- FindStat

Identifier

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001963: Graphs ⟶ ℤ

Values

[1,0] => [1] => [1] => ([],1) => 1

[1,0,1,0] => [1,2] => [1,2] => ([],2) => 1

[1,1,0,0] => [2,1] => [2,1] => ([(0,1)],2) => 2

[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => ([],3) => 1

[1,0,1,1,0,0] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3) => 2

[1,1,0,0,1,0] => [2,1,3] => [2,1,3] => ([(1,2)],3) => 2

[1,1,0,1,0,0] => [2,3,1] => [1,3,2] => ([(1,2)],3) => 2

[1,1,1,0,0,0] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 3

[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => ([],4) => 1

[1,0,1,0,1,1,0,0] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4) => 2

[1,0,1,1,0,0,1,0] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4) => 2

[1,0,1,1,0,1,0,0] => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 3

[1,0,1,1,1,0,0,0] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3

[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4) => 2

[1,1,0,0,1,1,0,0] => [2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4) => 2

[1,1,0,1,0,0,1,0] => [2,3,1,4] => [1,3,2,4] => ([(2,3)],4) => 2

[1,1,0,1,0,1,0,0] => [2,3,4,1] => [1,2,4,3] => ([(2,3)],4) => 2

[1,1,0,1,1,0,0,0] => [2,4,3,1] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4) => 3

[1,1,1,0,0,0,1,0] => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4) => 3

[1,1,1,0,0,1,0,0] => [3,2,4,1] => [2,1,4,3] => ([(0,3),(1,2)],4) => 2

[1,1,1,0,1,0,0,0] => [3,4,2,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4) => 3

[1,1,1,1,0,0,0,0] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4

[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5) => 1

[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2

[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5) => 2

[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 3

[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3

[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5) => 2

[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5) => 3

[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5) => 3

[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5) => 3

[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 4

[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3

[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 3

[1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 4

[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4

[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5) => 2

[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5) => 2

[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5) => 2

[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5) => 3

[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3

[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [1,3,2,4,5] => ([(3,4)],5) => 2

[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5) => 2

[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [1,2,4,3,5] => ([(3,4)],5) => 2

[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [1,2,3,5,4] => ([(3,4)],5) => 2

[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 3

[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5) => 3

[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5) => 2

[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 3

[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4

[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5) => 3

[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5) => 2

[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5) => 2

[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [2,1,3,5,4] => ([(1,4),(2,3)],5) => 2

[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5) => 3

[1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5) => 3

[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => [1,3,2,5,4] => ([(1,4),(2,3)],5) => 2

[1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5) => 3

[1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4

[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4

[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5) => 3

[1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5) => 3

[1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4

[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6) => 1

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [4,6,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 3

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [4,1,2,3,5,6] => ([(2,5),(3,5),(4,5)],6) => 2

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 3

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [3,5,1,2,4,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 3

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 4

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => [6,3,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => [5,4,1,2,3,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => [4,3,6,1,2,5] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 4

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => [3,6,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [3,1,2,4,5,6] => ([(3,5),(4,5)],6) => 2

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 3

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6) => 3

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 4

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => [6,2,5,1,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6) => 3

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 3

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6) => 3

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 3

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => [6,2,3,5,1,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => [5,2,4,1,3,6] => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => [4,2,3,6,1,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => 3

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => [3,6,2,5,1,4] => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => [6,5,2,4,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => [4,3,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => [3,2,6,1,4,5] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 3

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => [3,2,5,1,4,6] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 3

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => [3,2,4,6,1,5] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 3

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => [2,6,3,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 4

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,3,2,6] => [2,5,4,1,3,6] => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 4

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => [2,4,3,6,1,5] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 3

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => [2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 4

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => [6,2,5,4,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5

>>> Load all 613 entries. <<<

[1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => [5,4,3,1,2,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,4,3,6,2] => [4,3,2,6,1,5] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,0,1,1,1,1,0,0,1,0,0,0] => [1,5,4,6,3,2] => [3,2,6,5,1,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 4

[1,0,1,1,1,1,0,1,0,0,0,0] => [1,5,6,4,3,2] => [2,6,5,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5

[1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => [6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5

[1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(4,5)],6) => 2

[1,1,0,0,1,0,1,0,1,1,0,0] => [2,1,3,4,6,5] => [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2

[1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,3,5,4,6] => [1,5,2,3,4,6] => ([(2,5),(3,5),(4,5)],6) => 2

[1,1,0,0,1,0,1,1,0,1,0,0] => [2,1,3,5,6,4] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 3

[1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3,6,5,4] => [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3

[1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => [1,4,2,3,5,6] => ([(3,5),(4,5)],6) => 2

[1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5] => [1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6) => 3

[1,1,0,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => [1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6) => 3

[1,1,0,0,1,1,0,1,0,1,0,0] => [2,1,4,5,6,3] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6) => 3

[1,1,0,0,1,1,0,1,1,0,0,0] => [2,1,4,6,5,3] => [6,1,3,5,2,4] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,4,3,6] => [5,1,4,2,3,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3

[1,1,0,0,1,1,1,0,0,1,0,0] => [2,1,5,4,6,3] => [4,1,3,6,2,5] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => 3

[1,1,0,0,1,1,1,0,1,0,0,0] => [2,1,5,6,4,3] => [3,6,1,5,2,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,6,5,4,3] => [6,5,1,4,2,3] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,1,0,1,0,0,1,0,1,0,1,0] => [2,3,1,4,5,6] => [1,3,2,4,5,6] => ([(4,5)],6) => 2

[1,1,0,1,0,0,1,0,1,1,0,0] => [2,3,1,4,6,5] => [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6) => 2

[1,1,0,1,0,0,1,1,0,0,1,0] => [2,3,1,5,4,6] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6) => 2

[1,1,0,1,0,0,1,1,0,1,0,0] => [2,3,1,5,6,4] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6) => 3

[1,1,0,1,0,0,1,1,1,0,0,0] => [2,3,1,6,5,4] => [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3

[1,1,0,1,0,1,0,0,1,0,1,0] => [2,3,4,1,5,6] => [1,2,4,3,5,6] => ([(4,5)],6) => 2

[1,1,0,1,0,1,0,0,1,1,0,0] => [2,3,4,1,6,5] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6) => 2

[1,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,1,6] => [1,2,3,5,4,6] => ([(4,5)],6) => 2

[1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => [1,2,3,4,6,5] => ([(4,5)],6) => 2

[1,1,0,1,0,1,0,1,1,0,0,0] => [2,3,4,6,5,1] => [6,1,2,3,5,4] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 3

[1,1,0,1,0,1,1,0,0,0,1,0] => [2,3,5,4,1,6] => [5,1,2,4,3,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 3

[1,1,0,1,0,1,1,0,0,1,0,0] => [2,3,5,4,6,1] => [4,1,2,3,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6) => 2

[1,1,0,1,0,1,1,0,1,0,0,0] => [2,3,5,6,4,1] => [3,6,1,2,5,4] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => 3

[1,1,0,1,0,1,1,1,0,0,0,0] => [2,3,6,5,4,1] => [6,5,1,2,4,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,1,0,1,1,0,0,0,1,0,1,0] => [2,4,3,1,5,6] => [4,1,3,2,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6) => 3

[1,1,0,1,1,0,0,0,1,1,0,0] => [2,4,3,1,6,5] => [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6) => 2

[1,1,0,1,1,0,0,1,0,0,1,0] => [2,4,3,5,1,6] => [3,1,2,5,4,6] => ([(1,2),(3,5),(4,5)],6) => 2

[1,1,0,1,1,0,0,1,0,1,0,0] => [2,4,3,5,6,1] => [3,1,2,4,6,5] => ([(1,2),(3,5),(4,5)],6) => 2

[1,1,0,1,1,0,0,1,1,0,0,0] => [2,4,3,6,5,1] => [2,6,1,3,5,4] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 3

[1,1,0,1,1,0,1,0,0,0,1,0] => [2,4,5,3,1,6] => [2,5,1,4,3,6] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 3

[1,1,0,1,1,0,1,0,0,1,0,0] => [2,4,5,3,6,1] => [2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6) => 3

[1,1,0,1,1,0,1,0,1,0,0,0] => [2,4,5,6,3,1] => [2,3,6,1,5,4] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 3

