Identifier
-
Mp00031:
Dyck paths
—to 312-avoiding permutation⟶
Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000730: Set partitions ⟶ ℤ
Values
[1,0,1,0] => [1,2] => {{1},{2}} => 0
[1,1,0,0] => [2,1] => {{1,2}} => 1
[1,0,1,0,1,0] => [1,2,3] => {{1},{2},{3}} => 0
[1,0,1,1,0,0] => [1,3,2] => {{1},{2,3}} => 1
[1,1,0,0,1,0] => [2,1,3] => {{1,2},{3}} => 1
[1,1,0,1,0,0] => [2,3,1] => {{1,2,3}} => 1
[1,1,1,0,0,0] => [3,2,1] => {{1,3},{2}} => 2
[1,0,1,0,1,0,1,0] => [1,2,3,4] => {{1},{2},{3},{4}} => 0
[1,0,1,0,1,1,0,0] => [1,2,4,3] => {{1},{2},{3,4}} => 1
[1,0,1,1,0,0,1,0] => [1,3,2,4] => {{1},{2,3},{4}} => 1
[1,0,1,1,0,1,0,0] => [1,3,4,2] => {{1},{2,3,4}} => 1
[1,0,1,1,1,0,0,0] => [1,4,3,2] => {{1},{2,4},{3}} => 2
[1,1,0,0,1,0,1,0] => [2,1,3,4] => {{1,2},{3},{4}} => 1
[1,1,0,0,1,1,0,0] => [2,1,4,3] => {{1,2},{3,4}} => 1
[1,1,0,1,0,0,1,0] => [2,3,1,4] => {{1,2,3},{4}} => 1
[1,1,0,1,0,1,0,0] => [2,3,4,1] => {{1,2,3,4}} => 1
[1,1,0,1,1,0,0,0] => [2,4,3,1] => {{1,2,4},{3}} => 2
[1,1,1,0,0,0,1,0] => [3,2,1,4] => {{1,3},{2},{4}} => 2
[1,1,1,0,0,1,0,0] => [3,2,4,1] => {{1,3,4},{2}} => 2
[1,1,1,0,1,0,0,0] => [3,4,2,1] => {{1,2,3,4}} => 1
[1,1,1,1,0,0,0,0] => [4,3,2,1] => {{1,4},{2,3}} => 3
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}} => 0
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => {{1},{2},{3},{4,5}} => 1
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => {{1},{2},{3,4},{5}} => 1
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => {{1},{2},{3,4,5}} => 1
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => {{1},{2},{3,5},{4}} => 2
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => {{1},{2,3},{4},{5}} => 1
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => {{1},{2,3},{4,5}} => 1
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => {{1},{2,3,4},{5}} => 1
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => {{1},{2,3,4,5}} => 1
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => {{1},{2,3,5},{4}} => 2
[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => {{1},{2,4},{3},{5}} => 2
[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => {{1},{2,4,5},{3}} => 2
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => {{1},{2,3,4,5}} => 1
[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => {{1},{2,5},{3,4}} => 3
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => {{1,2},{3},{4},{5}} => 1
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => {{1,2},{3},{4,5}} => 1
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => {{1,2},{3,4},{5}} => 1
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => {{1,2},{3,4,5}} => 1
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => {{1,2},{3,5},{4}} => 2
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => {{1,2,3},{4},{5}} => 1
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => {{1,2,3},{4,5}} => 1
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => {{1,2,3,4},{5}} => 1
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => {{1,2,3,4,5}} => 1
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => {{1,2,3,5},{4}} => 2
[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => {{1,2,4},{3},{5}} => 2
[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => {{1,2,4,5},{3}} => 2
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => {{1,2,3,4,5}} => 1
[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => {{1,2,5},{3,4}} => 3
[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => {{1,3},{2},{4},{5}} => 2
[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => {{1,3},{2},{4,5}} => 2
[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => {{1,3,4},{2},{5}} => 2
[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => {{1,3,4,5},{2}} => 2
[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => {{1,3,5},{2},{4}} => 2
[1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => {{1,2,3,4},{5}} => 1
[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => {{1,2,3,4,5}} => 1
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => {{1,3,5},{2,4}} => 2
[1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => {{1,2,3,4,5}} => 1
[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => {{1,4},{2,3},{5}} => 3
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => {{1,4,5},{2,3}} => 3
[1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => {{1,2,3,4,5}} => 1
[1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => {{1,2,4,5},{3}} => 2
[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => {{1,5},{2,4},{3}} => 4
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => {{1},{2},{3},{4},{5},{6}} => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => {{1},{2},{3},{4},{5,6}} => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => {{1},{2},{3},{4,5},{6}} => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => {{1},{2},{3},{4,5,6}} => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => {{1},{2},{3},{4,6},{5}} => 2
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => {{1},{2},{3,4},{5},{6}} => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => {{1},{2},{3,4},{5,6}} => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => {{1},{2},{3,4,5},{6}} => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => {{1},{2},{3,4,5,6}} => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => {{1},{2},{3,4,6},{5}} => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => {{1},{2},{3,5},{4},{6}} => 2
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => {{1},{2},{3,5,6},{4}} => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => {{1},{2},{3,4,5,6}} => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => {{1},{2},{3,6},{4,5}} => 3
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => {{1},{2,3},{4},{5},{6}} => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => {{1},{2,3},{4},{5,6}} => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => {{1},{2,3},{4,5},{6}} => 1
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => {{1},{2,3},{4,5,6}} => 1
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => {{1},{2,3},{4,6},{5}} => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => {{1},{2,3,4},{5},{6}} => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => {{1},{2,3,4},{5,6}} => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => {{1},{2,3,4,5},{6}} => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => {{1},{2,3,4,5,6}} => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => {{1},{2,3,4,6},{5}} => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => {{1},{2,3,5},{4},{6}} => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => {{1},{2,3,5,6},{4}} => 2
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => {{1},{2,3,4,5,6}} => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => {{1},{2,3,6},{4,5}} => 3
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => {{1},{2,4},{3},{5},{6}} => 2
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => {{1},{2,4},{3},{5,6}} => 2
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => {{1},{2,4,5},{3},{6}} => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => {{1},{2,4,5,6},{3}} => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => {{1},{2,4,6},{3},{5}} => 2
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,3,2,6] => {{1},{2,3,4,5},{6}} => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => {{1},{2,3,4,5,6}} => 1
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => {{1},{2,4,6},{3,5}} => 2
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => {{1},{2,3,4,5,6}} => 1
[1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => {{1},{2,5},{3,4},{6}} => 3
>>> Load all 874 entries. <<<[1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,4,3,6,2] => {{1},{2,5,6},{3,4}} => 3
[1,0,1,1,1,1,0,0,1,0,0,0] => [1,5,4,6,3,2] => {{1},{2,3,4,5,6}} => 1
[1,0,1,1,1,1,0,1,0,0,0,0] => [1,5,6,4,3,2] => {{1},{2,3,5,6},{4}} => 2
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => {{1},{2,6},{3,5},{4}} => 4
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6] => {{1,2},{3},{4},{5},{6}} => 1
[1,1,0,0,1,0,1,0,1,1,0,0] => [2,1,3,4,6,5] => {{1,2},{3},{4},{5,6}} => 1
[1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,3,5,4,6] => {{1,2},{3},{4,5},{6}} => 1
[1,1,0,0,1,0,1,1,0,1,0,0] => [2,1,3,5,6,4] => {{1,2},{3},{4,5,6}} => 1
[1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3,6,5,4] => {{1,2},{3},{4,6},{5}} => 2
[1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => {{1,2},{3,4},{5},{6}} => 1
[1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5] => {{1,2},{3,4},{5,6}} => 1
[1,1,0,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => {{1,2},{3,4,5},{6}} => 1
[1,1,0,0,1,1,0,1,0,1,0,0] => [2,1,4,5,6,3] => {{1,2},{3,4,5,6}} => 1
[1,1,0,0,1,1,0,1,1,0,0,0] => [2,1,4,6,5,3] => {{1,2},{3,4,6},{5}} => 2
[1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,4,3,6] => {{1,2},{3,5},{4},{6}} => 2
[1,1,0,0,1,1,1,0,0,1,0,0] => [2,1,5,4,6,3] => {{1,2},{3,5,6},{4}} => 2
[1,1,0,0,1,1,1,0,1,0,0,0] => [2,1,5,6,4,3] => {{1,2},{3,4,5,6}} => 1
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,6,5,4,3] => {{1,2},{3,6},{4,5}} => 3
[1,1,0,1,0,0,1,0,1,0,1,0] => [2,3,1,4,5,6] => {{1,2,3},{4},{5},{6}} => 1
[1,1,0,1,0,0,1,0,1,1,0,0] => [2,3,1,4,6,5] => {{1,2,3},{4},{5,6}} => 1
[1,1,0,1,0,0,1,1,0,0,1,0] => [2,3,1,5,4,6] => {{1,2,3},{4,5},{6}} => 1
[1,1,0,1,0,0,1,1,0,1,0,0] => [2,3,1,5,6,4] => {{1,2,3},{4,5,6}} => 1
[1,1,0,1,0,0,1,1,1,0,0,0] => [2,3,1,6,5,4] => {{1,2,3},{4,6},{5}} => 2
[1,1,0,1,0,1,0,0,1,0,1,0] => [2,3,4,1,5,6] => {{1,2,3,4},{5},{6}} => 1
[1,1,0,1,0,1,0,0,1,1,0,0] => [2,3,4,1,6,5] => {{1,2,3,4},{5,6}} => 1
[1,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,1,6] => {{1,2,3,4,5},{6}} => 1
[1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => {{1,2,3,4,5,6}} => 1
