => [7,8,2,1,3,4,5,6] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,1,0,1,0,0,0,1,1,0,0,0,0] => [5,6,2,1,3,4,7,8] => {{1,2,3,4,5,6},{7},{8}} => {{1},{2},{3,4,5,6,7,8}} => 1
[1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0] => [8,4,2,1,3,5,6,7] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0] => [6,4,2,1,3,5,7,8] => {{1,2,3,4,5,6},{7},{8}} => {{1},{2},{3,4,5,6,7,8}} => 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0] => [8,3,2,1,4,5,6,7] => {{1,4,5,6,7,8},{2,3}} => {{1,2,3,4,5,8},{6,7}} => 0
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0] => [8,7,1,2,3,4,5,6] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0] => [6,7,1,2,3,4,5,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0] => [8,5,1,2,3,4,6,7] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0] => [7,5,1,2,3,4,6,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0] => [8,4,1,2,3,5,6,7] => {{1,3,5,6,7,8},{2,4}} => {{1,2,3,4,6,8},{5,7}} => 0
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0] => [8,3,1,2,4,5,6,7] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0] => [7,3,1,2,4,5,6,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0] => [6,3,1,2,4,5,7,8] => {{1,2,3,4,5,6},{7},{8}} => {{1},{2},{3,4,5,6,7,8}} => 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0] => [8,2,1,3,4,5,6,7] => {{1,3,4,5,6,7,8},{2}} => {{1,2,3,4,5,6,8},{7}} => 0
[1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0] => [7,2,1,3,4,5,6,8] => {{1,3,4,5,6,7},{2},{8}} => {{1},{2,3,4,5,6,8},{7}} => 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [8,1,2,3,4,5,6,7] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0] => [7,1,2,3,4,5,6,8] => {{1,2,3,4,5,6,7},{8}} => {{1},{2,3,4,5,6,7,8}} => 0
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0] => [6,1,2,3,4,5,7,8] => {{1,2,3,4,5,6},{7},{8}} => {{1},{2},{3,4,5,6,7,8}} => 1
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Generating function
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Sequences for the coefficients of a few of the
first generating functions, in the case at hand:
1,1 4,0,0,1 10,3,0,0,0,0,1 31,6,0,4,0,0,0,0,0,0,1 97,21,0,8,0,0,5,0,0,0,0,0,0,0,0,1 316,66,0,30,0,0,10,0,0,0,6,0,0,0,0,0,0,0,0,0,0,1
$F_{2} = 1 + q$
$F_{3} = 4 + q^{3}$
$F_{4} = 10 + 3\ q + q^{6}$
$F_{5} = 31 + 6\ q + 4\ q^{3} + q^{10}$
$F_{6} = 97 + 21\ q + 8\ q^{3} + 5\ q^{6} + q^{15}$
$F_{7} = 316 + 66\ q + 30\ q^{3} + 10\ q^{6} + 6\ q^{10} + q^{21}$
Description
The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton.
This is the number of pairs $i\lt j$ in different blocks such that $i$ is the maximal element of a block and $j$ is a singleton block.
This is the number of pairs $i\lt j$ in different blocks such that $i$ is the maximal element of a block and $j$ is a singleton block.
Map
complement
Description
The complement of a set partition obtained by replacing $i$ with $n+1-i$.
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].
This bijection is defined in [1, Section 2].
Map
to cycle type
Description
Let $\pi=c_1\dots c_r$ a permutation of size $n$ decomposed in its cyclic parts. The associated set partition of $[n]$ then is $S=S_1\cup\dots\cup S_r$ such that $S_i$ is the set of integers in the cycle $c_i$.
A permutation is cyclic [1] if and only if its cycle type is a hook partition [2].
A permutation is cyclic [1] if and only if its cycle type is a hook partition [2].
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