[1,1,0,1,1,0,1,1,0,0,0,0] => [2,4,6,5,3,1] => [6,2,5,1,4,3] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,1,0,1,1,1,0,0,0,0,1,0] => [2,5,4,3,1,6] => [5,4,1,3,2,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,1,0,1,1,1,0,0,0,1,0,0] => [2,5,4,3,6,1] => [4,3,1,2,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3

[1,1,0,1,1,1,0,0,1,0,0,0] => [2,5,4,6,3,1] => [3,2,6,1,5,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) => 4

[1,1,0,1,1,1,0,1,0,0,0,0] => [2,5,6,4,3,1] => [2,6,5,1,4,3] => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,1,0,1,1,1,1,0,0,0,0,0] => [2,6,5,4,3,1] => [6,5,4,1,3,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5

[1,1,1,0,0,0,1,0,1,0,1,0] => [3,2,1,4,5,6] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6) => 3

[1,1,1,0,0,0,1,0,1,1,0,0] => [3,2,1,4,6,5] => [2,1,6,3,4,5] => ([(0,1),(2,5),(3,5),(4,5)],6) => 2

[1,1,1,0,0,0,1,1,0,0,1,0] => [3,2,1,5,4,6] => [2,1,5,3,4,6] => ([(1,2),(3,5),(4,5)],6) => 2

[1,1,1,0,0,0,1,1,0,1,0,0] => [3,2,1,5,6,4] => [2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6) => 3

[1,1,1,0,0,0,1,1,1,0,0,0] => [3,2,1,6,5,4] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3

[1,1,1,0,0,1,0,0,1,0,1,0] => [3,2,4,1,5,6] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6) => 2

[1,1,1,0,0,1,0,0,1,1,0,0] => [3,2,4,1,6,5] => [2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6) => 2

[1,1,1,0,0,1,0,1,0,0,1,0] => [3,2,4,5,1,6] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6) => 2

[1,1,1,0,0,1,0,1,0,1,0,0] => [3,2,4,5,6,1] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6) => 2

[1,1,1,0,0,1,0,1,1,0,0,0] => [3,2,4,6,5,1] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 3

[1,1,1,0,0,1,1,0,0,0,1,0] => [3,2,5,4,1,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6) => 3

[1,1,1,0,0,1,1,0,0,1,0,0] => [3,2,5,4,6,1] => [1,4,2,3,6,5] => ([(1,2),(3,5),(4,5)],6) => 2

[1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,5,6,4,1] => [1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 3

[1,1,1,0,0,1,1,1,0,0,0,0] => [3,2,6,5,4,1] => [6,1,5,2,4,3] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,1,1,0,1,0,0,0,1,0,1,0] => [3,4,2,1,5,6] => [1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6) => 3

[1,1,1,0,1,0,0,0,1,1,0,0] => [3,4,2,1,6,5] => [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6) => 2

[1,1,1,0,1,0,0,1,0,0,1,0] => [3,4,2,5,1,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6) => 2

[1,1,1,0,1,0,0,1,0,1,0,0] => [3,4,2,5,6,1] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6) => 2

[1,1,1,0,1,0,0,1,1,0,0,0] => [3,4,2,6,5,1] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6) => 3

[1,1,1,0,1,0,1,0,0,0,1,0] => [3,4,5,2,1,6] => [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6) => 3

[1,1,1,0,1,0,1,0,0,1,0,0] => [3,4,5,2,6,1] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6) => 2

[1,1,1,0,1,0,1,0,1,0,0,0] => [3,4,5,6,2,1] => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6) => 3

[1,1,1,0,1,0,1,1,0,0,0,0] => [3,4,6,5,2,1] => [6,1,2,5,4,3] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,1,1,0,1,1,0,0,0,0,1,0] => [3,5,4,2,1,6] => [5,1,4,3,2,6] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,1,1,0,1,1,0,0,0,1,0,0] => [3,5,4,2,6,1] => [4,1,3,2,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => 3

[1,1,1,0,1,1,0,0,1,0,0,0] => [3,5,4,6,2,1] => [3,1,2,6,5,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => 3

[1,1,1,0,1,1,0,1,0,0,0,0] => [3,5,6,4,2,1] => [2,6,1,5,4,3] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,1,1,0,1,1,1,0,0,0,0,0] => [3,6,5,4,2,1] => [6,5,1,4,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5

[1,1,1,1,0,0,0,0,1,0,1,0] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,1,1,1,0,0,0,0,1,1,0,0] => [4,3,2,1,6,5] => [3,2,1,6,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => 3

[1,1,1,1,0,0,0,1,0,0,1,0] => [4,3,2,5,1,6] => [3,2,1,5,4,6] => ([(1,2),(3,4),(3,5),(4,5)],6) => 3

[1,1,1,1,0,0,0,1,0,1,0,0] => [4,3,2,5,6,1] => [3,2,1,4,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6) => 3

[1,1,1,1,0,0,0,1,1,0,0,0] => [4,3,2,6,5,1] => [2,1,6,3,5,4] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => 3

[1,1,1,1,0,0,1,0,0,0,1,0] => [4,3,5,2,1,6] => [2,1,5,4,3,6] => ([(1,2),(3,4),(3,5),(4,5)],6) => 3

[1,1,1,1,0,0,1,0,0,1,0,0] => [4,3,5,2,6,1] => [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6) => 2

[1,1,1,1,0,0,1,0,1,0,0,0] => [4,3,5,6,2,1] => [2,1,3,6,5,4] => ([(1,2),(3,4),(3,5),(4,5)],6) => 3

[1,1,1,1,0,0,1,1,0,0,0,0] => [4,3,6,5,2,1] => [1,6,2,5,4,3] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,1,1,1,0,1,0,0,0,0,1,0] => [4,5,3,2,1,6] => [1,5,4,3,2,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,1,1,1,0,1,0,0,0,1,0,0] => [4,5,3,2,6,1] => [1,4,3,2,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6) => 3

[1,1,1,1,0,1,0,0,1,0,0,0] => [4,5,3,6,2,1] => [1,3,2,6,5,4] => ([(1,2),(3,4),(3,5),(4,5)],6) => 3

[1,1,1,1,0,1,0,1,0,0,0,0] => [4,5,6,3,2,1] => [1,2,6,5,4,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,1,1,1,0,1,1,0,0,0,0,0] => [4,6,5,3,2,1] => [6,1,5,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5

[1,1,1,1,1,0,0,0,0,0,1,0] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5

[1,1,1,1,1,0,0,0,0,1,0,0] => [5,4,3,2,6,1] => [4,3,2,1,6,5] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,1,1,1,1,0,0,0,1,0,0,0] => [5,4,3,6,2,1] => [3,2,1,6,5,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => 3

[1,1,1,1,1,0,0,1,0,0,0,0] => [5,4,6,3,2,1] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

[1,1,1,1,1,0,1,0,0,0,0,0] => [5,6,4,3,2,1] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5

[1,1,1,1,1,1,0,0,0,0,0,0] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6

[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] => ([],7) => 1

[1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2

[1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2

[1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 3

[1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2

[1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,3,5,4,7,6] => [4,7,1,2,3,5,6] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 3

[1,0,1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,3,5,6,4,7] => [4,6,1,2,3,5,7] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 3

[1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,3,5,6,7,4] => [4,5,7,1,2,3,6] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7) => 4

[1,0,1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,3,5,7,6,4] => [7,4,6,1,2,3,5] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 4

[1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,3,6,5,4,7] => [6,5,1,2,3,4,7] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,3,6,5,7,4] => [5,4,7,1,2,3,6] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5)],7) => 4

[1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,3,6,7,5,4] => [4,7,6,1,2,3,5] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [1,2,4,3,5,6,7] => [4,1,2,3,5,6,7] => ([(3,6),(4,6),(5,6)],7) => 2

[1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [1,2,4,3,5,7,6] => [3,7,1,2,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 3

[1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [1,2,4,3,6,5,7] => [3,6,1,2,4,5,7] => ([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 3

[1,0,1,0,1,1,0,0,1,1,0,1,0,0] => [1,2,4,3,6,7,5] => [3,5,7,1,2,4,6] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 4

[1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [1,2,4,3,7,6,5] => [7,3,6,1,2,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,0,1,1,0,1,0,0,1,0,1,0] => [1,2,4,5,3,6,7] => [3,5,1,2,4,6,7] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 3