[1,1,0,1,0,1,0,1,1,0,0,0] => [2,3,4,6,5,1] => {{1,2,3,4,6},{5}} => 2
[1,1,0,1,0,1,1,0,0,0,1,0] => [2,3,5,4,1,6] => {{1,2,3,5},{4},{6}} => 2
[1,1,0,1,0,1,1,0,0,1,0,0] => [2,3,5,4,6,1] => {{1,2,3,5,6},{4}} => 2
[1,1,0,1,0,1,1,0,1,0,0,0] => [2,3,5,6,4,1] => {{1,2,3,4,5,6}} => 1
[1,1,0,1,0,1,1,1,0,0,0,0] => [2,3,6,5,4,1] => {{1,2,3,6},{4,5}} => 3
[1,1,0,1,1,0,0,0,1,0,1,0] => [2,4,3,1,5,6] => {{1,2,4},{3},{5},{6}} => 2
[1,1,0,1,1,0,0,0,1,1,0,0] => [2,4,3,1,6,5] => {{1,2,4},{3},{5,6}} => 2
[1,1,0,1,1,0,0,1,0,0,1,0] => [2,4,3,5,1,6] => {{1,2,4,5},{3},{6}} => 2
[1,1,0,1,1,0,0,1,0,1,0,0] => [2,4,3,5,6,1] => {{1,2,4,5,6},{3}} => 2
[1,1,0,1,1,0,0,1,1,0,0,0] => [2,4,3,6,5,1] => {{1,2,4,6},{3},{5}} => 2
[1,1,0,1,1,0,1,0,0,0,1,0] => [2,4,5,3,1,6] => {{1,2,3,4,5},{6}} => 1
[1,1,0,1,1,0,1,0,0,1,0,0] => [2,4,5,3,6,1] => {{1,2,3,4,5,6}} => 1
[1,1,0,1,1,0,1,0,1,0,0,0] => [2,4,5,6,3,1] => {{1,2,4,6},{3,5}} => 2
[1,1,0,1,1,0,1,1,0,0,0,0] => [2,4,6,5,3,1] => {{1,2,3,4,5,6}} => 1
[1,1,0,1,1,1,0,0,0,0,1,0] => [2,5,4,3,1,6] => {{1,2,5},{3,4},{6}} => 3
[1,1,0,1,1,1,0,0,0,1,0,0] => [2,5,4,3,6,1] => {{1,2,5,6},{3,4}} => 3
[1,1,0,1,1,1,0,0,1,0,0,0] => [2,5,4,6,3,1] => {{1,2,3,4,5,6}} => 1
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,5,6,4,3,1] => {{1,2,3,5,6},{4}} => 2
[1,1,0,1,1,1,1,0,0,0,0,0] => [2,6,5,4,3,1] => {{1,2,6},{3,5},{4}} => 4
[1,1,1,0,0,0,1,0,1,0,1,0] => [3,2,1,4,5,6] => {{1,3},{2},{4},{5},{6}} => 2
[1,1,1,0,0,0,1,0,1,1,0,0] => [3,2,1,4,6,5] => {{1,3},{2},{4},{5,6}} => 2
[1,1,1,0,0,0,1,1,0,0,1,0] => [3,2,1,5,4,6] => {{1,3},{2},{4,5},{6}} => 2
[1,1,1,0,0,0,1,1,0,1,0,0] => [3,2,1,5,6,4] => {{1,3},{2},{4,5,6}} => 2
[1,1,1,0,0,0,1,1,1,0,0,0] => [3,2,1,6,5,4] => {{1,3},{2},{4,6},{5}} => 2
[1,1,1,0,0,1,0,0,1,0,1,0] => [3,2,4,1,5,6] => {{1,3,4},{2},{5},{6}} => 2
[1,1,1,0,0,1,0,0,1,1,0,0] => [3,2,4,1,6,5] => {{1,3,4},{2},{5,6}} => 2
[1,1,1,0,0,1,0,1,0,0,1,0] => [3,2,4,5,1,6] => {{1,3,4,5},{2},{6}} => 2
[1,1,1,0,0,1,0,1,0,1,0,0] => [3,2,4,5,6,1] => {{1,3,4,5,6},{2}} => 2
[1,1,1,0,0,1,0,1,1,0,0,0] => [3,2,4,6,5,1] => {{1,3,4,6},{2},{5}} => 2
[1,1,1,0,0,1,1,0,0,0,1,0] => [3,2,5,4,1,6] => {{1,3,5},{2},{4},{6}} => 2
[1,1,1,0,0,1,1,0,0,1,0,0] => [3,2,5,4,6,1] => {{1,3,5,6},{2},{4}} => 2
[1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,5,6,4,1] => {{1,3,4,5,6},{2}} => 2
[1,1,1,0,0,1,1,1,0,0,0,0] => [3,2,6,5,4,1] => {{1,3,6},{2},{4,5}} => 3
[1,1,1,0,1,0,0,0,1,0,1,0] => [3,4,2,1,5,6] => {{1,2,3,4},{5},{6}} => 1
[1,1,1,0,1,0,0,0,1,1,0,0] => [3,4,2,1,6,5] => {{1,2,3,4},{5,6}} => 1
[1,1,1,0,1,0,0,1,0,0,1,0] => [3,4,2,5,1,6] => {{1,2,3,4,5},{6}} => 1
[1,1,1,0,1,0,0,1,0,1,0,0] => [3,4,2,5,6,1] => {{1,2,3,4,5,6}} => 1
[1,1,1,0,1,0,0,1,1,0,0,0] => [3,4,2,6,5,1] => {{1,2,3,4,6},{5}} => 2
[1,1,1,0,1,0,1,0,0,0,1,0] => [3,4,5,2,1,6] => {{1,3,5},{2,4},{6}} => 2
[1,1,1,0,1,0,1,0,0,1,0,0] => [3,4,5,2,6,1] => {{1,3,5,6},{2,4}} => 2
[1,1,1,0,1,0,1,0,1,0,0,0] => [3,4,5,6,2,1] => {{1,2,3,4,5,6}} => 1
[1,1,1,0,1,0,1,1,0,0,0,0] => [3,4,6,5,2,1] => {{1,3,6},{2,4,5}} => 3
[1,1,1,0,1,1,0,0,0,0,1,0] => [3,5,4,2,1,6] => {{1,2,3,4,5},{6}} => 1
[1,1,1,0,1,1,0,0,0,1,0,0] => [3,5,4,2,6,1] => {{1,2,3,4,5,6}} => 1
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,5,4,6,2,1] => {{1,3,4,6},{2,5}} => 3
[1,1,1,0,1,1,0,1,0,0,0,0] => [3,5,6,4,2,1] => {{1,3,6},{2,5},{4}} => 3
[1,1,1,0,1,1,1,0,0,0,0,0] => [3,6,5,4,2,1] => {{1,2,3,5,6},{4}} => 2
[1,1,1,1,0,0,0,0,1,0,1,0] => [4,3,2,1,5,6] => {{1,4},{2,3},{5},{6}} => 3
[1,1,1,1,0,0,0,0,1,1,0,0] => [4,3,2,1,6,5] => {{1,4},{2,3},{5,6}} => 3
[1,1,1,1,0,0,0,1,0,0,1,0] => [4,3,2,5,1,6] => {{1,4,5},{2,3},{6}} => 3
[1,1,1,1,0,0,0,1,0,1,0,0] => [4,3,2,5,6,1] => {{1,4,5,6},{2,3}} => 3
[1,1,1,1,0,0,0,1,1,0,0,0] => [4,3,2,6,5,1] => {{1,4,6},{2,3},{5}} => 3
[1,1,1,1,0,0,1,0,0,0,1,0] => [4,3,5,2,1,6] => {{1,2,3,4,5},{6}} => 1
[1,1,1,1,0,0,1,0,0,1,0,0] => [4,3,5,2,6,1] => {{1,2,3,4,5,6}} => 1
[1,1,1,1,0,0,1,0,1,0,0,0] => [4,3,5,6,2,1] => {{1,4,6},{2,3,5}} => 3
[1,1,1,1,0,0,1,1,0,0,0,0] => [4,3,6,5,2,1] => {{1,2,3,4,5,6}} => 1
[1,1,1,1,0,1,0,0,0,0,1,0] => [4,5,3,2,1,6] => {{1,2,4,5},{3},{6}} => 2
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,5,3,2,6,1] => {{1,2,4,5,6},{3}} => 2
[1,1,1,1,0,1,0,0,1,0,0,0] => [4,5,3,6,2,1] => {{1,4,6},{2,5},{3}} => 3
[1,1,1,1,0,1,0,1,0,0,0,0] => [4,5,6,3,2,1] => {{1,3,4,6},{2,5}} => 3
[1,1,1,1,0,1,1,0,0,0,0,0] => [4,6,5,3,2,1] => {{1,2,3,4,5,6}} => 1
[1,1,1,1,1,0,0,0,0,0,1,0] => [5,4,3,2,1,6] => {{1,5},{2,4},{3},{6}} => 4
[1,1,1,1,1,0,0,0,0,1,0,0] => [5,4,3,2,6,1] => {{1,5,6},{2,4},{3}} => 4
[1,1,1,1,1,0,0,0,1,0,0,0] => [5,4,3,6,2,1] => {{1,2,4,5,6},{3}} => 2
[1,1,1,1,1,0,0,1,0,0,0,0] => [5,4,6,3,2,1] => {{1,2,3,4,5,6}} => 1
[1,1,1,1,1,0,1,0,0,0,0,0] => [5,6,4,3,2,1] => {{1,2,5,6},{3,4}} => 3
[1,1,1,1,1,1,0,0,0,0,0,0] => [6,5,4,3,2,1] => {{1,6},{2,5},{3,4}} => 5
[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7] => {{1},{2},{3},{4},{5},{6},{7}} => 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,7,6] => {{1},{2},{3},{4},{5},{6,7}} => 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,4,6,5,7] => {{1},{2},{3},{4},{5,6},{7}} => 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,4,6,7,5] => {{1},{2},{3},{4},{5,6,7}} => 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,4,7,6,5] => {{1},{2},{3},{4},{5,7},{6}} => 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,3,5,4,6,7] => {{1},{2},{3},{4,5},{6},{7}} => 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,3,5,4,7,6] => {{1},{2},{3},{4,5},{6,7}} => 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,3,5,6,4,7] => {{1},{2},{3},{4,5,6},{7}} => 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,3,5,6,7,4] => {{1},{2},{3},{4,5,6,7}} => 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,3,5,7,6,4] => {{1},{2},{3},{4,5,7},{6}} => 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,3,6,5,4,7] => {{1},{2},{3},{4,6},{5},{7}} => 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,3,6,5,7,4] => {{1},{2},{3},{4,6,7},{5}} => 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,3,6,7,5,4] => {{1},{2},{3},{4,5,6,7}} => 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,3,7,6,5,4] => {{1},{2},{3},{4,7},{5,6}} => 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [1,2,4,3,5,6,7] => {{1},{2},{3,4},{5},{6},{7}} => 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [1,2,4,3,5,7,6] => {{1},{2},{3,4},{5},{6,7}} => 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [1,2,4,3,6,5,7] => {{1},{2},{3,4},{5,6},{7}} => 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0] => [1,2,4,3,6,7,5] => {{1},{2},{3,4},{5,6,7}} => 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [1,2,4,3,7,6,5] => {{1},{2},{3,4},{5,7},{6}} => 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0] => [1,2,4,5,3,6,7] => {{1},{2},{3,4,5},{6},{7}} => 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0] => [1,2,4,5,3,7,6] => {{1},{2},{3,4,5},{6,7}} => 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0] => [1,2,4,5,6,3,7] => {{1},{2},{3,4,5,6},{7}} => 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,2,4,5,6,7,3] => {{1},{2},{3,4,5,6,7}} => 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0] => [1,2,4,5,7,6,3] => {{1},{2},{3,4,5,7},{6}} => 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0] => [1,2,4,6,5,3,7] => {{1},{2},{3,4,6},{5},{7}} => 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0] => [1,2,4,6,5,7,3] => {{1},{2},{3,4,6,7},{5}} => 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0] => [1,2,4,6,7,5,3] => {{1},{2},{3,4,5,6,7}} => 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0] => [1,2,4,7,6,5,3] => {{1},{2},{3,4,7},{5,6}} => 3
[1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [1,2,5,4,3,6,7] => {{1},{2},{3,5},{4},{6},{7}} => 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [1,2,5,4,3,7,6] => {{1},{2},{3,5},{4},{6,7}} => 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0] => [1,2,5,4,6,3,7] => {{1},{2},{3,5,6},{4},{7}} => 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [1,2,5,4,6,7,3] => {{1},{2},{3,5,6,7},{4}} => 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0] => [1,2,5,4,7,6,3] => {{1},{2},{3,5,7},{4},{6}} => 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0] => [1,2,5,6,4,3,7] => {{1},{2},{3,4,5,6},{7}} => 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [1,2,5,6,4,7,3] => {{1},{2},{3,4,5,6,7}} => 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [1,2,5,6,7,4,3] => {{1},{2},{3,5,7},{4,6}} => 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [1,2,5,7,6,4,3] => {{1},{2},{3,4,5,6,7}} => 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [1,2,6,5,4,3,7] => {{1},{2},{3,6},{4,5},{7}} => 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [1,2,6,5,4,7,3] => {{1},{2},{3,6,7},{4,5}} => 3
[1,0,1,0,1,1,1,1,0,0,1,0,0,0] => [1,2,6,5,7,4,3] => {{1},{2},{3,4,5,6,7}} => 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [1,2,6,7,5,4,3] => {{1},{2},{3,4,6,7},{5}} => 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [1,2,7,6,5,4,3] => {{1},{2},{3,7},{4,6},{5}} => 4
[1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [1,3,2,4,5,6,7] => {{1},{2,3},{4},{5},{6},{7}} => 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [1,3,2,4,5,7,6] => {{1},{2,3},{4},{5},{6,7}} => 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [1,3,2,4,6,5,7] => {{1},{2,3},{4},{5,6},{7}} => 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0] => [1,3,2,4,6,7,5] => {{1},{2,3},{4},{5,6,7}} => 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [1,3,2,4,7,6,5] => {{1},{2,3},{4},{5,7},{6}} => 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [1,3,2,5,4,6,7] => {{1},{2,3},{4,5},{6},{7}} => 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4,7,6] => {{1},{2,3},{4,5},{6,7}} => 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0] => [1,3,2,5,6,4,7] => {{1},{2,3},{4,5,6},{7}} => 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0] => [1,3,2,5,6,7,4] => {{1},{2,3},{4,5,6,7}} => 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0] => [1,3,2,5,7,6,4] => {{1},{2,3},{4,5,7},{6}} => 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [1,3,2,6,5,4,7] => {{1},{2,3},{4,6},{5},{7}} => 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0] => [1,3,2,6,5,7,4] => {{1},{2,3},{4,6,7},{5}} => 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0] => [1,3,2,6,7,5,4] => {{1},{2,3},{4,5,6,7}} => 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [1,3,2,7,6,5,4] => {{1},{2,3},{4,7},{5,6}} => 3
