Identifier
-
Mp00201:
Dyck paths
—Ringel⟶
Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000054: Permutations ⟶ ℤ
Values
[1,0] => [2,1] => {{1,2}} => [2,1] => 2
[1,0,1,0] => [3,1,2] => {{1,2,3}} => [2,3,1] => 2
[1,1,0,0] => [2,3,1] => {{1,2,3}} => [2,3,1] => 2
[1,0,1,0,1,0] => [4,1,2,3] => {{1,2,3,4}} => [2,3,4,1] => 2
[1,0,1,1,0,0] => [3,1,4,2] => {{1,2,3,4}} => [2,3,4,1] => 2
[1,1,0,0,1,0] => [2,4,1,3] => {{1,2,3,4}} => [2,3,4,1] => 2
[1,1,0,1,0,0] => [4,3,1,2] => {{1,2,3,4}} => [2,3,4,1] => 2
[1,1,1,0,0,0] => [2,3,4,1] => {{1,2,3,4}} => [2,3,4,1] => 2
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => {{1,2,3,4,5}} => [2,3,4,5,1] => 2
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => {{1,2,3,4,5}} => [2,3,4,5,1] => 2
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => {{1,2,3,4,5}} => [2,3,4,5,1] => 2
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => {{1,2,3,4,5}} => [2,3,4,5,1] => 2
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => {{1,2,3,4,5}} => [2,3,4,5,1] => 2
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => {{1,2,3,4,5}} => [2,3,4,5,1] => 2
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => {{1,2,3,4,5}} => [2,3,4,5,1] => 2
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => {{1,2,3,4,5}} => [2,3,4,5,1] => 2
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => {{1,3,5},{2,4}} => [3,4,5,2,1] => 3
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => {{1,2,3,4,5}} => [2,3,4,5,1] => 2
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => {{1,2,3,4,5}} => [2,3,4,5,1] => 2
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => {{1,2,3,4,5}} => [2,3,4,5,1] => 2
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => {{1,2,3,4,5}} => [2,3,4,5,1] => 2
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => {{1,2,3,4,5}} => [2,3,4,5,1] => 2
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => {{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => 2
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => {{1,3,5,6},{2,4}} => [3,4,5,2,6,1] => 3
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => {{1,3,5},{2,4,6}} => [3,4,5,6,1,2] => 3
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => {{1,3,5,6},{2,4}} => [3,4,5,2,6,1] => 3
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => {{1,3,6},{2,4,5}} => [3,4,6,5,2,1] => 3
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => {{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => 2
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => {{1,4,6},{2,3,5}} => [4,3,5,6,2,1] => 4
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => {{1,3,4,6},{2,5}} => [3,5,4,6,2,1] => 3
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 2
[1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,0,1,0,1,1,0,0,1,0] => [5,1,2,3,7,4,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [7,1,2,3,6,4,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,0,1,1,0,0,1,0,1,0] => [4,1,2,7,3,5,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [4,1,2,6,3,7,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [7,1,2,5,3,4,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => [6,1,2,5,3,7,4] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [4,1,2,5,7,3,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,0,1,1,1,0,0,1,0,0] => [4,1,2,7,6,3,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [7,1,2,5,6,3,4] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,0,1,1,1,1,0,0,0,0] => [4,1,2,5,6,7,3] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,1,0,0,1,0,1,0,1,0] => [3,1,7,2,4,5,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,1,0,0,1,0,1,1,0,0] => [3,1,6,2,4,7,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [3,1,5,2,7,4,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [3,1,7,2,6,4,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,1,0,0,1,1,1,0,0,0] => [3,1,5,2,6,7,4] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [7,1,4,2,3,5,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [6,1,4,2,3,7,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [5,1,4,2,7,3,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [7,1,4,2,6,3,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,1,0,1,1,1,0,0,0,0] => [5,1,4,2,6,7,3] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,1,1,0,0,0,1,0,1,0] => [3,1,4,7,2,5,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,1,1,0,0,0,1,1,0,0] => [3,1,4,6,2,7,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,1,1,0,0,1,0,0,1,0] => [3,1,7,5,2,4,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [3,1,6,5,2,7,4] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,1,1,0,1,0,0,0,1,0] => [7,1,4,5,2,3,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,1,1,0,1,1,0,0,0,0] => [6,1,4,5,2,7,3] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,1,1,1,0,0,0,0,1,0] => [3,1,4,5,7,2,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,1,1,1,0,0,0,1,0,0] => [3,1,4,7,6,2,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,7,5,6,2,4] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,1,1,1,0,1,0,0,0,0] => [7,1,4,5,6,2,3] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,0,1,1,1,1,1,0,0,0,0,0] => [3,1,4,5,6,7,2] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,0,1,0,1,0,1,1,0,0] => [2,6,1,3,4,7,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,0,1,0,1,1,0,0,1,0] => [2,5,1,3,7,4,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
>>> Load all 778 entries. <<<[1,1,0,0,1,0,1,1,0,1,0,0] => [2,7,1,3,6,4,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,0,1,0,1,1,1,0,0,0] => [2,5,1,3,6,7,4] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,0,1,1,0,0,1,0,1,0] => [2,4,1,7,3,5,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,0,1,1,0,0,1,1,0,0] => [2,4,1,6,3,7,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,0,1,1,0,1,0,0,1,0] => [2,7,1,5,3,4,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,0,1,1,0,1,1,0,0,0] => [2,6,1,5,3,7,4] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,0,1,1,1,0,0,0,1,0] => [2,4,1,5,7,3,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,0,1,1,1,0,0,1,0,0] => [2,4,1,7,6,3,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,0,1,1,1,0,1,0,0,0] => [2,7,1,5,6,3,4] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,4,1,5,6,7,3] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,1,0,0,1,0,1,0,1,0] => [7,3,1,2,4,5,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,1,0,0,1,0,1,1,0,0] => [6,3,1,2,4,7,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,1,0,0,1,1,0,0,1,0] => [5,3,1,2,7,4,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,1,0,0,1,1,0,1,0,0] => [7,3,1,2,6,4,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,1,0,0,1,1,1,0,0,0] => [5,3,1,2,6,7,4] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,1,0,1,0,0,1,0,1,0] => [7,4,1,2,3,5,6] => {{1,3,5,6,7},{2,4}} => [3,4,5,2,6,7,1] => 3
[1,1,0,1,0,1,0,0,1,1,0,0] => [6,4,1,2,3,7,5] => {{1,3,5,6,7},{2,4}} => [3,4,5,2,6,7,1] => 3
[1,1,0,1,0,1,1,0,0,0,1,0] => [5,4,1,2,7,3,6] => {{1,3,5,6,7},{2,4}} => [3,4,5,2,6,7,1] => 3
[1,1,0,1,0,1,1,0,0,1,0,0] => [7,4,1,2,6,3,5] => {{1,3,5,6,7},{2,4}} => [3,4,5,2,6,7,1] => 3
[1,1,0,1,0,1,1,1,0,0,0,0] => [5,4,1,2,6,7,3] => {{1,3,5,6,7},{2,4}} => [3,4,5,2,6,7,1] => 3