[1,0,1,0,1,1,0,1,0,0,1,1,0,0] => [1,2,4,5,3,7,6] => [3,4,7,1,2,5,6] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 4

[1,0,1,0,1,1,0,1,0,1,0,0,1,0] => [1,2,4,5,6,3,7] => [3,4,6,1,2,5,7] => ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 4

[1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,2,4,5,6,7,3] => [3,4,5,7,1,2,6] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 4

[1,0,1,0,1,1,0,1,0,1,1,0,0,0] => [1,2,4,5,7,6,3] => [7,3,4,6,1,2,5] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,0,1,0,1,1,0,1,1,0,0,0,1,0] => [1,2,4,6,5,3,7] => [6,3,5,1,2,4,7] => ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,0,1,1,0,1,1,0,0,1,0,0] => [1,2,4,6,5,7,3] => [5,3,4,7,1,2,6] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 4

[1,0,1,0,1,1,0,1,1,0,1,0,0,0] => [1,2,4,6,7,5,3] => [4,7,3,6,1,2,5] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,0,1,0,1,1,0,1,1,1,0,0,0,0] => [1,2,4,7,6,5,3] => [7,6,3,5,1,2,4] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [1,2,5,4,3,6,7] => [5,4,1,2,3,6,7] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [1,2,5,4,3,7,6] => [4,3,7,1,2,5,6] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 4

[1,0,1,0,1,1,1,0,0,1,0,0,1,0] => [1,2,5,4,6,3,7] => [4,3,6,1,2,5,7] => ([(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 4

[1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [1,2,5,4,6,7,3] => [4,3,5,7,1,2,6] => ([(0,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 4

[1,0,1,0,1,1,1,0,0,1,1,0,0,0] => [1,2,5,4,7,6,3] => [3,7,4,6,1,2,5] => ([(0,4),(0,5),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 5

[1,0,1,0,1,1,1,0,1,0,0,0,1,0] => [1,2,5,6,4,3,7] => [3,6,5,1,2,4,7] => ([(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [1,2,5,6,4,7,3] => [3,5,4,7,1,2,6] => ([(0,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 4

[1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [1,2,5,6,7,4,3] => [3,4,7,6,1,2,5] => ([(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7) => 5

[1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [1,2,5,7,6,4,3] => [7,3,6,5,1,2,4] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [1,2,6,5,4,3,7] => [6,5,4,1,2,3,7] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [1,2,6,5,4,7,3] => [5,4,3,7,1,2,6] => ([(0,1),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7) => 5

[1,0,1,0,1,1,1,1,0,0,1,0,0,0] => [1,2,6,5,7,4,3] => [4,3,7,6,1,2,5] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [1,2,6,7,5,4,3] => [3,7,6,5,1,2,4] => ([(0,2),(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [1,2,7,6,5,4,3] => [7,6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [1,3,2,4,5,6,7] => [3,1,2,4,5,6,7] => ([(4,6),(5,6)],7) => 2

[1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [1,3,2,4,5,7,6] => [2,7,1,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 3

[1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [1,3,2,4,6,5,7] => [2,6,1,3,4,5,7] => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 3

[1,0,1,1,0,0,1,0,1,1,0,1,0,0] => [1,3,2,4,6,7,5] => [2,5,7,1,3,4,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 4

[1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [1,3,2,4,7,6,5] => [7,2,6,1,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [1,3,2,5,4,6,7] => [2,5,1,3,4,6,7] => ([(2,6),(3,6),(4,5),(5,6)],7) => 3

[1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4,7,6] => [2,4,7,1,3,5,6] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7) => 4

[1,0,1,1,0,0,1,1,0,1,0,0,1,0] => [1,3,2,5,6,4,7] => [2,4,6,1,3,5,7] => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 4

[1,0,1,1,0,0,1,1,0,1,0,1,0,0] => [1,3,2,5,6,7,4] => [2,4,5,7,1,3,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 4

[1,0,1,1,0,0,1,1,0,1,1,0,0,0] => [1,3,2,5,7,6,4] => [7,2,4,6,1,3,5] => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [1,3,2,6,5,4,7] => [6,2,5,1,3,4,7] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,0,0,1,1,1,0,0,1,0,0] => [1,3,2,6,5,7,4] => [5,2,4,7,1,3,6] => ([(0,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,6),(5,6)],7) => 4

[1,0,1,1,0,0,1,1,1,0,1,0,0,0] => [1,3,2,6,7,5,4] => [4,7,2,6,1,3,5] => ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [1,3,2,7,6,5,4] => [7,6,2,5,1,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,0,1,1,0,1,0,0,1,0,1,0,1,0] => [1,3,4,2,5,6,7] => [2,4,1,3,5,6,7] => ([(3,6),(4,5),(5,6)],7) => 3

[1,0,1,1,0,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5,7,6] => [2,3,7,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 3

[1,0,1,1,0,1,0,0,1,1,0,0,1,0] => [1,3,4,2,6,5,7] => [2,3,6,1,4,5,7] => ([(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 3

[1,0,1,1,0,1,0,0,1,1,0,1,0,0] => [1,3,4,2,6,7,5] => [2,3,5,7,1,4,6] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7) => 4

[1,0,1,1,0,1,0,0,1,1,1,0,0,0] => [1,3,4,2,7,6,5] => [7,2,3,6,1,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,0,1,0,1,0,0,1,0,1,0] => [1,3,4,5,2,6,7] => [2,3,5,1,4,6,7] => ([(2,6),(3,6),(4,5),(5,6)],7) => 3

[1,0,1,1,0,1,0,1,0,0,1,1,0,0] => [1,3,4,5,2,7,6] => [2,3,4,7,1,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 3

[1,0,1,1,0,1,0,1,0,1,0,0,1,0] => [1,3,4,5,6,2,7] => [2,3,4,6,1,5,7] => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 3

[1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,7,2] => [2,3,4,5,7,1,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 3

[1,0,1,1,0,1,0,1,0,1,1,0,0,0] => [1,3,4,5,7,6,2] => [7,2,3,4,6,1,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,0,1,0,1,1,0,0,0,1,0] => [1,3,4,6,5,2,7] => [6,2,3,5,1,4,7] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,0,1,0,1,1,0,0,1,0,0] => [1,3,4,6,5,7,2] => [5,2,3,4,7,1,6] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => 3

[1,0,1,1,0,1,0,1,1,0,1,0,0,0] => [1,3,4,6,7,5,2] => [4,7,2,3,6,1,5] => ([(0,4),(0,6),(1,2),(1,3),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,0,1,0,1,1,1,0,0,0,0] => [1,3,4,7,6,5,2] => [7,6,2,3,5,1,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,0,1,1,0,1,1,0,0,0,1,0,1,0] => [1,3,5,4,2,6,7] => [5,2,4,1,3,6,7] => ([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,0,1,1,0,0,0,1,1,0,0] => [1,3,5,4,2,7,6] => [4,2,3,7,1,5,6] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => 3

[1,0,1,1,0,1,1,0,0,1,0,0,1,0] => [1,3,5,4,6,2,7] => [4,2,3,6,1,5,7] => ([(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => 3

[1,0,1,1,0,1,1,0,0,1,0,1,0,0] => [1,3,5,4,6,7,2] => [4,2,3,5,7,1,6] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => 3

[1,0,1,1,0,1,1,0,0,1,1,0,0,0] => [1,3,5,4,7,6,2] => [3,7,2,4,6,1,5] => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,0,1,1,0,1,0,0,0,1,0] => [1,3,5,6,4,2,7] => [3,6,2,5,1,4,7] => ([(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,0,1,1,0,1,0,0,1,0,0] => [1,3,5,6,4,7,2] => [3,5,2,4,7,1,6] => ([(0,3),(1,5),(1,6),(2,4),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,0,1,1,0,1,0,1,0,0,0] => [1,3,5,6,7,4,2] => [3,4,7,2,6,1,5] => ([(0,4),(0,6),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,0,1,1,0,1,1,0,0,0,0] => [1,3,5,7,6,4,2] => [7,3,6,2,5,1,4] => ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,0,1,1,0,1,1,1,0,0,0,0,1,0] => [1,3,6,5,4,2,7] => [6,5,2,4,1,3,7] => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,0,1,1,0,1,1,1,0,0,0,1,0,0] => [1,3,6,5,4,7,2] => [5,4,2,3,7,1,6] => ([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,0,1,1,1,0,0,1,0,0,0] => [1,3,6,5,7,4,2] => [4,3,7,2,6,1,5] => ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,0,1,1,0,1,1,1,0,1,0,0,0,0] => [1,3,6,7,5,4,2] => [3,7,6,2,5,1,4] => ([(0,3),(0,6),(1,2),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [1,4,3,2,5,6,7] => [4,3,1,2,5,6,7] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [1,4,3,2,5,7,6] => [3,2,7,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7) => 3