[1,0,1,1,0,1,0,0,1,0,1,0,1,0] => [1,3,4,2,5,6,7] => {{1},{2,3,4},{5},{6},{7}} => 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5,7,6] => {{1},{2,3,4},{5},{6,7}} => 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0] => [1,3,4,2,6,5,7] => {{1},{2,3,4},{5,6},{7}} => 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0] => [1,3,4,2,6,7,5] => {{1},{2,3,4},{5,6,7}} => 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0] => [1,3,4,2,7,6,5] => {{1},{2,3,4},{5,7},{6}} => 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0] => [1,3,4,5,2,6,7] => {{1},{2,3,4,5},{6},{7}} => 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0] => [1,3,4,5,2,7,6] => {{1},{2,3,4,5},{6,7}} => 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0] => [1,3,4,5,6,2,7] => {{1},{2,3,4,5,6},{7}} => 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,7,2] => {{1},{2,3,4,5,6,7}} => 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0] => [1,3,4,5,7,6,2] => {{1},{2,3,4,5,7},{6}} => 2
[1,0,1,1,0,1,0,1,1,0,0,0,1,0] => [1,3,4,6,5,2,7] => {{1},{2,3,4,6},{5},{7}} => 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0] => [1,3,4,6,5,7,2] => {{1},{2,3,4,6,7},{5}} => 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,0] => [1,3,4,6,7,5,2] => {{1},{2,3,4,5,6,7}} => 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0] => [1,3,4,7,6,5,2] => {{1},{2,3,4,7},{5,6}} => 3
[1,0,1,1,0,1,1,0,0,0,1,0,1,0] => [1,3,5,4,2,6,7] => {{1},{2,3,5},{4},{6},{7}} => 2
[1,0,1,1,0,1,1,0,0,0,1,1,0,0] => [1,3,5,4,2,7,6] => {{1},{2,3,5},{4},{6,7}} => 2
[1,0,1,1,0,1,1,0,0,1,0,0,1,0] => [1,3,5,4,6,2,7] => {{1},{2,3,5,6},{4},{7}} => 2
[1,0,1,1,0,1,1,0,0,1,0,1,0,0] => [1,3,5,4,6,7,2] => {{1},{2,3,5,6,7},{4}} => 2
[1,0,1,1,0,1,1,0,0,1,1,0,0,0] => [1,3,5,4,7,6,2] => {{1},{2,3,5,7},{4},{6}} => 2
[1,0,1,1,0,1,1,0,1,0,0,0,1,0] => [1,3,5,6,4,2,7] => {{1},{2,3,4,5,6},{7}} => 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0] => [1,3,5,6,4,7,2] => {{1},{2,3,4,5,6,7}} => 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0] => [1,3,5,6,7,4,2] => {{1},{2,3,5,7},{4,6}} => 2
[1,0,1,1,0,1,1,0,1,1,0,0,0,0] => [1,3,5,7,6,4,2] => {{1},{2,3,4,5,6,7}} => 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0] => [1,3,6,5,4,2,7] => {{1},{2,3,6},{4,5},{7}} => 3
[1,0,1,1,0,1,1,1,0,0,0,1,0,0] => [1,3,6,5,4,7,2] => {{1},{2,3,6,7},{4,5}} => 3
[1,0,1,1,0,1,1,1,0,0,1,0,0,0] => [1,3,6,5,7,4,2] => {{1},{2,3,4,5,6,7}} => 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0] => [1,3,6,7,5,4,2] => {{1},{2,3,4,6,7},{5}} => 2
[1,0,1,1,0,1,1,1,1,0,0,0,0,0] => [1,3,7,6,5,4,2] => {{1},{2,3,7},{4,6},{5}} => 4
[1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [1,4,3,2,5,6,7] => {{1},{2,4},{3},{5},{6},{7}} => 2
[1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [1,4,3,2,5,7,6] => {{1},{2,4},{3},{5},{6,7}} => 2
[1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [1,4,3,2,6,5,7] => {{1},{2,4},{3},{5,6},{7}} => 2
[1,0,1,1,1,0,0,0,1,1,0,1,0,0] => [1,4,3,2,6,7,5] => {{1},{2,4},{3},{5,6,7}} => 2
[1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [1,4,3,2,7,6,5] => {{1},{2,4},{3},{5,7},{6}} => 2
[1,0,1,1,1,0,0,1,0,0,1,0,1,0] => [1,4,3,5,2,6,7] => {{1},{2,4,5},{3},{6},{7}} => 2
[1,0,1,1,1,0,0,1,0,0,1,1,0,0] => [1,4,3,5,2,7,6] => {{1},{2,4,5},{3},{6,7}} => 2
[1,0,1,1,1,0,0,1,0,1,0,0,1,0] => [1,4,3,5,6,2,7] => {{1},{2,4,5,6},{3},{7}} => 2
[1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [1,4,3,5,6,7,2] => {{1},{2,4,5,6,7},{3}} => 2
[1,0,1,1,1,0,0,1,0,1,1,0,0,0] => [1,4,3,5,7,6,2] => {{1},{2,4,5,7},{3},{6}} => 2
[1,0,1,1,1,0,0,1,1,0,0,0,1,0] => [1,4,3,6,5,2,7] => {{1},{2,4,6},{3},{5},{7}} => 2
[1,0,1,1,1,0,0,1,1,0,0,1,0,0] => [1,4,3,6,5,7,2] => {{1},{2,4,6,7},{3},{5}} => 2
[1,0,1,1,1,0,0,1,1,0,1,0,0,0] => [1,4,3,6,7,5,2] => {{1},{2,4,5,6,7},{3}} => 2
[1,0,1,1,1,0,0,1,1,1,0,0,0,0] => [1,4,3,7,6,5,2] => {{1},{2,4,7},{3},{5,6}} => 3
[1,0,1,1,1,0,1,0,0,0,1,0,1,0] => [1,4,5,3,2,6,7] => {{1},{2,3,4,5},{6},{7}} => 1
[1,0,1,1,1,0,1,0,0,0,1,1,0,0] => [1,4,5,3,2,7,6] => {{1},{2,3,4,5},{6,7}} => 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,0] => [1,4,5,3,6,2,7] => {{1},{2,3,4,5,6},{7}} => 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [1,4,5,3,6,7,2] => {{1},{2,3,4,5,6,7}} => 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0] => [1,4,5,3,7,6,2] => {{1},{2,3,4,5,7},{6}} => 2
[1,0,1,1,1,0,1,0,1,0,0,0,1,0] => [1,4,5,6,3,2,7] => {{1},{2,4,6},{3,5},{7}} => 2
[1,0,1,1,1,0,1,0,1,0,0,1,0,0] => [1,4,5,6,3,7,2] => {{1},{2,4,6,7},{3,5}} => 2
[1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [1,4,5,6,7,3,2] => {{1},{2,3,4,5,6,7}} => 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [1,4,5,7,6,3,2] => {{1},{2,4,7},{3,5,6}} => 3
[1,0,1,1,1,0,1,1,0,0,0,0,1,0] => [1,4,6,5,3,2,7] => {{1},{2,3,4,5,6},{7}} => 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [1,4,6,5,3,7,2] => {{1},{2,3,4,5,6,7}} => 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0] => [1,4,6,5,7,3,2] => {{1},{2,4,5,7},{3,6}} => 3
[1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [1,4,6,7,5,3,2] => {{1},{2,4,7},{3,6},{5}} => 3
[1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [1,4,7,6,5,3,2] => {{1},{2,3,4,6,7},{5}} => 2
[1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [1,5,4,3,2,6,7] => {{1},{2,5},{3,4},{6},{7}} => 3
[1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [1,5,4,3,2,7,6] => {{1},{2,5},{3,4},{6,7}} => 3
[1,0,1,1,1,1,0,0,0,1,0,0,1,0] => [1,5,4,3,6,2,7] => {{1},{2,5,6},{3,4},{7}} => 3
[1,0,1,1,1,1,0,0,0,1,0,1,0,0] => [1,5,4,3,6,7,2] => {{1},{2,5,6,7},{3,4}} => 3
[1,0,1,1,1,1,0,0,0,1,1,0,0,0] => [1,5,4,3,7,6,2] => {{1},{2,5,7},{3,4},{6}} => 3
[1,0,1,1,1,1,0,0,1,0,0,0,1,0] => [1,5,4,6,3,2,7] => {{1},{2,3,4,5,6},{7}} => 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0] => [1,5,4,6,3,7,2] => {{1},{2,3,4,5,6,7}} => 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,0] => [1,5,4,6,7,3,2] => {{1},{2,5,7},{3,4,6}} => 3
[1,0,1,1,1,1,0,0,1,1,0,0,0,0] => [1,5,4,7,6,3,2] => {{1},{2,3,4,5,6,7}} => 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0] => [1,5,6,4,3,2,7] => {{1},{2,3,5,6},{4},{7}} => 2
[1,0,1,1,1,1,0,1,0,0,0,1,0,0] => [1,5,6,4,3,7,2] => {{1},{2,3,5,6,7},{4}} => 2
[1,0,1,1,1,1,0,1,0,0,1,0,0,0] => [1,5,6,4,7,3,2] => {{1},{2,5,7},{3,6},{4}} => 3
[1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [1,5,6,7,4,3,2] => {{1},{2,4,5,7},{3,6}} => 3
[1,0,1,1,1,1,0,1,1,0,0,0,0,0] => [1,5,7,6,4,3,2] => {{1},{2,3,4,5,6,7}} => 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [1,6,5,4,3,2,7] => {{1},{2,6},{3,5},{4},{7}} => 4
[1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [1,6,5,4,3,7,2] => {{1},{2,6,7},{3,5},{4}} => 4
[1,0,1,1,1,1,1,0,0,0,1,0,0,0] => [1,6,5,4,7,3,2] => {{1},{2,3,5,6,7},{4}} => 2
[1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [1,6,5,7,4,3,2] => {{1},{2,3,4,5,6,7}} => 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [1,6,7,5,4,3,2] => {{1},{2,3,6,7},{4,5}} => 3
[1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,7,6,5,4,3,2] => {{1},{2,7},{3,6},{4,5}} => 5
[1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6,7] => {{1,2},{3},{4},{5},{6},{7}} => 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [2,1,3,4,5,7,6] => {{1,2},{3},{4},{5},{6,7}} => 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [2,1,3,4,6,5,7] => {{1,2},{3},{4},{5,6},{7}} => 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0] => [2,1,3,4,6,7,5] => {{1,2},{3},{4},{5,6,7}} => 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [2,1,3,4,7,6,5] => {{1,2},{3},{4},{5,7},{6}} => 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [2,1,3,5,4,6,7] => {{1,2},{3},{4,5},{6},{7}} => 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [2,1,3,5,4,7,6] => {{1,2},{3},{4,5},{6,7}} => 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0] => [2,1,3,5,6,4,7] => {{1,2},{3},{4,5,6},{7}} => 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0] => [2,1,3,5,6,7,4] => {{1,2},{3},{4,5,6,7}} => 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0] => [2,1,3,5,7,6,4] => {{1,2},{3},{4,5,7},{6}} => 2
[1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [2,1,3,6,5,4,7] => {{1,2},{3},{4,6},{5},{7}} => 2
[1,1,0,0,1,0,1,1,1,0,0,1,0,0] => [2,1,3,6,5,7,4] => {{1,2},{3},{4,6,7},{5}} => 2
[1,1,0,0,1,0,1,1,1,0,1,0,0,0] => [2,1,3,6,7,5,4] => {{1,2},{3},{4,5,6,7}} => 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [2,1,3,7,6,5,4] => {{1,2},{3},{4,7},{5,6}} => 3
[1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,3,5,6,7] => {{1,2},{3,4},{5},{6},{7}} => 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [2,1,4,3,5,7,6] => {{1,2},{3,4},{5},{6,7}} => 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5,7] => {{1,2},{3,4},{5,6},{7}} => 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0] => [2,1,4,3,6,7,5] => {{1,2},{3,4},{5,6,7}} => 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [2,1,4,3,7,6,5] => {{1,2},{3,4},{5,7},{6}} => 2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0] => [2,1,4,5,3,6,7] => {{1,2},{3,4,5},{6},{7}} => 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,0] => [2,1,4,5,3,7,6] => {{1,2},{3,4,5},{6,7}} => 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0] => [2,1,4,5,6,3,7] => {{1,2},{3,4,5,6},{7}} => 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0] => [2,1,4,5,6,7,3] => {{1,2},{3,4,5,6,7}} => 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0] => [2,1,4,5,7,6,3] => {{1,2},{3,4,5,7},{6}} => 2