[1,1,0,1,1,0,0,0,1,0,1,0] => [4,3,1,7,2,5,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,1,1,0,0,0,1,1,0,0] => [4,3,1,6,2,7,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,1,1,0,0,1,0,0,1,0] => [7,3,1,5,2,4,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,1,1,0,0,1,1,0,0,0] => [6,3,1,5,2,7,4] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,1,1,0,1,0,0,1,0,0] => [7,4,1,6,2,3,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,1,1,1,0,0,0,0,1,0] => [4,3,1,5,7,2,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,1,1,1,0,0,0,1,0,0] => [4,3,1,7,6,2,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,1,1,1,0,0,1,0,0,0] => [7,3,1,5,6,2,4] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,0,1,1,1,1,0,0,0,0,0] => [4,3,1,5,6,7,2] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,0,0,0,1,0,1,0,1,0] => [2,3,7,1,4,5,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,0,0,0,1,0,1,1,0,0] => [2,3,6,1,4,7,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,0,0,0,1,1,0,0,1,0] => [2,3,5,1,7,4,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,0,0,0,1,1,0,1,0,0] => [2,3,7,1,6,4,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,0,0,0,1,1,1,0,0,0] => [2,3,5,1,6,7,4] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,0,0,1,0,0,1,0,1,0] => [2,7,4,1,3,5,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,0,0,1,0,0,1,1,0,0] => [2,6,4,1,3,7,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,0,0,1,1,0,0,0,1,0] => [2,5,4,1,7,3,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,0,0,1,1,0,0,1,0,0] => [2,7,4,1,6,3,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,0,0,1,1,1,0,0,0,0] => [2,5,4,1,6,7,3] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,0,1,0,0,0,1,0,1,0] => [7,3,4,1,2,5,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,0,1,0,0,0,1,1,0,0] => [6,3,4,1,2,7,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,0,1,0,1,0,1,0,0,0] => [6,7,5,1,2,3,4] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,0,1,1,0,0,0,0,1,0] => [5,3,4,1,7,2,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,0,1,1,0,0,0,1,0,0] => [7,3,4,1,6,2,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,0,1,1,1,0,0,0,0,0] => [5,3,4,1,6,7,2] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,1,0,0,0,0,1,0,1,0] => [2,3,4,7,1,5,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,1,0,0,0,0,1,1,0,0] => [2,3,4,6,1,7,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,1,0,0,0,1,0,0,1,0] => [2,3,7,5,1,4,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,1,0,0,0,1,1,0,0,0] => [2,3,6,5,1,7,4] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,1,0,0,1,0,0,0,1,0] => [2,7,4,5,1,3,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,6,4,5,1,7,3] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,1,0,1,0,0,0,0,1,0] => [7,3,4,5,1,2,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,1,0,1,1,0,0,0,0,0] => [6,3,4,5,1,7,2] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,1,1,0,0,0,0,1,0,0] => [2,3,4,7,6,1,5] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,1,1,0,0,0,1,0,0,0] => [2,3,7,5,6,1,4] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,1,1,0,0,1,0,0,0,0] => [2,7,4,5,6,1,3] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[1,1,1,1,1,0,1,0,0,0,0,0] => [7,3,4,5,6,1,2] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 2
[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}} => [2,3,4,5,6,7,1] => 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,1,2,3,4,5,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [7,1,2,3,4,5,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [6,1,2,3,4,8,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [8,1,2,3,4,7,5,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [6,1,2,3,4,7,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [5,1,2,3,8,4,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [5,1,2,3,7,4,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0] => [8,1,2,3,6,4,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0] => [7,1,2,3,6,4,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [5,1,2,3,6,8,4,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [5,1,2,3,8,7,4,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [8,1,2,3,6,7,4,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [5,1,2,3,6,7,8,4] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [4,1,2,8,3,5,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [4,1,2,7,3,5,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [4,1,2,6,3,8,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0] => [4,1,2,8,3,7,5,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [4,1,2,6,3,7,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0] => [8,1,2,5,3,4,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0] => [7,1,2,5,3,4,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0] => [6,1,2,5,3,8,4,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0] => [8,1,2,5,3,7,4,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0] => [8,1,2,6,3,7,4,5] => {{1,2,3,5,8},{4,6,7}} => [2,3,5,6,8,7,4,1] => 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0] => [6,1,2,5,3,7,8,4] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [4,1,2,5,8,3,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [4,1,2,5,7,3,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0] => [4,1,2,8,6,3,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0] => [4,1,2,7,6,3,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0] => [8,1,2,5,6,3,4,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [7,1,2,5,6,3,8,4] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [4,1,2,5,6,8,3,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,8,7,3,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0] => [4,1,2,8,6,7,3,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [8,1,2,5,6,7,3,4] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [4,1,2,5,6,7,8,3] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [3,1,8,2,4,5,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [3,1,7,2,4,5,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [3,1,6,2,4,8,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0] => [3,1,8,2,4,7,5,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [3,1,6,2,4,7,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [3,1,5,2,8,4,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [3,1,5,2,7,4,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0] => [3,1,8,2,6,4,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0] => [3,1,7,2,6,4,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [3,1,5,2,6,8,4,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0] => [3,1,5,2,8,7,4,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0] => [3,1,8,2,6,7,4,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [3,1,5,2,6,7,8,4] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,0] => [8,1,4,2,3,5,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0] => [7,1,4,2,3,5,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0] => [6,1,4,2,3,8,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0] => [8,1,4,2,3,7,5,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,1,0,0,1,1,1,0,0,0] => [6,1,4,2,3,7,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0] => [6,1,8,2,3,4,5,7] => {{1,2,4,6},{3,5,7,8}} => [2,4,5,6,7,1,8,3] => 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,0] => [6,1,7,2,3,4,8,5] => {{1,2,4,6},{3,5,7,8}} => [2,4,5,6,7,1,8,3] => 2