[1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [1,4,3,2,6,5,7] => [3,2,6,1,4,5,7] => ([(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7) => 3

[1,0,1,1,1,0,0,0,1,1,0,1,0,0] => [1,4,3,2,6,7,5] => [3,2,5,7,1,4,6] => ([(0,5),(1,2),(1,6),(2,6),(3,4),(3,6),(4,5),(5,6)],7) => 4

[1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [1,4,3,2,7,6,5] => [2,7,3,6,1,4,5] => ([(0,5),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,1,0,0,1,0,0,1,0,1,0] => [1,4,3,5,2,6,7] => [3,2,5,1,4,6,7] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7) => 3

[1,0,1,1,1,0,0,1,0,0,1,1,0,0] => [1,4,3,5,2,7,6] => [3,2,4,7,1,5,6] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7) => 3

[1,0,1,1,1,0,0,1,0,1,0,0,1,0] => [1,4,3,5,6,2,7] => [3,2,4,6,1,5,7] => ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 3

[1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [1,4,3,5,6,7,2] => [3,2,4,5,7,1,6] => ([(0,6),(1,6),(2,3),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,0,1,1,1,0,0,1,0,1,1,0,0,0] => [1,4,3,5,7,6,2] => [2,7,3,4,6,1,5] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,1,0,0,1,1,0,0,0,1,0] => [1,4,3,6,5,2,7] => [2,6,3,5,1,4,7] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,1,0,0,1,1,0,0,1,0,0] => [1,4,3,6,5,7,2] => [2,5,3,4,7,1,6] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => 3

[1,0,1,1,1,0,0,1,1,0,1,0,0,0] => [1,4,3,6,7,5,2] => [2,4,7,3,6,1,5] => ([(0,6),(1,3),(1,6),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,1,0,0,1,1,1,0,0,0,0] => [1,4,3,7,6,5,2] => [7,2,6,3,5,1,4] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,0,1,1,1,0,1,0,0,0,1,0,1,0] => [1,4,5,3,2,6,7] => [2,5,4,1,3,6,7] => ([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,1,0,1,0,0,0,1,1,0,0] => [1,4,5,3,2,7,6] => [2,4,3,7,1,5,6] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7) => 3

[1,0,1,1,1,0,1,0,0,1,0,0,1,0] => [1,4,5,3,6,2,7] => [2,4,3,6,1,5,7] => ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 3

[1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [1,4,5,3,6,7,2] => [2,4,3,5,7,1,6] => ([(0,6),(1,6),(2,3),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,0,1,1,1,0,1,0,0,1,1,0,0,0] => [1,4,5,3,7,6,2] => [2,3,7,4,6,1,5] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,1,0,1,0,1,0,0,0,1,0] => [1,4,5,6,3,2,7] => [2,3,6,5,1,4,7] => ([(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,1,0,1,0,1,0,0,1,0,0] => [1,4,5,6,3,7,2] => [2,3,5,4,7,1,6] => ([(0,6),(1,6),(2,3),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [1,4,5,6,7,3,2] => [2,3,4,7,6,1,5] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [1,4,5,7,6,3,2] => [7,2,3,6,5,1,4] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,0,1,1,1,0,1,1,0,0,0,0,1,0] => [1,4,6,5,3,2,7] => [6,2,5,4,1,3,7] => ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [1,4,6,5,3,7,2] => [5,2,4,3,7,1,6] => ([(0,2),(1,5),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,1,0,1,1,0,0,1,0,0,0] => [1,4,6,5,7,3,2] => [4,2,3,7,6,1,5] => ([(0,3),(0,6),(1,3),(1,6),(2,4),(2,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [1,4,6,7,5,3,2] => [3,7,2,6,5,1,4] => ([(0,2),(0,6),(1,3),(1,4),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [1,5,4,3,2,6,7] => [5,4,3,1,2,6,7] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [1,5,4,3,2,7,6] => [4,3,2,7,1,5,6] => ([(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 4

[1,0,1,1,1,1,0,0,0,1,0,0,1,0] => [1,5,4,3,6,2,7] => [4,3,2,6,1,5,7] => ([(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,1,1,0,0,0,1,0,1,0,0] => [1,5,4,3,6,7,2] => [4,3,2,5,7,1,6] => ([(0,6),(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,1,1,0,0,0,1,1,0,0,0] => [1,5,4,3,7,6,2] => [3,2,7,4,6,1,5] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,1,1,0,0,1,0,0,0,1,0] => [1,5,4,6,3,2,7] => [3,2,6,5,1,4,7] => ([(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,1,1,0,0,1,0,0,1,0,0] => [1,5,4,6,3,7,2] => [3,2,5,4,7,1,6] => ([(0,5),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7) => 3

[1,0,1,1,1,1,0,0,1,0,1,0,0,0] => [1,5,4,6,7,3,2] => [3,2,4,7,6,1,5] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,1,1,0,0,1,1,0,0,0,0] => [1,5,4,7,6,3,2] => [2,7,3,6,5,1,4] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,0,1,1,1,1,0,1,0,0,0,0,1,0] => [1,5,6,4,3,2,7] => [2,6,5,4,1,3,7] => ([(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,0,1,1,1,1,0,1,0,0,0,1,0,0] => [1,5,6,4,3,7,2] => [2,5,4,3,7,1,6] => ([(0,6),(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,1,1,0,1,0,0,1,0,0,0] => [1,5,6,4,7,3,2] => [2,4,3,7,6,1,5] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [1,5,6,7,4,3,2] => [2,3,7,6,5,1,4] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [1,6,5,4,3,2,7] => [6,5,4,3,1,2,7] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [1,6,5,4,3,7,2] => [5,4,3,2,7,1,6] => ([(0,1),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,0,1,1,1,1,1,0,0,0,1,0,0,0] => [1,6,5,4,7,3,2] => [4,3,2,7,6,1,5] => ([(0,4),(0,5),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [1,6,5,7,4,3,2] => [3,2,7,6,5,1,4] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ([(5,6)],7) => 2

[1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [2,1,3,4,5,7,6] => [1,7,2,3,4,5,6] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2

[1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [2,1,3,4,6,5,7] => [1,6,2,3,4,5,7] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2

[1,1,0,0,1,0,1,0,1,1,0,1,0,0] => [2,1,3,4,6,7,5] => [1,5,7,2,3,4,6] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 3

[1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [2,1,3,4,7,6,5] => [7,1,6,2,3,4,5] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [2,1,3,5,4,6,7] => [1,5,2,3,4,6,7] => ([(3,6),(4,6),(5,6)],7) => 2

[1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [2,1,3,5,4,7,6] => [1,4,7,2,3,5,6] => ([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 3

[1,1,0,0,1,0,1,1,0,1,0,0,1,0] => [2,1,3,5,6,4,7] => [1,4,6,2,3,5,7] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 3

[1,1,0,0,1,0,1,1,0,1,0,1,0,0] => [2,1,3,5,6,7,4] => [1,4,5,7,2,3,6] => ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 4

[1,1,0,0,1,0,1,1,0,1,1,0,0,0] => [2,1,3,5,7,6,4] => [7,1,4,6,2,3,5] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [2,1,3,6,5,4,7] => [6,1,5,2,3,4,7] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,0,0,1,0,1,1,1,0,0,1,0,0] => [2,1,3,6,5,7,4] => [5,1,4,7,2,3,6] => ([(0,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 4

[1,1,0,0,1,0,1,1,1,0,1,0,0,0] => [2,1,3,6,7,5,4] => [4,7,1,6,2,3,5] => ([(0,4),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7) => 4

[1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [2,1,3,7,6,5,4] => [7,6,1,5,2,3,4] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,3,5,6,7] => [1,4,2,3,5,6,7] => ([(4,6),(5,6)],7) => 2