[1,1,0,0,1,1,0,1,1,0,0,0,1,0] => [2,1,4,6,5,3,7] => {{1,2},{3,4,6},{5},{7}} => 2
[1,1,0,0,1,1,0,1,1,0,0,1,0,0] => [2,1,4,6,5,7,3] => {{1,2},{3,4,6,7},{5}} => 2
[1,1,0,0,1,1,0,1,1,0,1,0,0,0] => [2,1,4,6,7,5,3] => {{1,2},{3,4,5,6,7}} => 1
[1,1,0,0,1,1,0,1,1,1,0,0,0,0] => [2,1,4,7,6,5,3] => {{1,2},{3,4,7},{5,6}} => 3
[1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [2,1,5,4,3,6,7] => {{1,2},{3,5},{4},{6},{7}} => 2
[1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [2,1,5,4,3,7,6] => {{1,2},{3,5},{4},{6,7}} => 2
[1,1,0,0,1,1,1,0,0,1,0,0,1,0] => [2,1,5,4,6,3,7] => {{1,2},{3,5,6},{4},{7}} => 2
[1,1,0,0,1,1,1,0,0,1,0,1,0,0] => [2,1,5,4,6,7,3] => {{1,2},{3,5,6,7},{4}} => 2
[1,1,0,0,1,1,1,0,0,1,1,0,0,0] => [2,1,5,4,7,6,3] => {{1,2},{3,5,7},{4},{6}} => 2
[1,1,0,0,1,1,1,0,1,0,0,0,1,0] => [2,1,5,6,4,3,7] => {{1,2},{3,4,5,6},{7}} => 1
[1,1,0,0,1,1,1,0,1,0,0,1,0,0] => [2,1,5,6,4,7,3] => {{1,2},{3,4,5,6,7}} => 1
[1,1,0,0,1,1,1,0,1,0,1,0,0,0] => [2,1,5,6,7,4,3] => {{1,2},{3,5,7},{4,6}} => 2
[1,1,0,0,1,1,1,0,1,1,0,0,0,0] => [2,1,5,7,6,4,3] => {{1,2},{3,4,5,6,7}} => 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [2,1,6,5,4,3,7] => {{1,2},{3,6},{4,5},{7}} => 3
[1,1,0,0,1,1,1,1,0,0,0,1,0,0] => [2,1,6,5,4,7,3] => {{1,2},{3,6,7},{4,5}} => 3
[1,1,0,0,1,1,1,1,0,0,1,0,0,0] => [2,1,6,5,7,4,3] => {{1,2},{3,4,5,6,7}} => 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0] => [2,1,6,7,5,4,3] => {{1,2},{3,4,6,7},{5}} => 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [2,1,7,6,5,4,3] => {{1,2},{3,7},{4,6},{5}} => 4
[1,1,0,1,0,0,1,0,1,0,1,0,1,0] => [2,3,1,4,5,6,7] => {{1,2,3},{4},{5},{6},{7}} => 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,0] => [2,3,1,4,5,7,6] => {{1,2,3},{4},{5},{6,7}} => 1
[1,1,0,1,0,0,1,0,1,1,0,0,1,0] => [2,3,1,4,6,5,7] => {{1,2,3},{4},{5,6},{7}} => 1
[1,1,0,1,0,0,1,0,1,1,0,1,0,0] => [2,3,1,4,6,7,5] => {{1,2,3},{4},{5,6,7}} => 1
[1,1,0,1,0,0,1,0,1,1,1,0,0,0] => [2,3,1,4,7,6,5] => {{1,2,3},{4},{5,7},{6}} => 2
[1,1,0,1,0,0,1,1,0,0,1,0,1,0] => [2,3,1,5,4,6,7] => {{1,2,3},{4,5},{6},{7}} => 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,0] => [2,3,1,5,4,7,6] => {{1,2,3},{4,5},{6,7}} => 1
[1,1,0,1,0,0,1,1,0,1,0,0,1,0] => [2,3,1,5,6,4,7] => {{1,2,3},{4,5,6},{7}} => 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0] => [2,3,1,5,6,7,4] => {{1,2,3},{4,5,6,7}} => 1
[1,1,0,1,0,0,1,1,0,1,1,0,0,0] => [2,3,1,5,7,6,4] => {{1,2,3},{4,5,7},{6}} => 2
[1,1,0,1,0,0,1,1,1,0,0,0,1,0] => [2,3,1,6,5,4,7] => {{1,2,3},{4,6},{5},{7}} => 2
[1,1,0,1,0,0,1,1,1,0,0,1,0,0] => [2,3,1,6,5,7,4] => {{1,2,3},{4,6,7},{5}} => 2
[1,1,0,1,0,0,1,1,1,0,1,0,0,0] => [2,3,1,6,7,5,4] => {{1,2,3},{4,5,6,7}} => 1
[1,1,0,1,0,0,1,1,1,1,0,0,0,0] => [2,3,1,7,6,5,4] => {{1,2,3},{4,7},{5,6}} => 3
[1,1,0,1,0,1,0,0,1,0,1,0,1,0] => [2,3,4,1,5,6,7] => {{1,2,3,4},{5},{6},{7}} => 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0] => [2,3,4,1,5,7,6] => {{1,2,3,4},{5},{6,7}} => 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,0] => [2,3,4,1,6,5,7] => {{1,2,3,4},{5,6},{7}} => 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,0] => [2,3,4,1,6,7,5] => {{1,2,3,4},{5,6,7}} => 1
[1,1,0,1,0,1,0,0,1,1,1,0,0,0] => [2,3,4,1,7,6,5] => {{1,2,3,4},{5,7},{6}} => 2
[1,1,0,1,0,1,0,1,0,0,1,0,1,0] => [2,3,4,5,1,6,7] => {{1,2,3,4,5},{6},{7}} => 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0] => [2,3,4,5,1,7,6] => {{1,2,3,4,5},{6,7}} => 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,6,1,7] => {{1,2,3,4,5,6},{7}} => 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,7,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0] => [2,3,4,5,7,6,1] => {{1,2,3,4,5,7},{6}} => 2
[1,1,0,1,0,1,0,1,1,0,0,0,1,0] => [2,3,4,6,5,1,7] => {{1,2,3,4,6},{5},{7}} => 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,0] => [2,3,4,6,5,7,1] => {{1,2,3,4,6,7},{5}} => 2
[1,1,0,1,0,1,0,1,1,0,1,0,0,0] => [2,3,4,6,7,5,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,0] => [2,3,4,7,6,5,1] => {{1,2,3,4,7},{5,6}} => 3
[1,1,0,1,0,1,1,0,0,0,1,0,1,0] => [2,3,5,4,1,6,7] => {{1,2,3,5},{4},{6},{7}} => 2
[1,1,0,1,0,1,1,0,0,0,1,1,0,0] => [2,3,5,4,1,7,6] => {{1,2,3,5},{4},{6,7}} => 2
[1,1,0,1,0,1,1,0,0,1,0,0,1,0] => [2,3,5,4,6,1,7] => {{1,2,3,5,6},{4},{7}} => 2
[1,1,0,1,0,1,1,0,0,1,0,1,0,0] => [2,3,5,4,6,7,1] => {{1,2,3,5,6,7},{4}} => 2
[1,1,0,1,0,1,1,0,0,1,1,0,0,0] => [2,3,5,4,7,6,1] => {{1,2,3,5,7},{4},{6}} => 2
[1,1,0,1,0,1,1,0,1,0,0,0,1,0] => [2,3,5,6,4,1,7] => {{1,2,3,4,5,6},{7}} => 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,0] => [2,3,5,6,4,7,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0] => [2,3,5,6,7,4,1] => {{1,2,3,5,7},{4,6}} => 2
[1,1,0,1,0,1,1,0,1,1,0,0,0,0] => [2,3,5,7,6,4,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,0,1,0,1,1,1,0,0,0,0,1,0] => [2,3,6,5,4,1,7] => {{1,2,3,6},{4,5},{7}} => 3
[1,1,0,1,0,1,1,1,0,0,0,1,0,0] => [2,3,6,5,4,7,1] => {{1,2,3,6,7},{4,5}} => 3
[1,1,0,1,0,1,1,1,0,0,1,0,0,0] => [2,3,6,5,7,4,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0] => [2,3,6,7,5,4,1] => {{1,2,3,4,6,7},{5}} => 2
[1,1,0,1,0,1,1,1,1,0,0,0,0,0] => [2,3,7,6,5,4,1] => {{1,2,3,7},{4,6},{5}} => 4
[1,1,0,1,1,0,0,0,1,0,1,0,1,0] => [2,4,3,1,5,6,7] => {{1,2,4},{3},{5},{6},{7}} => 2
[1,1,0,1,1,0,0,0,1,0,1,1,0,0] => [2,4,3,1,5,7,6] => {{1,2,4},{3},{5},{6,7}} => 2
[1,1,0,1,1,0,0,0,1,1,0,0,1,0] => [2,4,3,1,6,5,7] => {{1,2,4},{3},{5,6},{7}} => 2
[1,1,0,1,1,0,0,0,1,1,0,1,0,0] => [2,4,3,1,6,7,5] => {{1,2,4},{3},{5,6,7}} => 2
[1,1,0,1,1,0,0,0,1,1,1,0,0,0] => [2,4,3,1,7,6,5] => {{1,2,4},{3},{5,7},{6}} => 2
[1,1,0,1,1,0,0,1,0,0,1,0,1,0] => [2,4,3,5,1,6,7] => {{1,2,4,5},{3},{6},{7}} => 2
[1,1,0,1,1,0,0,1,0,0,1,1,0,0] => [2,4,3,5,1,7,6] => {{1,2,4,5},{3},{6,7}} => 2
[1,1,0,1,1,0,0,1,0,1,0,0,1,0] => [2,4,3,5,6,1,7] => {{1,2,4,5,6},{3},{7}} => 2
[1,1,0,1,1,0,0,1,0,1,0,1,0,0] => [2,4,3,5,6,7,1] => {{1,2,4,5,6,7},{3}} => 2
[1,1,0,1,1,0,0,1,0,1,1,0,0,0] => [2,4,3,5,7,6,1] => {{1,2,4,5,7},{3},{6}} => 2
[1,1,0,1,1,0,0,1,1,0,0,0,1,0] => [2,4,3,6,5,1,7] => {{1,2,4,6},{3},{5},{7}} => 2
[1,1,0,1,1,0,0,1,1,0,0,1,0,0] => [2,4,3,6,5,7,1] => {{1,2,4,6,7},{3},{5}} => 2
[1,1,0,1,1,0,0,1,1,0,1,0,0,0] => [2,4,3,6,7,5,1] => {{1,2,4,5,6,7},{3}} => 2
[1,1,0,1,1,0,0,1,1,1,0,0,0,0] => [2,4,3,7,6,5,1] => {{1,2,4,7},{3},{5,6}} => 3
[1,1,0,1,1,0,1,0,0,0,1,0,1,0] => [2,4,5,3,1,6,7] => {{1,2,3,4,5},{6},{7}} => 1
[1,1,0,1,1,0,1,0,0,0,1,1,0,0] => [2,4,5,3,1,7,6] => {{1,2,3,4,5},{6,7}} => 1
[1,1,0,1,1,0,1,0,0,1,0,0,1,0] => [2,4,5,3,6,1,7] => {{1,2,3,4,5,6},{7}} => 1
[1,1,0,1,1,0,1,0,0,1,0,1,0,0] => [2,4,5,3,6,7,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,0,1,1,0,1,0,0,1,1,0,0,0] => [2,4,5,3,7,6,1] => {{1,2,3,4,5,7},{6}} => 2
[1,1,0,1,1,0,1,0,1,0,0,0,1,0] => [2,4,5,6,3,1,7] => {{1,2,4,6},{3,5},{7}} => 2
[1,1,0,1,1,0,1,0,1,0,0,1,0,0] => [2,4,5,6,3,7,1] => {{1,2,4,6,7},{3,5}} => 2
[1,1,0,1,1,0,1,0,1,0,1,0,0,0] => [2,4,5,6,7,3,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,0,1,1,0,1,0,1,1,0,0,0,0] => [2,4,5,7,6,3,1] => {{1,2,4,7},{3,5,6}} => 3
[1,1,0,1,1,0,1,1,0,0,0,0,1,0] => [2,4,6,5,3,1,7] => {{1,2,3,4,5,6},{7}} => 1
[1,1,0,1,1,0,1,1,0,0,0,1,0,0] => [2,4,6,5,3,7,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,0] => [2,4,6,5,7,3,1] => {{1,2,4,5,7},{3,6}} => 3
[1,1,0,1,1,0,1,1,0,1,0,0,0,0] => [2,4,6,7,5,3,1] => {{1,2,4,7},{3,6},{5}} => 3
[1,1,0,1,1,0,1,1,1,0,0,0,0,0] => [2,4,7,6,5,3,1] => {{1,2,3,4,6,7},{5}} => 2
[1,1,0,1,1,1,0,0,0,0,1,0,1,0] => [2,5,4,3,1,6,7] => {{1,2,5},{3,4},{6},{7}} => 3
[1,1,0,1,1,1,0,0,0,0,1,1,0,0] => [2,5,4,3,1,7,6] => {{1,2,5},{3,4},{6,7}} => 3
[1,1,0,1,1,1,0,0,0,1,0,0,1,0] => [2,5,4,3,6,1,7] => {{1,2,5,6},{3,4},{7}} => 3
[1,1,0,1,1,1,0,0,0,1,0,1,0,0] => [2,5,4,3,6,7,1] => {{1,2,5,6,7},{3,4}} => 3
[1,1,0,1,1,1,0,0,0,1,1,0,0,0] => [2,5,4,3,7,6,1] => {{1,2,5,7},{3,4},{6}} => 3
[1,1,0,1,1,1,0,0,1,0,0,0,1,0] => [2,5,4,6,3,1,7] => {{1,2,3,4,5,6},{7}} => 1
[1,1,0,1,1,1,0,0,1,0,0,1,0,0] => [2,5,4,6,3,7,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,0,1,1,1,0,0,1,0,1,0,0,0] => [2,5,4,6,7,3,1] => {{1,2,5,7},{3,4,6}} => 3
[1,1,0,1,1,1,0,0,1,1,0,0,0,0] => [2,5,4,7,6,3,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,0,1,1,1,0,1,0,0,0,0,1,0] => [2,5,6,4,3,1,7] => {{1,2,3,5,6},{4},{7}} => 2
[1,1,0,1,1,1,0,1,0,0,0,1,0,0] => [2,5,6,4,3,7,1] => {{1,2,3,5,6,7},{4}} => 2
[1,1,0,1,1,1,0,1,0,0,1,0,0,0] => [2,5,6,4,7,3,1] => {{1,2,5,7},{3,6},{4}} => 3
[1,1,0,1,1,1,0,1,0,1,0,0,0,0] => [2,5,6,7,4,3,1] => {{1,2,4,5,7},{3,6}} => 3
[1,1,0,1,1,1,0,1,1,0,0,0,0,0] => [2,5,7,6,4,3,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,0,1,1,1,1,0,0,0,0,0,1,0] => [2,6,5,4,3,1,7] => {{1,2,6},{3,5},{4},{7}} => 4
[1,1,0,1,1,1,1,0,0,0,0,1,0,0] => [2,6,5,4,3,7,1] => {{1,2,6,7},{3,5},{4}} => 4
[1,1,0,1,1,1,1,0,0,0,1,0,0,0] => [2,6,5,4,7,3,1] => {{1,2,3,5,6,7},{4}} => 2
[1,1,0,1,1,1,1,0,0,1,0,0,0,0] => [2,6,5,7,4,3,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0] => [2,6,7,5,4,3,1] => {{1,2,3,6,7},{4,5}} => 3
[1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [2,7,6,5,4,3,1] => {{1,2,7},{3,6},{4,5}} => 5
[1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [3,2,1,4,5,6,7] => {{1,3},{2},{4},{5},{6},{7}} => 2
[1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [3,2,1,4,5,7,6] => {{1,3},{2},{4},{5},{6,7}} => 2
[1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [3,2,1,4,6,5,7] => {{1,3},{2},{4},{5,6},{7}} => 2
[1,1,1,0,0,0,1,0,1,1,0,1,0,0] => [3,2,1,4,6,7,5] => {{1,3},{2},{4},{5,6,7}} => 2
[1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [3,2,1,4,7,6,5] => {{1,3},{2},{4},{5,7},{6}} => 2