[1,0,1,1,0,1,1,0,0,0,1,0,1,0] => [5,1,4,2,8,3,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,1,1,0,0,0,1,1,0,0] => [5,1,4,2,7,3,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,1,1,0,0,1,0,0,1,0] => [8,1,4,2,6,3,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,1,1,0,0,1,1,0,0,0] => [7,1,4,2,6,3,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,1,1,0,1,0,0,1,0,0] => [8,1,5,2,7,3,4,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,1,1,1,0,0,0,0,1,0] => [5,1,4,2,6,8,3,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,0] => [5,1,4,2,8,7,3,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,1,1,1,0,0,1,0,0,0] => [8,1,4,2,6,7,3,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,0,1,1,1,1,0,0,0,0,0] => [5,1,4,2,6,7,8,3] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [3,1,4,8,2,5,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [3,1,4,7,2,5,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [3,1,4,6,2,8,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,0,0,0,1,1,0,1,0,0] => [3,1,4,8,2,7,5,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [3,1,4,6,2,7,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,0,0,1,0,0,1,0,1,0] => [3,1,8,5,2,4,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,0,0,1,0,0,1,1,0,0] => [3,1,7,5,2,4,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,0,0,1,1,0,0,0,1,0] => [3,1,6,5,2,8,4,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,0,0,1,1,0,0,1,0,0] => [3,1,8,5,2,7,4,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,0,0,1,1,0,1,0,0,0] => [3,1,8,6,2,7,4,5] => {{1,2,3,5,8},{4,6,7}} => [2,3,5,6,8,7,4,1] => 2
[1,0,1,1,1,0,0,1,1,1,0,0,0,0] => [3,1,6,5,2,7,8,4] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,0,1,0,0,0,1,0,1,0] => [8,1,4,5,2,3,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,0,1,0,0,0,1,1,0,0] => [7,1,4,5,2,3,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [7,1,8,6,2,3,4,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,0,1,1,0,0,0,0,1,0] => [6,1,4,5,2,8,3,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [8,1,4,5,2,7,3,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [6,1,4,5,2,7,8,3] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [3,1,4,5,8,2,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [3,1,4,5,7,2,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,1,0,0,0,1,0,0,1,0] => [3,1,4,8,6,2,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,1,0,0,0,1,1,0,0,0] => [3,1,4,7,6,2,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,1,0,0,1,0,0,0,1,0] => [3,1,8,5,6,2,4,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,1,0,0,1,1,0,0,0,0] => [3,1,7,5,6,2,8,4] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,1,0,1,0,0,0,0,1,0] => [8,1,4,5,6,2,3,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,1,0,1,1,0,0,0,0,0] => [7,1,4,5,6,2,8,3] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [3,1,4,5,6,8,2,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [3,1,4,5,8,7,2,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,1,1,0,0,0,1,0,0,0] => [3,1,4,8,6,7,2,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [3,1,8,5,6,7,2,4] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [8,1,4,5,6,7,2,3] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [3,1,4,5,6,7,8,2] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [2,8,1,3,4,5,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [2,7,1,3,4,5,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [2,6,1,3,4,8,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0] => [2,8,1,3,4,7,5,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [2,6,1,3,4,7,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [2,5,1,3,8,4,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,7,4,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,0,1,1,0,1,0,0,1,0] => [2,8,1,3,6,4,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,0,1,1,0,1,1,0,0,0] => [2,7,1,3,6,4,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [2,5,1,3,6,8,4,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,0,1,1,1,0,0,1,0,0] => [2,5,1,3,8,7,4,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,0,1,1,1,0,1,0,0,0] => [2,8,1,3,6,7,4,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [2,5,1,3,6,7,8,4] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [2,4,1,8,3,5,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [2,4,1,7,3,5,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,8,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0] => [2,4,1,8,3,7,5,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [2,4,1,6,3,7,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0] => [2,8,1,5,3,4,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,1,0,1,0,0,1,1,0,0] => [2,7,1,5,3,4,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,1,0,1,1,0,0,0,1,0] => [2,6,1,5,3,8,4,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,1,0,1,1,0,0,1,0,0] => [2,8,1,5,3,7,4,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,1,0,1,1,1,0,0,0,0] => [2,6,1,5,3,7,8,4] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [2,4,1,5,8,3,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [2,4,1,5,7,3,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,1,1,0,0,1,0,0,1,0] => [2,4,1,8,6,3,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,1,1,0,0,1,1,0,0,0] => [2,4,1,7,6,3,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,1,1,0,1,0,0,0,1,0] => [2,8,1,5,6,3,4,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,1,1,0,1,1,0,0,0,0] => [2,7,1,5,6,3,8,4] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [2,4,1,5,6,8,3,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,1,1,1,0,0,0,1,0,0] => [2,4,1,5,8,7,3,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,1,1,1,0,0,1,0,0,0] => [2,4,1,8,6,7,3,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,1,1,1,0,1,0,0,0,0] => [2,8,1,5,6,7,3,4] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [2,4,1,5,6,7,8,3] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,0] => [8,3,1,2,4,5,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,0,0,1,0,1,0,1,1,0,0] => [7,3,1,2,4,5,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,0] => [6,3,1,2,4,8,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,0,0,1,0,1,1,0,1,0,0] => [8,3,1,2,4,7,5,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,0,0,1,0,1,1,1,0,0,0] => [6,3,1,2,4,7,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,0,0,1,1,0,0,1,0,1,0] => [5,3,1,2,8,4,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,0,0,1,1,0,0,1,1,0,0] => [5,3,1,2,7,4,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,0,0,1,1,0,1,0,0,1,0] => [8,3,1,2,6,4,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,0,0,1,1,0,1,1,0,0,0] => [7,3,1,2,6,4,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,0,0,1,1,1,0,0,0,1,0] => [5,3,1,2,6,8,4,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,0,0,1,1,1,0,0,1,0,0] => [5,3,1,2,8,7,4,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,0,0,1,1,1,0,1,0,0,0] => [8,3,1,2,6,7,4,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,0,0,1,1,1,1,0,0,0,0] => [5,3,1,2,6,7,8,4] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0] => [8,4,1,2,3,5,6,7] => {{1,3,5,6,7,8},{2,4}} => [3,4,5,2,6,7,8,1] => 3
[1,1,0,1,0,1,0,0,1,0,1,1,0,0] => [7,4,1,2,3,5,8,6] => {{1,3,5,6,7,8},{2,4}} => [3,4,5,2,6,7,8,1] => 3
[1,1,0,1,0,1,0,0,1,1,0,0,1,0] => [6,4,1,2,3,8,5,7] => {{1,3,5,6,7,8},{2,4}} => [3,4,5,2,6,7,8,1] => 3
[1,1,0,1,0,1,0,0,1,1,0,1,0,0] => [8,4,1,2,3,7,5,6] => {{1,3,5,6,7,8},{2,4}} => [3,4,5,2,6,7,8,1] => 3
[1,1,0,1,0,1,0,0,1,1,1,0,0,0] => [6,4,1,2,3,7,8,5] => {{1,3,5,6,7,8},{2,4}} => [3,4,5,2,6,7,8,1] => 3
[1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [8,7,1,2,3,4,5,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,0,1,1,0,0,0,1,0,1,0] => [5,4,1,2,8,3,6,7] => {{1,3,5,6,7,8},{2,4}} => [3,4,5,2,6,7,8,1] => 3
[1,1,0,1,0,1,1,0,0,0,1,1,0,0] => [5,4,1,2,7,3,8,6] => {{1,3,5,6,7,8},{2,4}} => [3,4,5,2,6,7,8,1] => 3
[1,1,0,1,0,1,1,0,0,1,0,0,1,0] => [8,4,1,2,6,3,5,7] => {{1,3,5,6,7,8},{2,4}} => [3,4,5,2,6,7,8,1] => 3
[1,1,0,1,0,1,1,0,0,1,1,0,0,0] => [7,4,1,2,6,3,8,5] => {{1,3,5,6,7,8},{2,4}} => [3,4,5,2,6,7,8,1] => 3
[1,1,0,1,0,1,1,0,1,0,0,1,0,0] => [5,8,1,2,7,3,4,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,0,1,1,1,0,0,0,0,1,0] => [5,4,1,2,6,8,3,7] => {{1,3,5,6,7,8},{2,4}} => [3,4,5,2,6,7,8,1] => 3