[1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [2,1,4,3,5,7,6] => [1,3,7,2,4,5,6] => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 3

[1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5,7] => [1,3,6,2,4,5,7] => ([(2,6),(3,6),(4,5),(5,6)],7) => 3

[1,1,0,0,1,1,0,0,1,1,0,1,0,0] => [2,1,4,3,6,7,5] => [1,3,5,7,2,4,6] => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 4

[1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [2,1,4,3,7,6,5] => [7,1,3,6,2,4,5] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,0,1,1,0,1,0,0,1,0,1,0] => [2,1,4,5,3,6,7] => [1,3,5,2,4,6,7] => ([(3,6),(4,5),(5,6)],7) => 3

[1,1,0,0,1,1,0,1,0,0,1,1,0,0] => [2,1,4,5,3,7,6] => [1,3,4,7,2,5,6] => ([(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 3

[1,1,0,0,1,1,0,1,0,1,0,0,1,0] => [2,1,4,5,6,3,7] => [1,3,4,6,2,5,7] => ([(2,6),(3,6),(4,5),(5,6)],7) => 3

[1,1,0,0,1,1,0,1,0,1,0,1,0,0] => [2,1,4,5,6,7,3] => [1,3,4,5,7,2,6] => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 3

[1,1,0,0,1,1,0,1,0,1,1,0,0,0] => [2,1,4,5,7,6,3] => [7,1,3,4,6,2,5] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,0,1,1,0,1,1,0,0,0,1,0] => [2,1,4,6,5,3,7] => [6,1,3,5,2,4,7] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,0,1,1,0,1,1,0,0,1,0,0] => [2,1,4,6,5,7,3] => [5,1,3,4,7,2,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(5,6)],7) => 3

[1,1,0,0,1,1,0,1,1,0,1,0,0,0] => [2,1,4,6,7,5,3] => [4,7,1,3,6,2,5] => ([(0,3),(0,6),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,0,1,1,0,1,1,1,0,0,0,0] => [2,1,4,7,6,5,3] => [7,6,1,3,5,2,4] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [2,1,5,4,3,6,7] => [5,1,4,2,3,6,7] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [2,1,5,4,3,7,6] => [4,1,3,7,2,5,6] => ([(0,4),(1,4),(2,5),(3,5),(3,6),(4,6),(5,6)],7) => 3

[1,1,0,0,1,1,1,0,0,1,0,0,1,0] => [2,1,5,4,6,3,7] => [4,1,3,6,2,5,7] => ([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,0,0,1,1,1,0,0,1,0,1,0,0] => [2,1,5,4,6,7,3] => [4,1,3,5,7,2,6] => ([(0,6),(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,0,0,1,1,1,0,0,1,1,0,0,0] => [2,1,5,4,7,6,3] => [3,7,1,4,6,2,5] => ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,0,1,1,1,0,1,0,0,0,1,0] => [2,1,5,6,4,3,7] => [3,6,1,5,2,4,7] => ([(1,2),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,0,1,1,1,0,1,0,0,1,0,0] => [2,1,5,6,4,7,3] => [3,5,1,4,7,2,6] => ([(0,4),(1,2),(1,6),(2,5),(3,5),(3,6),(4,6),(5,6)],7) => 4

[1,1,0,0,1,1,1,0,1,0,1,0,0,0] => [2,1,5,6,7,4,3] => [3,4,7,1,6,2,5] => ([(0,4),(0,6),(1,3),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,0,1,1,1,0,1,1,0,0,0,0] => [2,1,5,7,6,4,3] => [7,3,6,1,5,2,4] => ([(0,1),(0,5),(0,6),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [2,1,6,5,4,3,7] => [6,5,1,4,2,3,7] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,0,1,1,1,1,0,0,0,1,0,0] => [2,1,6,5,4,7,3] => [5,4,1,3,7,2,6] => ([(0,2),(1,5),(1,6),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,0,1,1,1,1,0,0,1,0,0,0] => [2,1,6,5,7,4,3] => [4,3,7,1,6,2,5] => ([(0,4),(0,6),(1,2),(1,3),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6),(5,6)],7) => 5

[1,1,0,0,1,1,1,1,0,1,0,0,0,0] => [2,1,6,7,5,4,3] => [3,7,6,1,5,2,4] => ([(0,2),(0,4),(1,3),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [2,1,7,6,5,4,3] => [7,6,5,1,4,2,3] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,1,0,1,0,0,1,0,1,0,1,0,1,0] => [2,3,1,4,5,6,7] => [1,3,2,4,5,6,7] => ([(5,6)],7) => 2

[1,1,0,1,0,0,1,0,1,0,1,1,0,0] => [2,3,1,4,5,7,6] => [1,2,7,3,4,5,6] => ([(2,6),(3,6),(4,6),(5,6)],7) => 2

[1,1,0,1,0,0,1,0,1,1,0,0,1,0] => [2,3,1,4,6,5,7] => [1,2,6,3,4,5,7] => ([(3,6),(4,6),(5,6)],7) => 2

[1,1,0,1,0,0,1,0,1,1,0,1,0,0] => [2,3,1,4,6,7,5] => [1,2,5,7,3,4,6] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 3

[1,1,0,1,0,0,1,0,1,1,1,0,0,0] => [2,3,1,4,7,6,5] => [7,1,2,6,3,4,5] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,0,1,0,0,1,1,0,0,1,0,1,0] => [2,3,1,5,4,6,7] => [1,2,5,3,4,6,7] => ([(4,6),(5,6)],7) => 2

[1,1,0,1,0,0,1,1,0,0,1,1,0,0] => [2,3,1,5,4,7,6] => [1,2,4,7,3,5,6] => ([(2,6),(3,6),(4,5),(5,6)],7) => 3

[1,1,0,1,0,0,1,1,0,1,0,0,1,0] => [2,3,1,5,6,4,7] => [1,2,4,6,3,5,7] => ([(3,6),(4,5),(5,6)],7) => 3

[1,1,0,1,0,0,1,1,0,1,0,1,0,0] => [2,3,1,5,6,7,4] => [1,2,4,5,7,3,6] => ([(2,6),(3,6),(4,5),(5,6)],7) => 3

[1,1,0,1,0,0,1,1,0,1,1,0,0,0] => [2,3,1,5,7,6,4] => [7,1,2,4,6,3,5] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,1,0,0,1,1,1,0,0,0,1,0] => [2,3,1,6,5,4,7] => [6,1,2,5,3,4,7] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,0,1,0,0,1,1,1,0,0,1,0,0] => [2,3,1,6,5,7,4] => [5,1,2,4,7,3,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6),(5,6)],7) => 3

[1,1,0,1,0,0,1,1,1,0,1,0,0,0] => [2,3,1,6,7,5,4] => [4,7,1,2,6,3,5] => ([(0,3),(0,6),(1,3),(1,6),(2,4),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,1,0,0,1,1,1,1,0,0,0,0] => [2,3,1,7,6,5,4] => [7,6,1,2,5,3,4] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,1,0,1,0,0,1,0,1,0,1,0] => [2,3,4,1,5,6,7] => [1,2,4,3,5,6,7] => ([(5,6)],7) => 2

[1,1,0,1,0,1,0,0,1,0,1,1,0,0] => [2,3,4,1,5,7,6] => [1,2,3,7,4,5,6] => ([(3,6),(4,6),(5,6)],7) => 2

[1,1,0,1,0,1,0,0,1,1,0,0,1,0] => [2,3,4,1,6,5,7] => [1,2,3,6,4,5,7] => ([(4,6),(5,6)],7) => 2

[1,1,0,1,0,1,0,0,1,1,0,1,0,0] => [2,3,4,1,6,7,5] => [1,2,3,5,7,4,6] => ([(3,6),(4,5),(5,6)],7) => 3

[1,1,0,1,0,1,0,0,1,1,1,0,0,0] => [2,3,4,1,7,6,5] => [7,1,2,3,6,4,5] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,0,1,0,1,0,1,0,0,1,0,1,0] => [2,3,4,5,1,6,7] => [1,2,3,5,4,6,7] => ([(5,6)],7) => 2

[1,1,0,1,0,1,0,1,0,0,1,1,0,0] => [2,3,4,5,1,7,6] => [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7) => 2