[1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [3,2,1,5,4,6,7] => {{1,3},{2},{4,5},{6},{7}} => 2
[1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [3,2,1,5,4,7,6] => {{1,3},{2},{4,5},{6,7}} => 2
[1,1,1,0,0,0,1,1,0,1,0,0,1,0] => [3,2,1,5,6,4,7] => {{1,3},{2},{4,5,6},{7}} => 2
[1,1,1,0,0,0,1,1,0,1,0,1,0,0] => [3,2,1,5,6,7,4] => {{1,3},{2},{4,5,6,7}} => 2
[1,1,1,0,0,0,1,1,0,1,1,0,0,0] => [3,2,1,5,7,6,4] => {{1,3},{2},{4,5,7},{6}} => 2
[1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [3,2,1,6,5,4,7] => {{1,3},{2},{4,6},{5},{7}} => 2
[1,1,1,0,0,0,1,1,1,0,0,1,0,0] => [3,2,1,6,5,7,4] => {{1,3},{2},{4,6,7},{5}} => 2
[1,1,1,0,0,0,1,1,1,0,1,0,0,0] => [3,2,1,6,7,5,4] => {{1,3},{2},{4,5,6,7}} => 2
[1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [3,2,1,7,6,5,4] => {{1,3},{2},{4,7},{5,6}} => 3
[1,1,1,0,0,1,0,0,1,0,1,0,1,0] => [3,2,4,1,5,6,7] => {{1,3,4},{2},{5},{6},{7}} => 2
[1,1,1,0,0,1,0,0,1,0,1,1,0,0] => [3,2,4,1,5,7,6] => {{1,3,4},{2},{5},{6,7}} => 2
[1,1,1,0,0,1,0,0,1,1,0,0,1,0] => [3,2,4,1,6,5,7] => {{1,3,4},{2},{5,6},{7}} => 2
[1,1,1,0,0,1,0,0,1,1,0,1,0,0] => [3,2,4,1,6,7,5] => {{1,3,4},{2},{5,6,7}} => 2
[1,1,1,0,0,1,0,0,1,1,1,0,0,0] => [3,2,4,1,7,6,5] => {{1,3,4},{2},{5,7},{6}} => 2
[1,1,1,0,0,1,0,1,0,0,1,0,1,0] => [3,2,4,5,1,6,7] => {{1,3,4,5},{2},{6},{7}} => 2
[1,1,1,0,0,1,0,1,0,0,1,1,0,0] => [3,2,4,5,1,7,6] => {{1,3,4,5},{2},{6,7}} => 2
[1,1,1,0,0,1,0,1,0,1,0,0,1,0] => [3,2,4,5,6,1,7] => {{1,3,4,5,6},{2},{7}} => 2
[1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [3,2,4,5,6,7,1] => {{1,3,4,5,6,7},{2}} => 2
[1,1,1,0,0,1,0,1,0,1,1,0,0,0] => [3,2,4,5,7,6,1] => {{1,3,4,5,7},{2},{6}} => 2
[1,1,1,0,0,1,0,1,1,0,0,0,1,0] => [3,2,4,6,5,1,7] => {{1,3,4,6},{2},{5},{7}} => 2
[1,1,1,0,0,1,0,1,1,0,0,1,0,0] => [3,2,4,6,5,7,1] => {{1,3,4,6,7},{2},{5}} => 2
[1,1,1,0,0,1,0,1,1,0,1,0,0,0] => [3,2,4,6,7,5,1] => {{1,3,4,5,6,7},{2}} => 2
[1,1,1,0,0,1,0,1,1,1,0,0,0,0] => [3,2,4,7,6,5,1] => {{1,3,4,7},{2},{5,6}} => 3
[1,1,1,0,0,1,1,0,0,0,1,0,1,0] => [3,2,5,4,1,6,7] => {{1,3,5},{2},{4},{6},{7}} => 2
[1,1,1,0,0,1,1,0,0,0,1,1,0,0] => [3,2,5,4,1,7,6] => {{1,3,5},{2},{4},{6,7}} => 2
[1,1,1,0,0,1,1,0,0,1,0,0,1,0] => [3,2,5,4,6,1,7] => {{1,3,5,6},{2},{4},{7}} => 2
[1,1,1,0,0,1,1,0,0,1,0,1,0,0] => [3,2,5,4,6,7,1] => {{1,3,5,6,7},{2},{4}} => 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,0] => [3,2,5,4,7,6,1] => {{1,3,5,7},{2},{4},{6}} => 2
[1,1,1,0,0,1,1,0,1,0,0,0,1,0] => [3,2,5,6,4,1,7] => {{1,3,4,5,6},{2},{7}} => 2
[1,1,1,0,0,1,1,0,1,0,0,1,0,0] => [3,2,5,6,4,7,1] => {{1,3,4,5,6,7},{2}} => 2
[1,1,1,0,0,1,1,0,1,0,1,0,0,0] => [3,2,5,6,7,4,1] => {{1,3,5,7},{2},{4,6}} => 2
[1,1,1,0,0,1,1,0,1,1,0,0,0,0] => [3,2,5,7,6,4,1] => {{1,3,4,5,6,7},{2}} => 2
[1,1,1,0,0,1,1,1,0,0,0,0,1,0] => [3,2,6,5,4,1,7] => {{1,3,6},{2},{4,5},{7}} => 3
[1,1,1,0,0,1,1,1,0,0,0,1,0,0] => [3,2,6,5,4,7,1] => {{1,3,6,7},{2},{4,5}} => 3
[1,1,1,0,0,1,1,1,0,0,1,0,0,0] => [3,2,6,5,7,4,1] => {{1,3,4,5,6,7},{2}} => 2
[1,1,1,0,0,1,1,1,0,1,0,0,0,0] => [3,2,6,7,5,4,1] => {{1,3,4,6,7},{2},{5}} => 2
[1,1,1,0,0,1,1,1,1,0,0,0,0,0] => [3,2,7,6,5,4,1] => {{1,3,7},{2},{4,6},{5}} => 4
[1,1,1,0,1,0,0,0,1,0,1,0,1,0] => [3,4,2,1,5,6,7] => {{1,2,3,4},{5},{6},{7}} => 1
[1,1,1,0,1,0,0,0,1,0,1,1,0,0] => [3,4,2,1,5,7,6] => {{1,2,3,4},{5},{6,7}} => 1
[1,1,1,0,1,0,0,0,1,1,0,0,1,0] => [3,4,2,1,6,5,7] => {{1,2,3,4},{5,6},{7}} => 1
[1,1,1,0,1,0,0,0,1,1,0,1,0,0] => [3,4,2,1,6,7,5] => {{1,2,3,4},{5,6,7}} => 1
[1,1,1,0,1,0,0,0,1,1,1,0,0,0] => [3,4,2,1,7,6,5] => {{1,2,3,4},{5,7},{6}} => 2
[1,1,1,0,1,0,0,1,0,0,1,0,1,0] => [3,4,2,5,1,6,7] => {{1,2,3,4,5},{6},{7}} => 1
[1,1,1,0,1,0,0,1,0,0,1,1,0,0] => [3,4,2,5,1,7,6] => {{1,2,3,4,5},{6,7}} => 1
[1,1,1,0,1,0,0,1,0,1,0,0,1,0] => [3,4,2,5,6,1,7] => {{1,2,3,4,5,6},{7}} => 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,0] => [3,4,2,5,6,7,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,1,0,1,0,0,1,0,1,1,0,0,0] => [3,4,2,5,7,6,1] => {{1,2,3,4,5,7},{6}} => 2
[1,1,1,0,1,0,0,1,1,0,0,0,1,0] => [3,4,2,6,5,1,7] => {{1,2,3,4,6},{5},{7}} => 2
[1,1,1,0,1,0,0,1,1,0,0,1,0,0] => [3,4,2,6,5,7,1] => {{1,2,3,4,6,7},{5}} => 2
[1,1,1,0,1,0,0,1,1,0,1,0,0,0] => [3,4,2,6,7,5,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,1,0,1,0,0,1,1,1,0,0,0,0] => [3,4,2,7,6,5,1] => {{1,2,3,4,7},{5,6}} => 3
[1,1,1,0,1,0,1,0,0,0,1,0,1,0] => [3,4,5,2,1,6,7] => {{1,3,5},{2,4},{6},{7}} => 2
[1,1,1,0,1,0,1,0,0,0,1,1,0,0] => [3,4,5,2,1,7,6] => {{1,3,5},{2,4},{6,7}} => 2
[1,1,1,0,1,0,1,0,0,1,0,0,1,0] => [3,4,5,2,6,1,7] => {{1,3,5,6},{2,4},{7}} => 2
[1,1,1,0,1,0,1,0,0,1,0,1,0,0] => [3,4,5,2,6,7,1] => {{1,3,5,6,7},{2,4}} => 2
[1,1,1,0,1,0,1,0,0,1,1,0,0,0] => [3,4,5,2,7,6,1] => {{1,3,5,7},{2,4},{6}} => 2
[1,1,1,0,1,0,1,0,1,0,0,0,1,0] => [3,4,5,6,2,1,7] => {{1,2,3,4,5,6},{7}} => 1
[1,1,1,0,1,0,1,0,1,0,0,1,0,0] => [3,4,5,6,2,7,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [3,4,5,6,7,2,1] => {{1,3,5,7},{2,4,6}} => 2
[1,1,1,0,1,0,1,0,1,1,0,0,0,0] => [3,4,5,7,6,2,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,1,0,1,0,1,1,0,0,0,0,1,0] => [3,4,6,5,2,1,7] => {{1,3,6},{2,4,5},{7}} => 3
[1,1,1,0,1,0,1,1,0,0,0,1,0,0] => [3,4,6,5,2,7,1] => {{1,3,6,7},{2,4,5}} => 3
[1,1,1,0,1,0,1,1,0,0,1,0,0,0] => [3,4,6,5,7,2,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,1,0,1,0,1,1,0,1,0,0,0,0] => [3,4,6,7,5,2,1] => {{1,2,3,4,6,7},{5}} => 2
[1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [3,4,7,6,5,2,1] => {{1,3,7},{2,4,6},{5}} => 4
[1,1,1,0,1,1,0,0,0,0,1,0,1,0] => [3,5,4,2,1,6,7] => {{1,2,3,4,5},{6},{7}} => 1
[1,1,1,0,1,1,0,0,0,0,1,1,0,0] => [3,5,4,2,1,7,6] => {{1,2,3,4,5},{6,7}} => 1
[1,1,1,0,1,1,0,0,0,1,0,0,1,0] => [3,5,4,2,6,1,7] => {{1,2,3,4,5,6},{7}} => 1
[1,1,1,0,1,1,0,0,0,1,0,1,0,0] => [3,5,4,2,6,7,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,1,0,1,1,0,0,0,1,1,0,0,0] => [3,5,4,2,7,6,1] => {{1,2,3,4,5,7},{6}} => 2
[1,1,1,0,1,1,0,0,1,0,0,0,1,0] => [3,5,4,6,2,1,7] => {{1,3,4,6},{2,5},{7}} => 3
[1,1,1,0,1,1,0,0,1,0,0,1,0,0] => [3,5,4,6,2,7,1] => {{1,3,4,6,7},{2,5}} => 3
[1,1,1,0,1,1,0,0,1,0,1,0,0,0] => [3,5,4,6,7,2,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,1,0,1,1,0,0,1,1,0,0,0,0] => [3,5,4,7,6,2,1] => {{1,3,4,7},{2,5,6}} => 3
[1,1,1,0,1,1,0,1,0,0,0,0,1,0] => [3,5,6,4,2,1,7] => {{1,3,6},{2,5},{4},{7}} => 3
[1,1,1,0,1,1,0,1,0,0,0,1,0,0] => [3,5,6,4,2,7,1] => {{1,3,6,7},{2,5},{4}} => 3
[1,1,1,0,1,1,0,1,0,0,1,0,0,0] => [3,5,6,4,7,2,1] => {{1,2,3,5,6,7},{4}} => 2
[1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [3,5,6,7,4,2,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,1,0,1,1,0,1,1,0,0,0,0,0] => [3,5,7,6,4,2,1] => {{1,3,7},{2,4,5,6}} => 4
[1,1,1,0,1,1,1,0,0,0,0,0,1,0] => [3,6,5,4,2,1,7] => {{1,2,3,5,6},{4},{7}} => 2
[1,1,1,0,1,1,1,0,0,0,0,1,0,0] => [3,6,5,4,2,7,1] => {{1,2,3,5,6,7},{4}} => 2
[1,1,1,0,1,1,1,0,0,0,1,0,0,0] => [3,6,5,4,7,2,1] => {{1,3,5,7},{2,6},{4}} => 4
[1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [3,6,5,7,4,2,1] => {{1,3,4,5,7},{2,6}} => 4
[1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [3,6,7,5,4,2,1] => {{1,3,7},{2,6},{4,5}} => 4
[1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [3,7,6,5,4,2,1] => {{1,2,3,6,7},{4,5}} => 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [4,3,2,1,5,6,7] => {{1,4},{2,3},{5},{6},{7}} => 3
[1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [4,3,2,1,5,7,6] => {{1,4},{2,3},{5},{6,7}} => 3
[1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [4,3,2,1,6,5,7] => {{1,4},{2,3},{5,6},{7}} => 3
[1,1,1,1,0,0,0,0,1,1,0,1,0,0] => [4,3,2,1,6,7,5] => {{1,4},{2,3},{5,6,7}} => 3
[1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [4,3,2,1,7,6,5] => {{1,4},{2,3},{5,7},{6}} => 3
[1,1,1,1,0,0,0,1,0,0,1,0,1,0] => [4,3,2,5,1,6,7] => {{1,4,5},{2,3},{6},{7}} => 3
[1,1,1,1,0,0,0,1,0,0,1,1,0,0] => [4,3,2,5,1,7,6] => {{1,4,5},{2,3},{6,7}} => 3
[1,1,1,1,0,0,0,1,0,1,0,0,1,0] => [4,3,2,5,6,1,7] => {{1,4,5,6},{2,3},{7}} => 3
[1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [4,3,2,5,6,7,1] => {{1,4,5,6,7},{2,3}} => 3
[1,1,1,1,0,0,0,1,0,1,1,0,0,0] => [4,3,2,5,7,6,1] => {{1,4,5,7},{2,3},{6}} => 3
[1,1,1,1,0,0,0,1,1,0,0,0,1,0] => [4,3,2,6,5,1,7] => {{1,4,6},{2,3},{5},{7}} => 3
[1,1,1,1,0,0,0,1,1,0,0,1,0,0] => [4,3,2,6,5,7,1] => {{1,4,6,7},{2,3},{5}} => 3
[1,1,1,1,0,0,0,1,1,0,1,0,0,0] => [4,3,2,6,7,5,1] => {{1,4,5,6,7},{2,3}} => 3
[1,1,1,1,0,0,0,1,1,1,0,0,0,0] => [4,3,2,7,6,5,1] => {{1,4,7},{2,3},{5,6}} => 3
[1,1,1,1,0,0,1,0,0,0,1,0,1,0] => [4,3,5,2,1,6,7] => {{1,2,3,4,5},{6},{7}} => 1
[1,1,1,1,0,0,1,0,0,0,1,1,0,0] => [4,3,5,2,1,7,6] => {{1,2,3,4,5},{6,7}} => 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,0] => [4,3,5,2,6,1,7] => {{1,2,3,4,5,6},{7}} => 1
[1,1,1,1,0,0,1,0,0,1,0,1,0,0] => [4,3,5,2,6,7,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,0] => [4,3,5,2,7,6,1] => {{1,2,3,4,5,7},{6}} => 2
[1,1,1,1,0,0,1,0,1,0,0,0,1,0] => [4,3,5,6,2,1,7] => {{1,4,6},{2,3,5},{7}} => 3
[1,1,1,1,0,0,1,0,1,0,0,1,0,0] => [4,3,5,6,2,7,1] => {{1,4,6,7},{2,3,5}} => 3
[1,1,1,1,0,0,1,0,1,0,1,0,0,0] => [4,3,5,6,7,2,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,1,1,0,0,1,0,1,1,0,0,0,0] => [4,3,5,7,6,2,1] => {{1,4,7},{2,3,5,6}} => 3
[1,1,1,1,0,0,1,1,0,0,0,0,1,0] => [4,3,6,5,2,1,7] => {{1,2,3,4,5,6},{7}} => 1
[1,1,1,1,0,0,1,1,0,0,0,1,0,0] => [4,3,6,5,2,7,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,1,1,0,0,1,1,0,0,1,0,0,0] => [4,3,6,5,7,2,1] => {{1,4,5,7},{2,3,6}} => 3
[1,1,1,1,0,0,1,1,0,1,0,0,0,0] => [4,3,6,7,5,2,1] => {{1,4,7},{2,3,6},{5}} => 3
[1,1,1,1,0,0,1,1,1,0,0,0,0,0] => [4,3,7,6,5,2,1] => {{1,2,3,4,6,7},{5}} => 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,0] => [4,5,3,2,1,6,7] => {{1,2,4,5},{3},{6},{7}} => 2
[1,1,1,1,0,1,0,0,0,0,1,1,0,0] => [4,5,3,2,1,7,6] => {{1,2,4,5},{3},{6,7}} => 2
[1,1,1,1,0,1,0,0,0,1,0,0,1,0] => [4,5,3,2,6,1,7] => {{1,2,4,5,6},{3},{7}} => 2