[1,1,0,1,0,1,1,1,0,0,0,1,0,0] => [5,4,1,2,8,7,3,6] => {{1,3,5,6,7,8},{2,4}} => [3,4,5,2,6,7,8,1] => 3
[1,1,0,1,0,1,1,1,0,0,1,0,0,0] => [8,4,1,2,6,7,3,5] => {{1,3,5,6,7,8},{2,4}} => [3,4,5,2,6,7,8,1] => 3
[1,1,0,1,0,1,1,1,1,0,0,0,0,0] => [5,4,1,2,6,7,8,3] => {{1,3,5,6,7,8},{2,4}} => [3,4,5,2,6,7,8,1] => 3
[1,1,0,1,1,0,0,0,1,0,1,0,1,0] => [4,3,1,8,2,5,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,0,0,0,1,0,1,1,0,0] => [4,3,1,7,2,5,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,0,0,0,1,1,0,0,1,0] => [4,3,1,6,2,8,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,0,0,0,1,1,0,1,0,0] => [4,3,1,8,2,7,5,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,0,0,0,1,1,1,0,0,0] => [4,3,1,6,2,7,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,0,0,1,0,0,1,0,1,0] => [8,3,1,5,2,4,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,0,0,1,0,0,1,1,0,0] => [7,3,1,5,2,4,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,0,0,1,1,0,0,0,1,0] => [6,3,1,5,2,8,4,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,0,0,1,1,0,0,1,0,0] => [8,3,1,5,2,7,4,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,0,0,1,1,0,1,0,0,0] => [8,3,1,6,2,7,4,5] => {{1,2,3,5,8},{4,6,7}} => [2,3,5,6,8,7,4,1] => 2
[1,1,0,1,1,0,0,1,1,1,0,0,0,0] => [6,3,1,5,2,7,8,4] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,0,1,0,0,1,0,0,1,0] => [8,4,1,6,2,3,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,0,1,0,0,1,0,1,0,0] => [7,4,1,8,2,3,5,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,0,1,0,0,1,1,0,0,0] => [7,4,1,6,2,3,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,0,1,0,1,0,1,0,0,0] => [8,7,1,6,2,3,4,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,0,1,1,0,0,1,0,0,0] => [8,4,1,6,2,7,3,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,1,0,0,0,0,1,0,1,0] => [4,3,1,5,8,2,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,1,0,0,0,0,1,1,0,0] => [4,3,1,5,7,2,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,1,0,0,0,1,0,0,1,0] => [4,3,1,8,6,2,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,1,0,0,0,1,1,0,0,0] => [4,3,1,7,6,2,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,1,0,0,1,0,0,0,1,0] => [8,3,1,5,6,2,4,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,1,0,0,1,1,0,0,0,0] => [7,3,1,5,6,2,8,4] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,1,0,1,0,0,0,1,0,0] => [8,4,1,5,7,2,3,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,1,0,1,0,0,1,0,0,0] => [8,4,1,7,6,2,3,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,1,1,0,0,0,0,0,1,0] => [4,3,1,5,6,8,2,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,1,1,0,0,0,0,1,0,0] => [4,3,1,5,8,7,2,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,1,1,0,0,0,1,0,0,0] => [4,3,1,8,6,7,2,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,1,1,0,0,1,0,0,0,0] => [8,3,1,5,6,7,2,4] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,0,1,1,1,1,0,1,0,0,0,0,0] => [8,4,1,5,6,7,2,3] => {{1,3,8},{2,4,5,6,7}} => [3,4,8,5,6,7,2,1] => 3
[1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [4,3,1,5,6,7,8,2] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [2,3,8,1,4,5,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [2,3,7,1,4,5,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [2,3,6,1,4,8,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,0,1,0,1,1,0,1,0,0] => [2,3,8,1,4,7,5,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [2,3,6,1,4,7,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [2,3,5,1,8,4,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [2,3,5,1,7,4,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,0,1,1,0,1,0,0,1,0] => [2,3,8,1,6,4,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,0,1,1,0,1,1,0,0,0] => [2,3,7,1,6,4,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [2,3,5,1,6,8,4,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,0,1,1,1,0,0,1,0,0] => [2,3,5,1,8,7,4,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,0,1,1,1,0,1,0,0,0] => [2,3,8,1,6,7,4,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [2,3,5,1,6,7,8,4] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,1,0,0,1,0,1,0,1,0] => [2,8,4,1,3,5,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,1,0,0,1,0,1,1,0,0] => [2,7,4,1,3,5,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,1,0,0,1,1,0,0,1,0] => [2,6,4,1,3,8,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,1,0,0,1,1,0,1,0,0] => [2,8,4,1,3,7,5,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,1,0,0,1,1,1,0,0,0] => [2,6,4,1,3,7,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,1,0,1,0,1,0,0,1,0] => [2,6,8,1,3,4,5,7] => {{1,2,4,6},{3,5,7,8}} => [2,4,5,6,7,1,8,3] => 2
[1,1,1,0,0,1,0,1,0,1,1,0,0,0] => [2,6,7,1,3,4,8,5] => {{1,2,4,6},{3,5,7,8}} => [2,4,5,6,7,1,8,3] => 2
[1,1,1,0,0,1,1,0,0,0,1,0,1,0] => [2,5,4,1,8,3,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,1,1,0,0,0,1,1,0,0] => [2,5,4,1,7,3,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,1,1,0,0,1,0,0,1,0] => [2,8,4,1,6,3,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,0] => [2,7,4,1,6,3,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,1,1,0,1,0,0,1,0,0] => [2,8,5,1,7,3,4,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,1,1,1,0,0,0,0,1,0] => [2,5,4,1,6,8,3,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,1,1,1,0,0,0,1,0,0] => [2,5,4,1,8,7,3,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,1,1,1,0,0,1,0,0,0] => [2,8,4,1,6,7,3,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,0,1,1,1,1,0,0,0,0,0] => [2,5,4,1,6,7,8,3] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,1,0,0,0,1,0,1,0,1,0] => [8,3,4,1,2,5,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,1,0,0,0,1,0,1,1,0,0] => [7,3,4,1,2,5,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,1,0,0,0,1,1,0,0,1,0] => [6,3,4,1,2,8,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,1,0,0,0,1,1,0,1,0,0] => [8,3,4,1,2,7,5,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,1,0,0,0,1,1,1,0,0,0] => [6,3,4,1,2,7,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,1,0,0,1,1,0,1,0,0,0] => [6,3,8,1,2,7,4,5] => {{1,4,6,7},{2,3,5,8}} => [4,3,5,6,8,7,1,2] => 4
[1,1,1,0,1,0,1,0,1,0,0,0,1,0] => [6,8,5,1,2,3,4,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,1,0,1,0,1,0,0,1,0,0] => [8,7,5,1,2,3,4,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [6,7,8,1,2,3,4,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,1,0,1,0,1,1,0,0,0,0] => [6,7,5,1,2,3,8,4] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,1,0,1,1,0,1,0,0,0,0] => [6,8,5,1,2,7,3,4] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,1,1,0,0,0,0,1,0,1,0] => [5,3,4,1,8,2,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,1,1,0,0,0,0,1,1,0,0] => [5,3,4,1,7,2,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,1,1,0,0,0,1,0,0,1,0] => [8,3,4,1,6,2,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,1,1,0,0,0,1,1,0,0,0] => [7,3,4,1,6,2,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,1,1,0,0,1,0,0,1,0,0] => [8,3,5,1,7,2,4,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,1,1,0,1,0,0,0,1,0,0] => [8,5,4,1,7,2,3,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,1,1,0,1,0,0,1,0,0,0] => [7,8,4,1,6,2,3,5] => {{1,3,4,7},{2,5,6,8}} => [3,5,4,7,6,8,1,2] => 3