[1,1,0,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,6,1,7] => [1,2,3,4,6,5,7] => ([(5,6)],7) => 2

[1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,7,1] => [1,2,3,4,5,7,6] => ([(5,6)],7) => 2

[1,1,0,1,0,1,0,1,0,1,1,0,0,0] => [2,3,4,5,7,6,1] => [7,1,2,3,4,6,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,0,1,0,1,0,1,1,0,0,0,1,0] => [2,3,4,6,5,1,7] => [6,1,2,3,5,4,7] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,0,1,0,1,0,1,1,0,0,1,0,0] => [2,3,4,6,5,7,1] => [5,1,2,3,4,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => 2

[1,1,0,1,0,1,0,1,1,0,1,0,0,0] => [2,3,4,6,7,5,1] => [4,7,1,2,3,6,5] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 3

[1,1,0,1,0,1,0,1,1,1,0,0,0,0] => [2,3,4,7,6,5,1] => [7,6,1,2,3,5,4] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,1,0,1,1,0,0,0,1,0,1,0] => [2,3,5,4,1,6,7] => [5,1,2,4,3,6,7] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,0,1,0,1,1,0,0,0,1,1,0,0] => [2,3,5,4,1,7,6] => [4,1,2,3,7,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7) => 2

[1,1,0,1,0,1,1,0,0,1,0,0,1,0] => [2,3,5,4,6,1,7] => [4,1,2,3,6,5,7] => ([(1,2),(3,6),(4,6),(5,6)],7) => 2

[1,1,0,1,0,1,1,0,0,1,0,1,0,0] => [2,3,5,4,6,7,1] => [4,1,2,3,5,7,6] => ([(1,2),(3,6),(4,6),(5,6)],7) => 2

[1,1,0,1,0,1,1,0,0,1,1,0,0,0] => [2,3,5,4,7,6,1] => [3,7,1,2,4,6,5] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7) => 3

[1,1,0,1,0,1,1,0,1,0,0,0,1,0] => [2,3,5,6,4,1,7] => [3,6,1,2,5,4,7] => ([(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7) => 3

[1,1,0,1,0,1,1,0,1,0,0,1,0,0] => [2,3,5,6,4,7,1] => [3,5,1,2,4,7,6] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 3

[1,1,0,1,0,1,1,0,1,0,1,0,0,0] => [2,3,5,6,7,4,1] => [3,4,7,1,2,6,5] => ([(0,1),(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 4

[1,1,0,1,0,1,1,0,1,1,0,0,0,0] => [2,3,5,7,6,4,1] => [7,3,6,1,2,5,4] => ([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,1,0,1,1,1,0,0,0,0,1,0] => [2,3,6,5,4,1,7] => [6,5,1,2,4,3,7] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,1,0,1,1,1,0,0,0,1,0,0] => [2,3,6,5,4,7,1] => [5,4,1,2,3,7,6] => ([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,0,1,0,1,1,1,0,0,1,0,0,0] => [2,3,6,5,7,4,1] => [4,3,7,1,2,6,5] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 4

[1,1,0,1,0,1,1,1,0,1,0,0,0,0] => [2,3,6,7,5,4,1] => [3,7,6,1,2,5,4] => ([(0,3),(0,4),(1,2),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,1,0,1,1,1,1,0,0,0,0,0] => [2,3,7,6,5,4,1] => [7,6,5,1,2,4,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,1,0,1,1,0,0,0,1,0,1,0,1,0] => [2,4,3,1,5,6,7] => [4,1,3,2,5,6,7] => ([(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,0,1,1,0,0,0,1,0,1,1,0,0] => [2,4,3,1,5,7,6] => [3,1,2,7,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7) => 2

[1,1,0,1,1,0,0,0,1,1,0,0,1,0] => [2,4,3,1,6,5,7] => [3,1,2,6,4,5,7] => ([(1,6),(2,6),(3,5),(4,5)],7) => 2

[1,1,0,1,1,0,0,0,1,1,0,1,0,0] => [2,4,3,1,6,7,5] => [3,1,2,5,7,4,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7) => 3

[1,1,0,1,1,0,0,0,1,1,1,0,0,0] => [2,4,3,1,7,6,5] => [2,7,1,3,6,4,5] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => 3

[1,1,0,1,1,0,0,1,0,0,1,0,1,0] => [2,4,3,5,1,6,7] => [3,1,2,5,4,6,7] => ([(2,3),(4,6),(5,6)],7) => 2

[1,1,0,1,1,0,0,1,0,0,1,1,0,0] => [2,4,3,5,1,7,6] => [3,1,2,4,7,5,6] => ([(1,6),(2,6),(3,5),(4,5)],7) => 2

[1,1,0,1,1,0,0,1,0,1,0,0,1,0] => [2,4,3,5,6,1,7] => [3,1,2,4,6,5,7] => ([(2,3),(4,6),(5,6)],7) => 2

[1,1,0,1,1,0,0,1,0,1,0,1,0,0] => [2,4,3,5,6,7,1] => [3,1,2,4,5,7,6] => ([(2,3),(4,6),(5,6)],7) => 2

[1,1,0,1,1,0,0,1,0,1,1,0,0,0] => [2,4,3,5,7,6,1] => [2,7,1,3,4,6,5] => ([(0,6),(1,6),(2,3),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,0,1,1,0,0,1,1,0,0,0,1,0] => [2,4,3,6,5,1,7] => [2,6,1,3,5,4,7] => ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 3

[1,1,0,1,1,0,0,1,1,0,0,1,0,0] => [2,4,3,6,5,7,1] => [2,5,1,3,4,7,6] => ([(0,1),(2,6),(3,6),(4,5),(5,6)],7) => 3

[1,1,0,1,1,0,0,1,1,0,1,0,0,0] => [2,4,3,6,7,5,1] => [2,4,7,1,3,6,5] => ([(0,5),(1,2),(1,6),(2,6),(3,4),(3,6),(4,5),(5,6)],7) => 4

[1,1,0,1,1,0,0,1,1,1,0,0,0,0] => [2,4,3,7,6,5,1] => [7,2,6,1,3,5,4] => ([(0,4),(0,6),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,1,1,0,1,0,0,0,1,0,1,0] => [2,4,5,3,1,6,7] => [2,5,1,4,3,6,7] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7) => 3

[1,1,0,1,1,0,1,0,0,0,1,1,0,0] => [2,4,5,3,1,7,6] => [2,4,1,3,7,5,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7) => 3

[1,1,0,1,1,0,1,0,0,1,0,0,1,0] => [2,4,5,3,6,1,7] => [2,4,1,3,6,5,7] => ([(1,2),(3,6),(4,5),(5,6)],7) => 3

[1,1,0,1,1,0,1,0,0,1,0,1,0,0] => [2,4,5,3,6,7,1] => [2,4,1,3,5,7,6] => ([(1,2),(3,6),(4,5),(5,6)],7) => 3

[1,1,0,1,1,0,1,0,0,1,1,0,0,0] => [2,4,5,3,7,6,1] => [2,3,7,1,4,6,5] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7) => 3

[1,1,0,1,1,0,1,0,1,0,0,0,1,0] => [2,4,5,6,3,1,7] => [2,3,6,1,5,4,7] => ([(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7) => 3

[1,1,0,1,1,0,1,0,1,0,0,1,0,0] => [2,4,5,6,3,7,1] => [2,3,5,1,4,7,6] => ([(0,1),(2,6),(3,6),(4,5),(5,6)],7) => 3

[1,1,0,1,1,0,1,0,1,0,1,0,0,0] => [2,4,5,6,7,3,1] => [2,3,4,7,1,6,5] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7) => 3

[1,1,0,1,1,0,1,0,1,1,0,0,0,0] => [2,4,5,7,6,3,1] => [7,2,3,6,1,5,4] => ([(0,4),(0,6),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,1,1,0,1,1,0,0,0,0,1,0] => [2,4,6,5,3,1,7] => [6,2,5,1,4,3,7] => ([(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,1,1,0,1,1,0,0,0,1,0,0] => [2,4,6,5,3,7,1] => [5,2,4,1,3,7,6] => ([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,1,1,0,1,1,0,0,1,0,0,0] => [2,4,6,5,7,3,1] => [4,2,3,7,1,6,5] => ([(0,1),(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => 4