[1,1,1,1,0,1,0,0,0,1,0,1,0,0] => [4,5,3,2,6,7,1] => {{1,2,4,5,6,7},{3}} => 2
[1,1,1,1,0,1,0,0,0,1,1,0,0,0] => [4,5,3,2,7,6,1] => {{1,2,4,5,7},{3},{6}} => 2
[1,1,1,1,0,1,0,0,1,0,0,0,1,0] => [4,5,3,6,2,1,7] => {{1,4,6},{2,5},{3},{7}} => 3
[1,1,1,1,0,1,0,0,1,0,0,1,0,0] => [4,5,3,6,2,7,1] => {{1,4,6,7},{2,5},{3}} => 3
[1,1,1,1,0,1,0,0,1,0,1,0,0,0] => [4,5,3,6,7,2,1] => {{1,2,4,5,6,7},{3}} => 2
[1,1,1,1,0,1,0,0,1,1,0,0,0,0] => [4,5,3,7,6,2,1] => {{1,4,7},{2,5,6},{3}} => 3
[1,1,1,1,0,1,0,1,0,0,0,0,1,0] => [4,5,6,3,2,1,7] => {{1,3,4,6},{2,5},{7}} => 3
[1,1,1,1,0,1,0,1,0,0,0,1,0,0] => [4,5,6,3,2,7,1] => {{1,3,4,6,7},{2,5}} => 3
[1,1,1,1,0,1,0,1,0,0,1,0,0,0] => [4,5,6,3,7,2,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [4,5,6,7,3,2,1] => {{1,4,7},{2,3,5,6}} => 3
[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [4,5,7,6,3,2,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [4,6,5,3,2,1,7] => {{1,2,3,4,5,6},{7}} => 1
[1,1,1,1,0,1,1,0,0,0,0,1,0,0] => [4,6,5,3,2,7,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,1,1,0,1,1,0,0,0,1,0,0,0] => [4,6,5,3,7,2,1] => {{1,3,4,5,7},{2,6}} => 4
[1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [4,6,5,7,3,2,1] => {{1,4,7},{2,6},{3,5}} => 4
[1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [4,6,7,5,3,2,1] => {{1,3,4,5,7},{2,6}} => 4
[1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [4,7,6,5,3,2,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [5,4,3,2,1,6,7] => {{1,5},{2,4},{3},{6},{7}} => 4
[1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [5,4,3,2,1,7,6] => {{1,5},{2,4},{3},{6,7}} => 4
[1,1,1,1,1,0,0,0,0,1,0,0,1,0] => [5,4,3,2,6,1,7] => {{1,5,6},{2,4},{3},{7}} => 4
[1,1,1,1,1,0,0,0,0,1,0,1,0,0] => [5,4,3,2,6,7,1] => {{1,5,6,7},{2,4},{3}} => 4
[1,1,1,1,1,0,0,0,0,1,1,0,0,0] => [5,4,3,2,7,6,1] => {{1,5,7},{2,4},{3},{6}} => 4
[1,1,1,1,1,0,0,0,1,0,0,0,1,0] => [5,4,3,6,2,1,7] => {{1,2,4,5,6},{3},{7}} => 2
[1,1,1,1,1,0,0,0,1,0,0,1,0,0] => [5,4,3,6,2,7,1] => {{1,2,4,5,6,7},{3}} => 2
[1,1,1,1,1,0,0,0,1,0,1,0,0,0] => [5,4,3,6,7,2,1] => {{1,5,7},{2,4,6},{3}} => 4
[1,1,1,1,1,0,0,0,1,1,0,0,0,0] => [5,4,3,7,6,2,1] => {{1,2,4,5,6,7},{3}} => 2
[1,1,1,1,1,0,0,1,0,0,0,0,1,0] => [5,4,6,3,2,1,7] => {{1,2,3,4,5,6},{7}} => 1
[1,1,1,1,1,0,0,1,0,0,0,1,0,0] => [5,4,6,3,2,7,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,1,1,1,0,0,1,0,0,1,0,0,0] => [5,4,6,3,7,2,1] => {{1,5,7},{2,3,4,6}} => 4
[1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [5,4,6,7,3,2,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [5,4,7,6,3,2,1] => {{1,3,5,7},{2,4,6}} => 2
[1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [5,6,4,3,2,1,7] => {{1,2,5,6},{3,4},{7}} => 3
[1,1,1,1,1,0,1,0,0,0,0,1,0,0] => [5,6,4,3,2,7,1] => {{1,2,5,6,7},{3,4}} => 3
[1,1,1,1,1,0,1,0,0,0,1,0,0,0] => [5,6,4,3,7,2,1] => {{1,5,7},{2,6},{3,4}} => 4
[1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [5,6,4,7,3,2,1] => {{1,3,4,5,7},{2,6}} => 4
[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [5,6,7,4,3,2,1] => {{1,3,5,7},{2,6},{4}} => 4
[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [5,7,6,4,3,2,1] => {{1,2,3,5,6,7},{4}} => 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [6,5,4,3,2,1,7] => {{1,6},{2,5},{3,4},{7}} => 5
[1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [6,5,4,3,2,7,1] => {{1,6,7},{2,5},{3,4}} => 5
[1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [6,5,4,3,7,2,1] => {{1,2,5,6,7},{3,4}} => 3
[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [6,5,4,7,3,2,1] => {{1,2,3,4,5,6,7}} => 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [6,5,7,4,3,2,1] => {{1,2,3,5,6,7},{4}} => 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [6,7,5,4,3,2,1] => {{1,2,6,7},{3,5},{4}} => 4
[1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [7,6,5,4,3,2,1] => {{1,7},{2,6},{3,5},{4}} => 6
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,2,4,5,6,7,8,3] => {{1},{2},{3,4,5,6,7,8}} => 1
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0] => [1,2,4,5,7,8,6,3] => {{1},{2},{3,4,5,6,7,8}} => 1
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0] => [1,2,4,6,8,7,5,3] => {{1},{2},{3,4,5,6,7,8}} => 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [1,2,5,6,4,7,8,3] => {{1},{2},{3,4,5,6,7,8}} => 1
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [1,2,5,6,7,8,4,3] => {{1},{2},{3,4,5,6,7,8}} => 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [1,2,5,7,6,4,8,3] => {{1},{2},{3,4,5,6,7,8}} => 1
[1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0] => [1,2,6,5,8,7,4,3] => {{1},{2},{3,4,5,6,7,8}} => 1
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [1,2,7,6,8,5,4,3] => {{1},{2},{3,4,5,6,7,8}} => 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0] => [1,3,4,5,6,7,2,8] => {{1},{2,3,4,5,6,7},{8}} => 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,7,8,2] => {{1},{2,3,4,5,6,7,8}} => 1
[1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0] => [1,3,4,5,6,8,7,2] => {{1},{2,3,4,5,6,8},{7}} => 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0] => [1,3,4,5,7,6,8,2] => {{1},{2,3,4,5,7,8},{6}} => 2
[1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0] => [1,3,4,5,7,8,6,2] => {{1},{2,3,4,5,6,7,8}} => 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0] => [1,3,4,6,5,7,8,2] => {{1},{2,3,4,6,7,8},{5}} => 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0] => [1,3,4,6,7,5,2,8] => {{1},{2,3,4,5,6,7},{8}} => 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0] => [1,3,4,6,7,5,8,2] => {{1},{2,3,4,5,6,7,8}} => 1
[1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0] => [1,3,4,7,6,8,5,2] => {{1},{2,3,4,5,6,7,8}} => 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0] => [1,3,5,6,4,7,2,8] => {{1},{2,3,4,5,6,7},{8}} => 1
[1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0] => [1,3,5,6,7,8,4,2] => {{1},{2,3,4,5,6,7,8}} => 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0] => [1,3,5,7,6,4,2,8] => {{1},{2,3,4,5,6,7},{8}} => 1
[1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0] => [1,3,5,8,7,6,4,2] => {{1},{2,3,4,5,7,8},{6}} => 2
[1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0] => [1,3,6,5,7,4,2,8] => {{1},{2,3,4,5,6,7},{8}} => 1
[1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0] => [1,3,6,8,7,5,4,2] => {{1},{2,3,4,5,6,7,8}} => 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [1,4,3,5,6,7,8,2] => {{1},{2,4,5,6,7,8},{3}} => 2
[1,0,1,1,1,0,0,1,0,1,1,0,1,0,0,0] => [1,4,3,5,7,8,6,2] => {{1},{2,4,5,6,7,8},{3}} => 2
[1,0,1,1,1,0,0,1,1,0,1,1,0,0,0,0] => [1,4,3,6,8,7,5,2] => {{1},{2,4,5,6,7,8},{3}} => 2
[1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0] => [1,4,3,7,6,8,5,2] => {{1},{2,4,5,6,7,8},{3}} => 2
[1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0] => [1,4,5,3,6,7,2,8] => {{1},{2,3,4,5,6,7},{8}} => 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0] => [1,4,5,3,6,7,8,2] => {{1},{2,3,4,5,6,7,8}} => 1
[1,0,1,1,1,0,1,0,0,1,0,1,1,0,0,0] => [1,4,5,3,6,8,7,2] => {{1},{2,3,4,5,6,8},{7}} => 2
[1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0] => [1,4,5,3,7,8,6,2] => {{1},{2,3,4,5,6,7,8}} => 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0] => [1,4,5,6,7,3,2,8] => {{1},{2,3,4,5,6,7},{8}} => 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0] => [1,4,5,6,7,3,8,2] => {{1},{2,3,4,5,6,7,8}} => 1
[1,0,1,1,1,0,1,1,0,0,0,1,1,0,0,0] => [1,4,6,5,3,8,7,2] => {{1},{2,3,4,5,6,8},{7}} => 2
[1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0] => [1,4,6,7,5,8,3,2] => {{1},{2,3,4,6,7,8},{5}} => 2
[1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [1,4,6,7,8,5,3,2] => {{1},{2,3,4,5,6,7,8}} => 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0] => [1,5,4,6,3,7,2,8] => {{1},{2,3,4,5,6,7},{8}} => 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,1,0,0] => [1,5,4,6,3,7,8,2] => {{1},{2,3,4,5,6,7,8}} => 1
[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0] => [1,5,4,6,7,8,3,2] => {{1},{2,3,4,5,6,7,8}} => 1
[1,0,1,1,1,1,0,1,0,0,1,0,1,0,0,0] => [1,5,6,4,7,8,3,2] => {{1},{2,3,5,6,7,8},{4}} => 2
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [1,5,8,7,6,4,3,2] => {{1},{2,3,4,5,6,7,8}} => 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0,1,0] => [1,6,5,7,4,3,2,8] => {{1},{2,3,4,5,6,7},{8}} => 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0] => [1,6,5,7,4,3,8,2] => {{1},{2,3,4,5,6,7,8}} => 1
[1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [1,6,5,7,8,4,3,2] => {{1},{2,3,4,5,6,7,8}} => 1
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,6,8,7,5,4,3,2] => {{1},{2,3,4,6,7,8},{5}} => 2
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [1,7,6,5,8,4,3,2] => {{1},{2,3,4,5,6,7,8}} => 1
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [1,7,6,8,5,4,3,2] => {{1},{2,3,4,6,7,8},{5}} => 2
[1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0] => [2,1,4,5,6,7,8,3] => {{1,2},{3,4,5,6,7,8}} => 1
[1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0] => [2,1,4,5,7,8,6,3] => {{1,2},{3,4,5,6,7,8}} => 1
[1,1,0,0,1,1,0,1,1,0,1,0,0,1,0,0] => [2,1,4,6,7,5,8,3] => {{1,2},{3,4,5,6,7,8}} => 1
[1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0] => [2,1,4,6,8,7,5,3] => {{1,2},{3,4,5,6,7,8}} => 1
[1,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0] => [2,1,5,6,4,7,8,3] => {{1,2},{3,4,5,6,7,8}} => 1
[1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0] => [2,1,5,6,7,8,4,3] => {{1,2},{3,4,5,6,7,8}} => 1
[1,1,0,0,1,1,1,0,1,1,0,0,0,1,0,0] => [2,1,5,7,6,4,8,3] => {{1,2},{3,4,5,6,7,8}} => 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0] => [2,1,6,5,7,4,8,3] => {{1,2},{3,4,5,6,7,8}} => 1
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0] => [2,1,6,5,8,7,4,3] => {{1,2},{3,4,5,6,7,8}} => 1
[1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0] => [2,1,6,8,7,5,4,3] => {{1,2},{3,4,5,6,7,8}} => 1