[1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [7,8,5,1,6,2,3,4] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,1,1,1,0,0,0,0,0,1,0] => [5,3,4,1,6,8,2,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,1,1,1,0,0,0,0,1,0,0] => [5,3,4,1,8,7,2,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,1,1,1,0,0,0,1,0,0,0] => [8,3,4,1,6,7,2,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [5,3,4,1,6,7,8,2] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [2,3,4,8,1,5,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [2,3,4,7,1,5,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [2,3,4,6,1,8,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,0,0,0,0,1,1,0,1,0,0] => [2,3,4,8,1,7,5,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [2,3,4,6,1,7,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0] => [2,3,8,5,1,4,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,0,0,0,1,0,0,1,1,0,0] => [2,3,7,5,1,4,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,0,0,0,1,1,0,0,0,1,0] => [2,3,6,5,1,8,4,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,0,0,0,1,1,0,0,1,0,0] => [2,3,8,5,1,7,4,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,0,0,0,1,1,0,1,0,0,0] => [2,3,8,6,1,7,4,5] => {{1,2,3,5,8},{4,6,7}} => [2,3,5,6,8,7,4,1] => 2
[1,1,1,1,0,0,0,1,1,1,0,0,0,0] => [2,3,6,5,1,7,8,4] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,0,0,1,0,0,0,1,0,1,0] => [2,8,4,5,1,3,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,0,0,1,0,0,0,1,1,0,0] => [2,7,4,5,1,3,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,0,0,1,0,1,0,1,0,0,0] => [2,7,8,6,1,3,4,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,0,0,1,1,0,0,0,0,1,0] => [2,6,4,5,1,8,3,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,0,0,1,1,0,0,0,1,0,0] => [2,8,4,5,1,7,3,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,0,0,1,1,1,0,0,0,0,0] => [2,6,4,5,1,7,8,3] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,0] => [8,3,4,5,1,2,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,0,1,0,0,0,0,1,1,0,0] => [7,3,4,5,1,2,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,0,1,0,0,1,0,0,0,1,0] => [8,3,6,5,1,2,4,7] => {{1,4,5,7,8},{2,3,6}} => [4,3,6,5,7,2,8,1] => 4
[1,1,1,1,0,1,0,0,1,0,1,0,0,0] => [7,3,8,6,1,2,4,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,0,1,0,0,1,1,0,0,0,0] => [7,3,6,5,1,2,8,4] => {{1,4,5,7,8},{2,3,6}} => [4,3,6,5,7,2,8,1] => 4
[1,1,1,1,0,1,0,1,0,0,0,0,1,0] => [8,6,4,5,1,2,3,7] => {{1,3,4,5,7,8},{2,6}} => [3,6,4,5,7,2,8,1] => 3
[1,1,1,1,0,1,0,1,0,0,1,0,0,0] => [7,8,4,6,1,2,3,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [7,8,6,5,1,2,3,4] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [7,6,4,5,1,2,8,3] => {{1,3,4,5,7,8},{2,6}} => [3,6,4,5,7,2,8,1] => 3
[1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [6,3,4,5,1,8,2,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,0,1,1,0,0,0,0,1,0,0] => [8,3,4,5,1,7,2,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [8,3,6,5,1,7,2,4] => {{1,4,5,8},{2,3,6,7}} => [4,3,6,5,8,7,2,1] => 4
[1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [6,3,4,5,1,7,8,2] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [2,3,4,5,8,1,6,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [2,3,4,5,7,1,8,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,1,0,0,0,0,1,0,0,1,0] => [2,3,4,8,6,1,5,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,1,0,0,0,0,1,1,0,0,0] => [2,3,4,7,6,1,8,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,1,0,0,0,1,0,0,0,1,0] => [2,3,8,5,6,1,4,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,1,0,0,0,1,1,0,0,0,0] => [2,3,7,5,6,1,8,4] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,1,0,0,1,0,0,0,0,1,0] => [2,8,4,5,6,1,3,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [2,7,4,5,6,1,8,3] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [8,3,4,5,6,1,2,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [7,3,4,5,6,1,8,2] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [2,3,4,5,6,8,1,7] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [2,3,4,5,8,7,1,6] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [2,3,4,8,6,7,1,5] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [2,3,8,5,6,7,1,4] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [2,8,4,5,6,7,1,3] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [8,3,4,5,6,7,1,2] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 2
[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}} => [2,3,4,5,6,7,8,1] => 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [9,1,2,3,4,5,6,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [8,1,2,3,4,5,6,9,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [7,1,2,3,4,5,9,6,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [9,1,2,3,4,5,8,6,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [7,1,2,3,4,5,8,9,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [6,1,2,3,4,9,5,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [6,1,2,3,4,8,5,9,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0] => [9,1,2,3,4,7,5,6,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0] => [8,1,2,3,4,7,5,9,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [6,1,2,3,4,7,9,5,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [6,1,2,3,4,9,8,5,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [9,1,2,3,4,7,8,5,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [6,1,2,3,4,7,8,9,5] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [5,1,2,3,9,4,6,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [5,1,2,3,8,4,6,9,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0] => [9,1,2,3,6,4,5,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [5,1,2,3,6,9,8,4,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,6,7,8,9,4] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [4,1,2,8,3,5,6,9,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0] => [4,1,2,9,3,5,8,6,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0] => [9,1,2,5,3,4,6,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [4,1,2,5,9,3,6,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [4,1,2,5,6,7,9,3,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [4,1,2,9,6,7,8,3,5] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [9,1,2,5,6,7,8,3,4] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [4,1,2,5,6,7,8,9,3] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [3,1,9,2,4,5,6,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [3,1,8,2,4,5,6,9,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0] => [3,1,9,2,4,5,8,6,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [3,1,5,2,7,4,9,6,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0] => [8,1,4,2,3,5,6,9,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0] => [9,1,4,2,3,5,8,6,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0] => [6,1,4,2,3,7,8,9,5] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [9,1,8,2,3,4,5,6,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [3,1,4,9,2,5,6,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,1,0,0,0,1,0,1,1,0,1,0,0] => [3,1,4,9,2,5,8,6,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0] => [9,1,4,5,2,3,6,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,1,0,1,0,0,0,1,1,1,0,0,0] => [7,1,4,5,2,3,8,9,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0] => [7,1,9,6,2,3,4,5,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0] => [9,1,8,6,2,3,4,5,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [7,1,8,9,2,3,4,5,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0] => [3,1,8,9,7,2,4,5,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,1,1,0,1,0,0,0,0,1,1,0,0] => [8,1,4,5,6,2,3,9,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,1,1,0,1,0,0,1,0,1,0,0,0] => [8,1,4,9,7,2,3,5,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [8,1,9,7,6,2,3,4,5] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [7,1,4,5,6,2,8,9,3] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [3,1,4,5,6,9,2,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [3,1,4,5,6,8,2,9,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [9,1,4,5,6,7,2,3,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [8,1,4,5,6,7,2,9,3] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [3,1,4,5,6,7,9,2,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [3,1,4,9,6,7,8,2,5] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [3,1,9,5,6,7,8,2,4] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [9,1,4,5,6,7,8,2,3] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [3,1,4,5,6,7,8,9,2] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,9,1,3,4,5,6,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => [2,8,1,3,4,5,6,9,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0] => [2,7,1,3,4,5,9,6,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0] => [2,9,1,3,4,5,8,6,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0] => [2,7,1,3,4,5,8,9,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0] => [2,6,1,3,4,9,5,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,0,1,0,1,1,1,1,0,1,0,0,0,0] => [2,9,1,3,6,7,8,4,5] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0] => [2,5,1,3,6,7,8,9,4] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [2,4,1,6,3,8,5,9,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0] => [2,6,1,5,3,7,8,9,4] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0] => [2,4,1,5,9,3,6,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0] => [2,8,1,9,7,3,4,5,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0] => [2,4,1,5,6,7,9,3,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0] => [2,4,1,5,6,9,8,3,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0] => [2,9,1,5,6,7,8,3,4] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,4,1,5,6,7,8,9,3] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0] => [9,3,1,2,4,5,6,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0] => [8,3,1,2,4,5,6,9,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0] => [9,3,1,2,4,5,8,6,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,0,0,1,0,1,1,1,1,0,0,0,0] => [6,3,1,2,4,7,8,9,5] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,0,0,1,1,1,1,0,1,0,0,0,0] => [9,3,1,2,6,7,8,4,5] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0] => [5,3,1,2,6,7,8,9,4] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0] => [9,7,1,2,3,4,5,6,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [8,9,1,2,3,4,5,6,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0] => [8,7,1,2,3,4,5,9,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0] => [9,7,1,2,3,4,8,5,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0] => [9,8,1,2,3,7,4,5,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0] => [5,8,1,2,9,3,4,6,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0] => [4,3,1,9,2,5,6,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0] => [9,4,1,8,2,3,5,6,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0] => [9,7,1,6,2,3,4,5,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0] => [8,7,1,9,2,3,4,5,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0] => [8,7,1,6,2,3,4,9,5] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0] => [9,8,1,6,2,7,3,4,5] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0] => [4,3,1,5,9,2,6,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0] => [9,8,1,7,6,2,3,4,5] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,1,1,1,0,0,0,0,0,1,1,0,0] => [4,3,1,5,6,8,2,9,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0] => [4,3,1,5,8,7,2,9,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0] => [4,3,1,5,6,7,9,2,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0] => [4,3,1,5,6,9,8,2,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0] => [9,3,1,5,6,7,8,2,4] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0] => [4,3,1,5,6,7,8,9,2] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => [2,3,9,1,4,5,6,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0] => [2,3,8,1,4,5,6,9,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0] => [2,3,9,1,4,5,8,6,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,0,0,1,0,1,1,1,0,1,0,0,0] => [2,3,9,1,4,7,8,5,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0] => [2,3,8,1,6,7,4,9,5] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0] => [2,3,5,1,6,7,9,4,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0] => [2,3,9,1,6,7,8,4,5] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0] => [2,3,5,1,6,7,8,9,4] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0] => [2,9,4,1,3,5,6,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0] => [2,6,4,1,3,7,8,9,5] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0] => [2,9,8,1,3,4,5,6,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,0,1,1,0,0,1,0,0,1,0,1,0] => [2,9,4,1,6,3,5,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0] => [2,9,4,1,6,3,8,5,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0] => [2,9,8,1,7,3,4,5,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0] => [9,3,4,1,2,5,6,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0] => [6,3,4,1,2,7,8,9,5] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0] => [9,3,8,1,2,4,5,6,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0] => [9,8,4,1,2,3,5,6,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0] => [6,9,5,1,2,3,4,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0] => [6,8,5,1,2,3,4,9,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0] => [8,7,5,1,2,3,4,9,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0] => [6,7,9,1,2,3,4,5,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0] => [6,9,8,1,2,3,4,5,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0] => [9,7,8,1,2,3,4,5,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0] => [6,7,8,1,2,3,4,9,5] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0] => [7,8,9,1,6,2,3,4,5] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0] => [7,8,5,1,6,2,3,9,4] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0] => [5,3,4,1,6,8,2,9,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0] => [8,3,4,1,6,7,2,9,5] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0] => [9,5,4,1,8,7,2,3,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0] => [5,3,4,1,9,7,8,2,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0] => [5,3,4,1,6,7,8,9,2] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0] => [2,3,4,9,1,5,6,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0] => [2,3,4,9,1,5,8,6,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0] => [2,3,4,9,1,7,8,5,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0] => [2,3,4,6,1,7,8,9,5] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0] => [2,3,9,5,1,4,6,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,0,0,0,1,1,0,0,1,0,0,1,0] => [2,3,9,5,1,7,4,6,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0] => [2,9,4,5,1,3,6,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0] => [2,7,4,5,1,3,8,9,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0] => [2,9,8,6,1,3,4,5,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0] => [2,7,8,9,1,3,4,5,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0] => [2,8,9,6,1,7,3,4,5] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0] => [7,3,4,5,1,2,8,9,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0] => [7,3,8,9,1,2,4,5,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0] => [7,8,4,6,1,2,3,9,5] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0] => [9,8,6,5,1,2,3,4,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0] => [7,8,6,5,1,2,3,9,4] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0] => [8,3,9,6,1,7,2,4,5] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0] => [6,3,4,5,1,7,9,2,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0] => [6,3,4,5,1,9,8,2,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0] => [6,3,4,5,1,7,8,9,2] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0] => [2,3,4,5,9,1,6,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0] => [2,3,4,5,8,1,6,9,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0] => [2,3,4,5,9,1,8,6,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0] => [2,3,4,5,7,1,8,9,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0] => [2,3,4,9,6,1,5,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0] => [2,3,4,9,6,1,8,5,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0] => [2,3,9,5,6,1,4,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0] => [2,3,8,5,6,1,4,9,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0] => [2,3,9,5,6,1,8,4,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0] => [2,8,4,5,6,1,3,9,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0] => [2,8,9,5,7,1,3,4,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0] => [2,8,9,7,6,1,3,4,5] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0] => [8,3,4,5,6,1,2,9,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0] => [8,3,9,7,6,1,2,4,5] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [8,9,7,5,6,1,2,3,4] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0] => [7,3,4,5,6,1,9,2,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0] => [9,3,4,5,6,1,8,2,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0] => [7,3,4,5,6,1,8,9,2] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0] => [2,3,4,5,6,9,1,7,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => [2,3,4,5,6,8,1,9,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0] => [2,3,4,5,9,7,1,6,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0] => [2,3,4,9,6,7,1,5,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0] => [2,3,9,5,6,7,1,4,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0] => [2,9,4,5,6,7,1,3,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0] => [2,8,4,5,6,7,1,9,3] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0] => [9,3,4,5,6,7,1,2,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [8,3,4,5,6,7,1,9,2] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [2,3,4,5,6,7,9,1,8] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0] => [2,3,4,5,6,9,8,1,7] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0] => [2,3,4,5,9,7,8,1,6] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [2,3,4,9,6,7,8,1,5] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0] => [2,3,9,5,6,7,8,1,4] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [2,9,4,5,6,7,8,1,3] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [9,3,4,5,6,7,8,1,2] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [2,3,4,5,6,7,8,10,1,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [3,1,4,5,6,7,8,9,10,2] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0] => [10,3,4,5,6,7,8,1,2,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [10,1,4,5,6,7,8,9,2,3] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0] => [2,10,4,5,6,7,8,1,3,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0] => [8,3,4,5,6,7,1,10,2,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [3,1,10,5,6,7,8,9,2,4] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [9,1,4,5,6,7,8,2,10,3] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0] => [2,3,4,10,6,7,8,1,5,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0] => [2,3,4,5,10,7,8,1,6,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0] => [2,3,4,5,6,10,8,1,7,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [4,3,1,5,6,7,8,10,2,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0] => [2,3,4,5,6,7,9,1,10,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [9,1,10,8,6,7,2,3,4,5] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,4,1,5,6,7,8,9,10,3] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0] => [2,3,4,5,6,7,10,1,8,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [3,1,4,5,6,7,8,10,2,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,1,0,0] => [9,3,4,5,6,7,1,2,10,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [2,10,1,5,6,7,8,9,3,4] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [4,1,2,5,6,7,8,9,10,3] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0] => [3,1,9,10,8,7,2,4,5,6] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[] => [1] => {{1}} => [1] => 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [10,9,1,2,3,4,5,6,7,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0] => [8,9,1,2,3,4,5,6,10,7] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0] => [10,7,1,2,3,4,5,9,6,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0] => [8,10,1,2,3,4,5,9,6,7] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0] => [8,7,1,2,3,4,5,9,10,6] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0] => [9,10,1,2,3,4,8,5,6,7] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0] => [9,7,1,2,3,4,8,5,10,6] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0] => [10,7,1,2,3,4,8,9,5,6] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0] => [10,9,1,2,3,7,8,4,5,6] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0] => [10,3,1,9,2,4,5,6,7,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0] => [10,3,1,5,2,7,4,9,6,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0] => [10,7,1,9,2,3,4,5,6,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0] => [4,3,1,5,6,7,10,9,2,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [10,3,1,5,6,7,8,9,2,4] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [4,3,1,5,6,7,8,9,10,2] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [2,9,10,1,3,4,5,6,7,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0] => [9,3,10,1,2,4,5,6,7,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0] => [6,7,8,1,2,3,4,9,10,5] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0,0] => [7,8,9,1,6,2,3,4,10,5] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0] => [5,3,4,1,6,10,8,9,2,7] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [5,3,4,1,6,7,8,9,10,2] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0,0] => [2,3,10,5,1,7,4,9,6,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0,0] => [2,7,8,9,1,3,4,5,10,6] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0,0] => [2,8,9,10,1,7,3,4,5,6] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0,0] => [10,8,4,9,1,2,3,5,6,7] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0,0] => [7,8,9,5,1,2,3,4,10,6] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0,0] => [10,3,4,5,1,7,8,9,2,6] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0] => [6,3,4,5,1,7,8,9,10,2] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0,0] => [2,3,10,5,6,1,8,9,4,7] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0,0] => [2,8,9,10,6,1,3,4,5,7] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0,0] => [8,3,4,9,10,1,2,5,6,7] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0,0] => [8,9,4,5,10,1,2,3,6,7] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0,0] => [8,9,7,5,6,1,2,3,10,4] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0] => [7,3,4,5,6,1,8,9,10,2] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,0] => [2,3,4,5,10,7,1,9,6,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0] => [2,9,10,8,6,7,1,3,4,5] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0,0] => [10,3,4,5,6,7,1,9,2,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0] => [8,3,4,5,6,7,1,9,10,2] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0] => [2,9,4,5,6,7,8,1,10,3] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0] => [9,3,4,5,6,7,8,1,10,2] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0] => [2,3,4,5,6,7,10,9,1,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0] => [2,3,4,5,6,10,8,9,1,7] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0] => [2,3,4,5,10,7,8,9,1,6] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0] => [2,3,4,10,6,7,8,9,1,5] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0] => [2,3,10,5,6,7,8,9,1,4] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0] => [2,10,4,5,6,7,8,9,1,3] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [10,3,4,5,6,7,8,9,1,2] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,1] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [10,1,2,3,4,5,6,7,8,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [9,1,2,3,4,5,6,7,10,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [10,1,2,3,4,5,6,9,7,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [10,1,2,9,3,4,5,6,7,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [9,1,10,2,3,4,5,6,7,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [8,1,2,3,4,5,6,9,10,7] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [7,1,2,3,4,5,10,9,6,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0] => [3,1,10,9,2,4,5,6,7,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [7,1,2,3,4,5,8,9,10,6] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [6,1,2,3,4,7,8,9,10,5] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0] => [7,1,10,9,2,3,4,5,6,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0] => [10,1,8,9,2,3,4,5,6,7] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [5,1,2,3,6,7,8,9,10,4] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [7,1,2,3,4,5,9,6,10,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [5,1,2,3,10,4,6,7,8,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [4,1,2,10,3,5,6,7,8,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => [3,1,9,2,4,5,6,7,10,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0] => [3,1,5,2,10,4,6,7,8,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [3,1,5,2,7,4,9,6,10,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => [3,1,4,10,2,5,6,7,8,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => [3,1,4,5,6,7,9,2,10,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,10,1,3,4,5,6,7,8,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [2,9,1,3,4,5,6,7,10,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [2,8,1,3,4,5,6,10,7,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,5,1,3,6,7,8,9,10,4] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,8,5,10,7,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [2,4,1,5,6,7,8,10,3,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,10,1,4,5,6,7,8,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,5,1,6,7,8,9,10,4] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,10,1,5,6,7,8,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,6,1,7,8,9,10,5] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0] => [2,3,4,5,10,1,6,7,8,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0] => [2,3,4,5,7,1,8,9,10,6] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0] => [2,3,4,5,6,10,1,7,8,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0] => [2,3,4,5,6,8,1,9,10,7] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [10,3,1,2,4,5,6,7,8,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0] => [10,7,1,2,3,4,5,6,8,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0] => [8,10,1,2,3,4,5,6,7,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => [4,3,1,10,2,5,6,7,8,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0] => [8,7,1,2,3,4,5,10,6,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [2,10,1,3,4,5,6,9,7,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0] => [10,3,1,2,4,5,6,9,7,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0] => [3,1,10,2,4,5,6,9,7,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => [9,3,1,2,4,5,6,7,10,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0] => [10,7,8,1,2,3,4,5,6,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0] => [6,7,10,1,2,3,4,5,8,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0] => [6,10,5,1,2,3,4,7,8,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => [10,3,4,1,2,5,6,7,8,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0] => [2,3,4,5,10,7,1,6,8,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0] => [2,3,4,10,6,1,5,7,8,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0] => [2,3,10,5,1,4,6,7,8,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0] => [2,10,4,1,3,5,6,7,8,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,1,0,0] => [6,7,9,1,2,3,4,5,10,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [2,8,1,9,10,3,4,5,6,7] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0] => [8,7,1,10,2,3,4,5,6,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [5,3,1,2,6,7,8,9,10,4] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,0,1,0,0,0,1,1,1,1,1,0,0,0,0,0] => [6,3,4,1,2,7,8,9,10,5] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,0,1,0,0,0,0,1,1,1,1,0,0,0,0] => [7,3,4,5,1,2,8,9,10,6] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,0,1,0,0,0,0,0,1,1,1,0,0,0] => [8,3,4,5,6,1,2,9,10,7] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0] => [10,1,2,5,3,4,6,7,8,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0] => [2,10,1,5,3,4,6,7,8,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0] => [3,1,8,9,10,2,4,5,6,7] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,1,0] => [2,10,8,1,3,4,5,6,7,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0] => [9,1,8,2,3,4,5,6,10,7] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0] => [9,7,1,2,3,4,5,6,10,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [2,10,1,9,3,4,5,6,7,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [2,9,1,10,8,7,3,4,5,6] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,0,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [2,6,1,5,3,7,8,9,10,4] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => [5,3,4,1,6,7,9,2,10,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,0,0,0,1,1,1,0,1,1,1,0,0,0,0,0] => [2,3,8,1,6,7,4,9,10,5] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,0,1,1,0,0,0,0,0,1,1,1,0,0,0] => [6,3,4,5,1,8,2,9,10,7] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,1,0,0,0,0] => [2,3,4,10,1,7,8,9,5,6] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,1,0,0] => [2,3,4,5,6,10,1,9,7,8] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,1,0,0,0] => [2,3,4,5,10,1,8,9,6,7] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,0,0,0,1,1,1,1,1,0,1,0,0,0,0,0] => [2,3,10,1,6,7,8,9,4,5] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0] => [2,10,4,1,6,3,8,5,7,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0,1,0] => [2,3,4,10,6,7,1,5,8,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
[1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0,1,0] => [2,3,10,5,6,1,4,7,8,9] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 2
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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:
13,1 36,5,1
$F_{1} = q^{2}$
$F_{2} = 2\ q^{2}$
$F_{3} = 5\ q^{2}$
$F_{4} = 13\ q^{2} + q^{3}$
$F_{5} = 36\ q^{2} + 5\ q^{3} + q^{4}$
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies St000703The number of deficiencies of a permutation. as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$ \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1). $$
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies St000703The number of deficiencies of a permutation. as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$ \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1). $$
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
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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