[1,1,0,1,1,0,1,1,0,1,0,0,0,0] => [2,4,6,7,5,3,1] => [3,7,2,6,1,5,4] => ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,1,0,1,1,0,1,1,1,0,0,0,0,0] => [2,4,7,6,5,3,1] => [7,6,2,5,1,4,3] => ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,1,0,1,1,1,0,0,0,0,1,0,1,0] => [2,5,4,3,1,6,7] => [5,4,1,3,2,6,7] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,1,1,1,0,0,0,0,1,1,0,0] => [2,5,4,3,1,7,6] => [4,3,1,2,7,5,6] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7) => 3

[1,1,0,1,1,1,0,0,0,1,0,0,1,0] => [2,5,4,3,6,1,7] => [4,3,1,2,6,5,7] => ([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,0,1,1,1,0,0,0,1,0,1,0,0] => [2,5,4,3,6,7,1] => [4,3,1,2,5,7,6] => ([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,0,1,1,1,0,0,0,1,1,0,0,0] => [2,5,4,3,7,6,1] => [3,2,7,1,4,6,5] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7) => 4

[1,1,0,1,1,1,0,0,1,0,0,0,1,0] => [2,5,4,6,3,1,7] => [3,2,6,1,5,4,7] => ([(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7) => 4

[1,1,0,1,1,1,0,0,1,0,0,1,0,0] => [2,5,4,6,3,7,1] => [3,2,5,1,4,7,6] => ([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,0,1,1,1,0,0,1,0,1,0,0,0] => [2,5,4,6,7,3,1] => [3,2,4,7,1,6,5] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7) => 4

[1,1,0,1,1,1,0,0,1,1,0,0,0,0] => [2,5,4,7,6,3,1] => [2,7,3,6,1,5,4] => ([(0,5),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,1,1,1,0,1,0,0,0,0,1,0] => [2,5,6,4,3,1,7] => [2,6,5,1,4,3,7] => ([(1,4),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,1,1,1,0,1,0,0,0,1,0,0] => [2,5,6,4,3,7,1] => [2,5,4,1,3,7,6] => ([(0,1),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,1,1,1,0,1,0,0,1,0,0,0] => [2,5,6,4,7,3,1] => [2,4,3,7,1,6,5] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7) => 4

[1,1,0,1,1,1,0,1,0,1,0,0,0,0] => [2,5,6,7,4,3,1] => [2,3,7,6,1,5,4] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,1,1,1,0,1,1,0,0,0,0,0] => [2,5,7,6,4,3,1] => [7,2,6,5,1,4,3] => ([(0,3),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,1,0,1,1,1,1,0,0,0,0,0,1,0] => [2,6,5,4,3,1,7] => [6,5,4,1,3,2,7] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,1,0,1,1,1,1,0,0,0,0,1,0,0] => [2,6,5,4,3,7,1] => [5,4,3,1,2,7,6] => ([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,0,1,1,1,1,0,0,0,1,0,0,0] => [2,6,5,4,7,3,1] => [4,3,2,7,1,6,5] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 4

[1,1,0,1,1,1,1,0,0,1,0,0,0,0] => [2,6,5,7,4,3,1] => [3,2,7,6,1,5,4] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 5

[1,1,0,1,1,1,1,0,1,0,0,0,0,0] => [2,6,7,5,4,3,1] => [2,7,6,5,1,4,3] => ([(0,3),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => ([(4,5),(4,6),(5,6)],7) => 3

[1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [3,2,1,4,5,7,6] => [2,1,7,3,4,5,6] => ([(0,1),(2,6),(3,6),(4,6),(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] => [2,1,6,3,4,5,7] => ([(1,2),(3,6),(4,6),(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] => [2,1,5,7,3,4,6] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 3

[1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [3,2,1,4,7,6,5] => [1,7,2,6,3,4,5] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [3,2,1,5,4,6,7] => [2,1,5,3,4,6,7] => ([(2,3),(4,6),(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] => [2,1,4,7,3,5,6] => ([(0,1),(2,6),(3,6),(4,5),(5,6)],7) => 3

[1,1,1,0,0,0,1,1,0,1,0,0,1,0] => [3,2,1,5,6,4,7] => [2,1,4,6,3,5,7] => ([(1,2),(3,6),(4,5),(5,6)],7) => 3

[1,1,1,0,0,0,1,1,0,1,0,1,0,0] => [3,2,1,5,6,7,4] => [2,1,4,5,7,3,6] => ([(0,1),(2,6),(3,6),(4,5),(5,6)],7) => 3

[1,1,1,0,0,0,1,1,0,1,1,0,0,0] => [3,2,1,5,7,6,4] => [1,7,2,4,6,3,5] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [3,2,1,6,5,4,7] => [1,6,2,5,3,4,7] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,0,0,0,1,1,1,0,0,1,0,0] => [3,2,1,6,5,7,4] => [1,5,2,4,7,3,6] => ([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,0,0,0,1,1,1,0,1,0,0,0] => [3,2,1,6,7,5,4] => [1,4,7,2,6,3,5] => ([(1,2),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [3,2,1,7,6,5,4] => [7,1,6,2,5,3,4] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,0,0,1,0,0,1,0,1,0,1,0] => [3,2,4,1,5,6,7] => [2,1,4,3,5,6,7] => ([(3,6),(4,5)],7) => 2

[1,1,1,0,0,1,0,0,1,0,1,1,0,0] => [3,2,4,1,5,7,6] => [2,1,3,7,4,5,6] => ([(1,2),(3,6),(4,6),(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] => [2,1,3,6,4,5,7] => ([(2,3),(4,6),(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] => [2,1,3,5,7,4,6] => ([(1,2),(3,6),(4,5),(5,6)],7) => 3

[1,1,1,0,0,1,0,0,1,1,1,0,0,0] => [3,2,4,1,7,6,5] => [1,7,2,3,6,4,5] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,0,0,1,0,1,0,0,1,0,1,0] => [3,2,4,5,1,6,7] => [2,1,3,5,4,6,7] => ([(3,6),(4,5)],7) => 2

[1,1,1,0,0,1,0,1,0,0,1,1,0,0] => [3,2,4,5,1,7,6] => [2,1,3,4,7,5,6] => ([(2,3),(4,6),(5,6)],7) => 2

[1,1,1,0,0,1,0,1,0,1,0,0,1,0] => [3,2,4,5,6,1,7] => [2,1,3,4,6,5,7] => ([(3,6),(4,5)],7) => 2

[1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [3,2,4,5,6,7,1] => [2,1,3,4,5,7,6] => ([(3,6),(4,5)],7) => 2

[1,1,1,0,0,1,0,1,0,1,1,0,0,0] => [3,2,4,5,7,6,1] => [1,7,2,3,4,6,5] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,0,0,1,0,1,1,0,0,0,1,0] => [3,2,4,6,5,1,7] => [1,6,2,3,5,4,7] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,0,0,1,0,1,1,0,0,1,0,0] => [3,2,4,6,5,7,1] => [1,5,2,3,4,7,6] => ([(1,2),(3,6),(4,6),(5,6)],7) => 2

[1,1,1,0,0,1,0,1,1,0,1,0,0,0] => [3,2,4,6,7,5,1] => [1,4,7,2,3,6,5] => ([(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7) => 3

[1,1,1,0,0,1,0,1,1,1,0,0,0,0] => [3,2,4,7,6,5,1] => [7,1,6,2,3,5,4] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,0,0,1,1,0,0,0,1,0,1,0] => [3,2,5,4,1,6,7] => [1,5,2,4,3,6,7] => ([(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,0,0,1,1,0,0,0,1,1,0,0] => [3,2,5,4,1,7,6] => [1,4,2,3,7,5,6] => ([(1,6),(2,6),(3,5),(4,5)],7) => 2

[1,1,1,0,0,1,1,0,0,1,0,0,1,0] => [3,2,5,4,6,1,7] => [1,4,2,3,6,5,7] => ([(2,3),(4,6),(5,6)],7) => 2

[1,1,1,0,0,1,1,0,0,1,0,1,0,0] => [3,2,5,4,6,7,1] => [1,4,2,3,5,7,6] => ([(2,3),(4,6),(5,6)],7) => 2

[1,1,1,0,0,1,1,0,0,1,1,0,0,0] => [3,2,5,4,7,6,1] => [1,3,7,2,4,6,5] => ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 3