[1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0] => [2,1,7,6,8,5,4,3] => {{1,2},{3,4,5,6,7,8}} => 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0] => [2,3,4,5,6,1,8,7] => {{1,2,3,4,5,6},{7,8}} => 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,6,7,1,8] => {{1,2,3,4,5,6,7},{8}} => 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,7,8,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0] => [2,3,4,5,6,8,7,1] => {{1,2,3,4,5,6,8},{7}} => 2
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0] => [2,3,4,5,7,6,8,1] => {{1,2,3,4,5,7,8},{6}} => 2
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0] => [2,3,4,5,7,8,6,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0] => [2,3,4,5,8,7,6,1] => {{1,2,3,4,5,8},{6,7}} => 3
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0] => [2,3,4,6,5,7,8,1] => {{1,2,3,4,6,7,8},{5}} => 2
[1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0] => [2,3,4,6,7,5,1,8] => {{1,2,3,4,5,6,7},{8}} => 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0] => [2,3,4,6,7,5,8,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0] => [2,3,4,6,7,8,5,1] => {{1,2,3,4,6,8},{5,7}} => 2
[1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0] => [2,3,4,6,8,7,5,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0] => [2,3,4,7,6,5,8,1] => {{1,2,3,4,7,8},{5,6}} => 3
[1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0] => [2,3,4,7,6,8,5,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0] => [2,3,4,7,8,6,5,1] => {{1,2,3,4,5,7,8},{6}} => 2
[1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0] => [2,3,5,4,6,7,8,1] => {{1,2,3,5,6,7,8},{4}} => 2
[1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0] => [2,3,5,4,7,8,6,1] => {{1,2,3,5,6,7,8},{4}} => 2
[1,1,0,1,0,1,1,0,1,0,0,0,1,1,0,0] => [2,3,5,6,4,1,8,7] => {{1,2,3,4,5,6},{7,8}} => 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0] => [2,3,5,6,4,7,8,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0] => [2,3,5,6,4,8,7,1] => {{1,2,3,4,5,6,8},{7}} => 2
[1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0] => [2,3,5,6,7,4,8,1] => {{1,2,3,5,7,8},{4,6}} => 2
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0] => [2,3,5,6,7,8,4,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,0,1,0] => [2,3,5,7,6,4,1,8] => {{1,2,3,4,5,6,7},{8}} => 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0] => [2,3,5,7,6,4,8,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0] => [2,3,5,7,6,8,4,1] => {{1,2,3,5,6,8},{4,7}} => 3
[1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0] => [2,3,5,8,7,6,4,1] => {{1,2,3,4,5,7,8},{6}} => 2
[1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0] => [2,3,6,5,4,7,8,1] => {{1,2,3,6,7,8},{4,5}} => 3
[1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0] => [2,3,6,5,7,4,1,8] => {{1,2,3,4,5,6,7},{8}} => 1
[1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0] => [2,3,6,5,8,7,4,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0] => [2,3,6,7,8,5,4,1] => {{1,2,3,5,6,8},{4,7}} => 3
[1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0] => [2,3,7,6,5,8,4,1] => {{1,2,3,4,6,7,8},{5}} => 2
[1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0] => [2,4,3,5,6,7,8,1] => {{1,2,4,5,6,7,8},{3}} => 2
[1,1,0,1,1,0,0,1,1,0,1,0,0,1,0,0] => [2,4,3,6,7,5,8,1] => {{1,2,4,5,6,7,8},{3}} => 2
[1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0] => [2,4,3,6,8,7,5,1] => {{1,2,4,5,6,7,8},{3}} => 2
[1,1,0,1,1,0,1,0,0,1,0,0,1,1,0,0] => [2,4,5,3,6,1,8,7] => {{1,2,3,4,5,6},{7,8}} => 1
[1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0] => [2,4,5,3,6,7,8,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,0,1,1,0,1,0,0,1,0,1,1,0,0,0] => [2,4,5,3,6,8,7,1] => {{1,2,3,4,5,6,8},{7}} => 2
[1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0] => [2,4,5,3,7,8,6,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0] => [2,4,5,3,8,7,6,1] => {{1,2,3,4,5,8},{6,7}} => 3
[1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0] => [2,4,5,6,3,7,8,1] => {{1,2,4,6,7,8},{3,5}} => 2
[1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0] => [2,4,5,6,7,3,1,8] => {{1,2,3,4,5,6,7},{8}} => 1
[1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0] => [2,4,5,6,7,3,8,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0] => [2,4,5,6,8,7,3,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0] => [2,4,5,7,8,6,3,1] => {{1,2,3,4,5,7,8},{6}} => 2
[1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0] => [2,4,6,5,3,1,8,7] => {{1,2,3,4,5,6},{7,8}} => 1
[1,1,0,1,1,0,1,1,0,0,0,1,0,1,0,0] => [2,4,6,5,3,7,8,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0] => [2,4,6,5,3,8,7,1] => {{1,2,3,4,5,6,8},{7}} => 2
[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0] => [2,4,6,5,7,3,8,1] => {{1,2,4,5,7,8},{3,6}} => 3
[1,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0] => [2,4,6,5,7,8,3,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0] => [2,4,6,7,5,8,3,1] => {{1,2,3,4,6,7,8},{5}} => 2
[1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0] => [2,4,6,7,8,5,3,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0] => [2,4,8,7,6,5,3,1] => {{1,2,3,4,7,8},{5,6}} => 3
[1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0] => [2,5,4,3,6,7,8,1] => {{1,2,5,6,7,8},{3,4}} => 3
[1,1,0,1,1,1,0,0,0,1,1,0,1,0,0,0] => [2,5,4,3,7,8,6,1] => {{1,2,5,6,7,8},{3,4}} => 3
[1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0] => [2,5,4,6,3,7,1,8] => {{1,2,3,4,5,6,7},{8}} => 1
[1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0] => [2,5,4,6,7,8,3,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,0,1,1,1,0,0,1,1,0,0,0,0,1,0] => [2,5,4,7,6,3,1,8] => {{1,2,3,4,5,6,7},{8}} => 1
[1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0] => [2,5,4,7,6,3,8,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0] => [2,5,4,8,7,6,3,1] => {{1,2,3,4,5,7,8},{6}} => 2
[1,1,0,1,1,1,0,1,0,0,0,1,0,1,0,0] => [2,5,6,4,3,7,8,1] => {{1,2,3,5,6,7,8},{4}} => 2
[1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0] => [2,5,6,4,7,8,3,1] => {{1,2,3,5,6,7,8},{4}} => 2
[1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0] => [2,5,6,7,4,3,8,1] => {{1,2,4,5,7,8},{3,6}} => 3
[1,1,0,1,1,1,0,1,0,1,0,0,1,0,0,0] => [2,5,6,7,4,8,3,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0] => [2,5,7,6,4,3,1,8] => {{1,2,3,4,5,6,7},{8}} => 1
[1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0] => [2,5,8,7,6,4,3,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0] => [2,6,5,4,8,7,3,1] => {{1,2,3,5,6,7,8},{4}} => 2
[1,1,0,1,1,1,1,0,0,1,0,0,0,1,0,0] => [2,6,5,7,4,3,8,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,0,1,1,1,1,0,1,0,0,1,0,0,0,0] => [2,6,7,5,8,4,3,1] => {{1,2,4,5,6,8},{3,7}} => 4
[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0] => [2,6,8,7,5,4,3,1] => {{1,2,3,4,6,7,8},{5}} => 2
[1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0] => [2,7,6,5,4,8,3,1] => {{1,2,3,6,7,8},{4,5}} => 3
[1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0] => [2,7,6,8,5,4,3,1] => {{1,2,3,4,6,7,8},{5}} => 2
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0] => [3,2,4,5,6,7,8,1] => {{1,3,4,5,6,7,8},{2}} => 2
[1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0] => [3,2,4,5,7,8,6,1] => {{1,3,4,5,6,7,8},{2}} => 2
[1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0] => [3,2,5,4,6,7,8,1] => {{1,3,5,6,7,8},{2},{4}} => 2
[1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0] => [3,2,5,4,7,8,6,1] => {{1,3,5,6,7,8},{2},{4}} => 2
[1,1,1,0,0,1,1,0,1,0,0,1,0,1,0,0] => [3,2,5,6,4,7,8,1] => {{1,3,4,5,6,7,8},{2}} => 2
[1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0] => [3,2,5,6,7,8,4,1] => {{1,3,4,5,6,7,8},{2}} => 2
[1,1,1,0,0,1,1,0,1,1,0,0,0,1,0,0] => [3,2,5,7,6,4,8,1] => {{1,3,4,5,6,7,8},{2}} => 2
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0] => [3,2,6,5,7,4,8,1] => {{1,3,4,5,6,7,8},{2}} => 2
[1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0] => [3,2,6,8,7,5,4,1] => {{1,3,4,5,6,7,8},{2}} => 2
[1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0] => [3,4,2,5,6,1,8,7] => {{1,2,3,4,5,6},{7,8}} => 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0] => [3,4,2,5,6,7,1,8] => {{1,2,3,4,5,6,7},{8}} => 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0] => [3,4,2,5,6,7,8,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,0,1,0,0,1,0,1,0,1,1,0,0,0] => [3,4,2,5,6,8,7,1] => {{1,2,3,4,5,6,8},{7}} => 2
[1,1,1,0,1,0,0,1,0,1,1,0,0,1,0,0] => [3,4,2,5,7,6,8,1] => {{1,2,3,4,5,7,8},{6}} => 2
[1,1,1,0,1,0,0,1,0,1,1,0,1,0,0,0] => [3,4,2,5,7,8,6,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,0,1,0,0,1,0,1,1,1,0,0,0,0] => [3,4,2,5,8,7,6,1] => {{1,2,3,4,5,8},{6,7}} => 3
[1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0] => [3,4,2,6,5,7,8,1] => {{1,2,3,4,6,7,8},{5}} => 2
[1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0] => [3,4,2,6,7,5,1,8] => {{1,2,3,4,5,6,7},{8}} => 1
[1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0] => [3,4,2,6,7,5,8,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,0,1,0,0,1,1,1,0,0,0,1,0,0] => [3,4,2,7,6,5,8,1] => {{1,2,3,4,7,8},{5,6}} => 3
[1,1,1,0,1,0,0,1,1,1,0,1,0,0,0,0] => [3,4,2,7,8,6,5,1] => {{1,2,3,4,5,7,8},{6}} => 2
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0] => [3,4,5,2,6,7,8,1] => {{1,3,5,6,7,8},{2,4}} => 2
[1,1,1,0,1,0,1,0,0,1,1,0,1,0,0,0] => [3,4,5,2,7,8,6,1] => {{1,3,5,6,7,8},{2,4}} => 2
[1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0] => [3,4,5,6,2,1,8,7] => {{1,2,3,4,5,6},{7,8}} => 1
[1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0] => [3,4,5,6,2,7,1,8] => {{1,2,3,4,5,6,7},{8}} => 1
[1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0] => [3,4,5,6,2,7,8,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0] => [3,4,5,6,2,8,7,1] => {{1,2,3,4,5,6,8},{7}} => 2
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0] => [3,4,5,6,7,8,2,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0] => [3,4,5,7,6,2,8,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0] => [3,4,5,8,7,6,2,1] => {{1,2,3,4,5,7,8},{6}} => 2
[1,1,1,0,1,0,1,1,0,0,1,0,0,1,0,0] => [3,4,6,5,7,2,8,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0] => [3,4,6,7,5,2,8,1] => {{1,2,3,4,6,7,8},{5}} => 2
[1,1,1,0,1,0,1,1,1,0,0,0,1,0,0,0] => [3,4,7,6,5,8,2,1] => {{1,2,3,4,6,7,8},{5}} => 2
[1,1,1,0,1,1,0,0,0,1,0,0,1,1,0,0] => [3,5,4,2,6,1,8,7] => {{1,2,3,4,5,6},{7,8}} => 1
[1,1,1,0,1,1,0,0,0,1,0,1,0,0,1,0] => [3,5,4,2,6,7,1,8] => {{1,2,3,4,5,6,7},{8}} => 1
[1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0] => [3,5,4,2,6,7,8,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,0,1,1,0,0,0,1,1,0,0,1,0,0] => [3,5,4,2,7,6,8,1] => {{1,2,3,4,5,7,8},{6}} => 2
[1,1,1,0,1,1,0,0,0,1,1,0,1,0,0,0] => [3,5,4,2,7,8,6,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,0,1,1,0,0,1,0,0,1,0,1,0,0] => [3,5,4,6,2,7,8,1] => {{1,3,4,6,7,8},{2,5}} => 3
[1,1,1,0,1,1,0,0,1,0,1,0,0,1,0,0] => [3,5,4,6,7,2,8,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0] => [3,5,4,7,6,8,2,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,0,1,1,0,0,1,1,0,1,0,0,0,0] => [3,5,4,7,8,6,2,1] => {{1,2,3,4,5,7,8},{6}} => 2
[1,1,1,0,1,1,0,1,0,0,1,0,0,1,0,0] => [3,5,6,4,7,2,8,1] => {{1,2,3,5,6,7,8},{4}} => 2
[1,1,1,0,1,1,0,1,0,1,0,0,0,0,1,0] => [3,5,6,7,4,2,1,8] => {{1,2,3,4,5,6,7},{8}} => 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0] => [3,5,6,7,4,2,8,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0] => [3,5,6,7,8,4,2,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,0,1,1,0,1,1,0,0,0,1,0,0,0] => [3,5,7,6,4,8,2,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,0,1,1,0,1,1,0,0,1,0,0,0,0] => [3,5,7,6,8,4,2,1] => {{1,2,3,5,7,8},{4,6}} => 2
[1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0] => [3,5,7,8,6,4,2,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0] => [3,6,5,4,2,7,8,1] => {{1,2,3,5,6,7,8},{4}} => 2
[1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0] => [3,6,5,4,7,8,2,1] => {{1,2,3,5,6,7,8},{4}} => 2
[1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0] => [3,7,6,5,8,4,2,1] => {{1,3,4,5,6,8},{2,7}} => 5
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0] => [4,3,2,5,6,7,8,1] => {{1,4,5,6,7,8},{2,3}} => 3
[1,1,1,1,0,0,0,1,0,1,1,0,1,0,0,0] => [4,3,2,5,7,8,6,1] => {{1,4,5,6,7,8},{2,3}} => 3
[1,1,1,1,0,0,0,1,1,0,1,0,0,1,0,0] => [4,3,2,6,7,5,8,1] => {{1,4,5,6,7,8},{2,3}} => 3
[1,1,1,1,0,0,0,1,1,0,1,1,0,0,0,0] => [4,3,2,6,8,7,5,1] => {{1,4,5,6,7,8},{2,3}} => 3
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0] => [4,3,5,2,6,1,8,7] => {{1,2,3,4,5,6},{7,8}} => 1
[1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0] => [4,3,5,2,6,7,8,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0] => [4,3,5,2,7,6,8,1] => {{1,2,3,4,5,7,8},{6}} => 2
[1,1,1,1,0,0,1,0,0,1,1,1,0,0,0,0] => [4,3,5,2,8,7,6,1] => {{1,2,3,4,5,8},{6,7}} => 3
[1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0] => [4,3,5,6,7,2,8,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0] => [4,3,5,6,8,7,2,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,1,0,0,1,0,1,1,0,0,1,0,0,0] => [4,3,5,7,6,8,2,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0] => [4,3,5,7,8,6,2,1] => {{1,2,3,4,5,7,8},{6}} => 2
[1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0] => [4,3,6,5,2,1,8,7] => {{1,2,3,4,5,6},{7,8}} => 1
[1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0] => [4,3,6,5,2,8,7,1] => {{1,2,3,4,5,6,8},{7}} => 2
[1,1,1,1,0,0,1,1,0,0,1,0,1,0,0,0] => [4,3,6,5,7,8,2,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0] => [4,3,6,7,8,5,2,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,1,0,0] => [4,3,7,6,5,2,8,1] => {{1,2,3,4,6,7,8},{5}} => 2
[1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0] => [4,3,8,7,6,5,2,1] => {{1,2,3,4,7,8},{5,6}} => 3
[1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0] => [4,5,3,2,6,7,8,1] => {{1,2,4,5,6,7,8},{3}} => 2
[1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0] => [4,5,3,6,7,2,8,1] => {{1,2,4,5,6,7,8},{3}} => 2
[1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0] => [4,5,3,6,8,7,2,1] => {{1,2,4,5,6,7,8},{3}} => 2
[1,1,1,1,0,1,0,0,1,1,0,0,1,0,0,0] => [4,5,3,7,6,8,2,1] => {{1,2,4,5,6,7,8},{3}} => 2
[1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0] => [4,5,6,3,2,7,8,1] => {{1,3,4,6,7,8},{2,5}} => 3
[1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0] => [4,5,6,3,7,2,8,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0] => [4,5,6,3,8,7,2,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0] => [4,5,6,7,3,8,2,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0] => [4,5,6,7,8,3,2,1] => {{1,2,4,5,7,8},{3,6}} => 3
[1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0] => [4,5,7,6,3,2,1,8] => {{1,2,3,4,5,6,7},{8}} => 1
[1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0] => [4,5,7,6,8,3,2,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0] => [4,5,8,7,6,3,2,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0] => [4,6,5,3,2,1,8,7] => {{1,2,3,4,5,6},{7,8}} => 1
[1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0] => [4,6,5,3,2,8,7,1] => {{1,2,3,4,5,6,8},{7}} => 2
[1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0] => [4,6,5,3,7,2,8,1] => {{1,3,4,5,7,8},{2,6}} => 4
[1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0] => [4,6,5,7,8,3,2,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,1,0,0] => [4,6,7,5,3,2,8,1] => {{1,3,4,5,7,8},{2,6}} => 4
[1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0] => [4,6,8,7,5,3,2,1] => {{1,2,3,4,6,7,8},{5}} => 2
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0] => [4,7,6,5,3,2,1,8] => {{1,2,3,4,5,6,7},{8}} => 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0] => [4,7,6,5,3,2,8,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0] => [4,7,6,5,3,8,2,1] => {{1,3,4,5,6,8},{2,7}} => 5
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0] => [4,8,7,6,5,3,2,1] => {{1,2,3,4,6,7,8},{5}} => 2
[1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0] => [5,4,3,6,2,7,8,1] => {{1,2,4,5,6,7,8},{3}} => 2
[1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0] => [5,4,3,6,7,8,2,1] => {{1,2,4,5,6,7,8},{3}} => 2
[1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0] => [5,4,3,7,6,2,8,1] => {{1,2,4,5,6,7,8},{3}} => 2
[1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0] => [5,4,6,3,2,1,8,7] => {{1,2,3,4,5,6},{7,8}} => 1
[1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0] => [5,4,6,3,2,7,1,8] => {{1,2,3,4,5,6,7},{8}} => 1
[1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0] => [5,4,6,3,2,7,8,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0] => [5,4,6,3,2,8,7,1] => {{1,2,3,4,5,6,8},{7}} => 2
[1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0] => [5,4,6,7,3,2,1,8] => {{1,2,3,4,5,6,7},{8}} => 1
[1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0] => [5,4,7,6,3,8,2,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0] => [5,4,7,8,6,3,2,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0] => [5,6,4,8,7,3,2,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0] => [5,6,7,4,3,8,2,1] => {{1,2,3,5,6,7,8},{4}} => 2
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0] => [5,6,8,7,4,3,2,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0] => [5,7,6,4,3,2,8,1] => {{1,2,3,5,6,7,8},{4}} => 2
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0] => [5,7,8,6,4,3,2,1] => {{1,3,4,5,6,8},{2,7}} => 5
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0] => [5,8,7,6,4,3,2,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0] => [6,5,4,3,7,2,8,1] => {{1,2,5,6,7,8},{3,4}} => 3
[1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0] => [6,5,4,3,8,7,2,1] => {{1,2,5,6,7,8},{3,4}} => 3
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0] => [6,5,4,7,3,2,1,8] => {{1,2,3,4,5,6,7},{8}} => 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0] => [6,5,4,7,3,2,8,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0] => [6,5,4,7,8,3,2,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0] => [6,5,7,4,3,2,8,1] => {{1,2,3,5,6,7,8},{4}} => 2
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0] => [6,5,7,4,8,3,2,1] => {{1,2,3,5,6,7,8},{4}} => 2
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0] => [6,5,7,8,4,3,2,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0] => [6,7,5,8,4,3,2,1] => {{1,3,4,5,6,8},{2,7}} => 5
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [6,8,7,5,4,3,2,1] => {{1,2,3,6,7,8},{4,5}} => 3
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [7,6,5,4,8,3,2,1] => {{1,2,3,5,6,7,8},{4}} => 2
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0] => [7,6,5,8,4,3,2,1] => {{1,2,3,4,5,6,7,8}} => 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [7,6,8,5,4,3,2,1] => {{1,2,3,6,7,8},{4,5}} => 3
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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 1,3,1 1,8,4,1 1,21,15,4,1 1,55,50,21,4,1 1,146,166,83,28,4,1
$F_{2} = 1 + q$
$F_{3} = 1 + 3\ q + q^{2}$
$F_{4} = 1 + 8\ q + 4\ q^{2} + q^{3}$
$F_{5} = 1 + 21\ q + 15\ q^{2} + 4\ q^{3} + q^{4}$
$F_{6} = 1 + 55\ q + 50\ q^{2} + 21\ q^{3} + 4\ q^{4} + q^{5}$
$F_{7} = 1 + 146\ q + 166\ q^{2} + 83\ q^{3} + 28\ q^{4} + 4\ q^{5} + q^{6}$
Description
The maximal arc length of a set partition.
The arcs of a set partition are those $i < j$ that are consecutive elements in the blocks. If there are no arcs, the maximal arc length is $0$.
The arcs of a set partition are those $i < j$ that are consecutive elements in the blocks. If there are no arcs, the maximal arc length is $0$.
Map
to 312-avoiding permutation
Description
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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