[1,1,1,0,0,1,1,0,1,0,0,0,1,0] => [3,2,5,6,4,1,7] => [1,3,6,2,5,4,7] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7) => 3

[1,1,1,0,0,1,1,0,1,0,0,1,0,0] => [3,2,5,6,4,7,1] => [1,3,5,2,4,7,6] => ([(1,2),(3,6),(4,5),(5,6)],7) => 3

[1,1,1,0,0,1,1,0,1,0,1,0,0,0] => [3,2,5,6,7,4,1] => [1,3,4,7,2,6,5] => ([(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7) => 3

[1,1,1,0,0,1,1,0,1,1,0,0,0,0] => [3,2,5,7,6,4,1] => [7,1,3,6,2,5,4] => ([(0,6),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,0,0,1,1,1,0,0,0,0,1,0] => [3,2,6,5,4,1,7] => [6,1,5,2,4,3,7] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,0,0,1,1,1,0,0,0,1,0,0] => [3,2,6,5,4,7,1] => [5,1,4,2,3,7,6] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,0,0,1,1,1,0,0,1,0,0,0] => [3,2,6,5,7,4,1] => [4,1,3,7,2,6,5] => ([(0,6),(1,2),(1,4),(2,4),(3,5),(3,6),(4,5),(5,6)],7) => 4

[1,1,1,0,0,1,1,1,0,1,0,0,0,0] => [3,2,6,7,5,4,1] => [3,7,1,6,2,5,4] => ([(0,1),(0,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,0,0,1,1,1,1,0,0,0,0,0] => [3,2,7,6,5,4,1] => [7,6,1,5,2,4,3] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,1,1,0,1,0,0,0,1,0,1,0,1,0] => [3,4,2,1,5,6,7] => [1,4,3,2,5,6,7] => ([(4,5),(4,6),(5,6)],7) => 3

[1,1,1,0,1,0,0,0,1,0,1,1,0,0] => [3,4,2,1,5,7,6] => [1,3,2,7,4,5,6] => ([(1,2),(3,6),(4,6),(5,6)],7) => 2

[1,1,1,0,1,0,0,0,1,1,0,0,1,0] => [3,4,2,1,6,5,7] => [1,3,2,6,4,5,7] => ([(2,3),(4,6),(5,6)],7) => 2

[1,1,1,0,1,0,0,0,1,1,0,1,0,0] => [3,4,2,1,6,7,5] => [1,3,2,5,7,4,6] => ([(1,2),(3,6),(4,5),(5,6)],7) => 3

[1,1,1,0,1,0,0,0,1,1,1,0,0,0] => [3,4,2,1,7,6,5] => [1,2,7,3,6,4,5] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,0,1,0,0,1,0,0,1,0,1,0] => [3,4,2,5,1,6,7] => [1,3,2,5,4,6,7] => ([(3,6),(4,5)],7) => 2

[1,1,1,0,1,0,0,1,0,0,1,1,0,0] => [3,4,2,5,1,7,6] => [1,3,2,4,7,5,6] => ([(2,3),(4,6),(5,6)],7) => 2

[1,1,1,0,1,0,0,1,0,1,0,0,1,0] => [3,4,2,5,6,1,7] => [1,3,2,4,6,5,7] => ([(3,6),(4,5)],7) => 2

[1,1,1,0,1,0,0,1,0,1,0,1,0,0] => [3,4,2,5,6,7,1] => [1,3,2,4,5,7,6] => ([(3,6),(4,5)],7) => 2

[1,1,1,0,1,0,0,1,0,1,1,0,0,0] => [3,4,2,5,7,6,1] => [1,2,7,3,4,6,5] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,0,1,0,0,1,1,0,0,0,1,0] => [3,4,2,6,5,1,7] => [1,2,6,3,5,4,7] => ([(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,0,1,0,0,1,1,0,0,1,0,0] => [3,4,2,6,5,7,1] => [1,2,5,3,4,7,6] => ([(2,3),(4,6),(5,6)],7) => 2

[1,1,1,0,1,0,0,1,1,0,1,0,0,0] => [3,4,2,6,7,5,1] => [1,2,4,7,3,6,5] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7) => 3

[1,1,1,0,1,0,0,1,1,1,0,0,0,0] => [3,4,2,7,6,5,1] => [7,1,2,6,3,5,4] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,0,1,0,1,0,0,0,1,0,1,0] => [3,4,5,2,1,6,7] => [1,2,5,4,3,6,7] => ([(4,5),(4,6),(5,6)],7) => 3

[1,1,1,0,1,0,1,0,0,0,1,1,0,0] => [3,4,5,2,1,7,6] => [1,2,4,3,7,5,6] => ([(2,3),(4,6),(5,6)],7) => 2

[1,1,1,0,1,0,1,0,0,1,0,0,1,0] => [3,4,5,2,6,1,7] => [1,2,4,3,6,5,7] => ([(3,6),(4,5)],7) => 2

[1,1,1,0,1,0,1,0,0,1,0,1,0,0] => [3,4,5,2,6,7,1] => [1,2,4,3,5,7,6] => ([(3,6),(4,5)],7) => 2

[1,1,1,0,1,0,1,0,0,1,1,0,0,0] => [3,4,5,2,7,6,1] => [1,2,3,7,4,6,5] => ([(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,0,1,0,1,0,1,0,0,0,1,0] => [3,4,5,6,2,1,7] => [1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7) => 3

[1,1,1,0,1,0,1,0,1,0,0,1,0,0] => [3,4,5,6,2,7,1] => [1,2,3,5,4,7,6] => ([(3,6),(4,5)],7) => 2

[1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [3,4,5,6,7,2,1] => [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7) => 3

[1,1,1,0,1,0,1,0,1,1,0,0,0,0] => [3,4,5,7,6,2,1] => [7,1,2,3,6,5,4] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,0,1,0,1,1,0,0,0,0,1,0] => [3,4,6,5,2,1,7] => [6,1,2,5,4,3,7] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,0,1,0,1,1,0,0,0,1,0,0] => [3,4,6,5,2,7,1] => [5,1,2,4,3,7,6] => ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,0,1,0,1,1,0,0,1,0,0,0] => [3,4,6,5,7,2,1] => [4,1,2,3,7,6,5] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7) => 3

[1,1,1,0,1,0,1,1,0,1,0,0,0,0] => [3,4,6,7,5,2,1] => [3,7,1,2,6,5,4] => ([(0,1),(0,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [3,4,7,6,5,2,1] => [7,6,1,2,5,4,3] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,1,1,0,1,1,0,0,0,0,1,0,1,0] => [3,5,4,2,1,6,7] => [5,1,4,3,2,6,7] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,0,1,1,0,0,0,0,1,1,0,0] => [3,5,4,2,1,7,6] => [4,1,3,2,7,5,6] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,0,1,1,0,0,0,1,0,0,1,0] => [3,5,4,2,6,1,7] => [4,1,3,2,6,5,7] => ([(1,2),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,0,1,1,0,0,0,1,0,1,0,0] => [3,5,4,2,6,7,1] => [4,1,3,2,5,7,6] => ([(1,2),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,0,1,1,0,0,0,1,1,0,0,0] => [3,5,4,2,7,6,1] => [3,1,2,7,4,6,5] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,0,1,1,0,0,1,0,0,0,1,0] => [3,5,4,6,2,1,7] => [3,1,2,6,5,4,7] => ([(1,6),(2,6),(3,4),(3,5),(4,5)],7) => 3

[1,1,1,0,1,1,0,0,1,0,0,1,0,0] => [3,5,4,6,2,7,1] => [3,1,2,5,4,7,6] => ([(0,3),(1,2),(4,6),(5,6)],7) => 2

[1,1,1,0,1,1,0,0,1,0,1,0,0,0] => [3,5,4,6,7,2,1] => [3,1,2,4,7,6,5] => ([(1,6),(2,6),(3,4),(3,5),(4,5)],7) => 3

[1,1,1,0,1,1,0,0,1,1,0,0,0,0] => [3,5,4,7,6,2,1] => [2,7,1,3,6,5,4] => ([(0,6),(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,0,1,1,0,1,0,0,0,0,1,0] => [3,5,6,4,2,1,7] => [2,6,1,5,4,3,7] => ([(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,0,1,1,0,1,0,0,0,1,0,0] => [3,5,6,4,2,7,1] => [2,5,1,4,3,7,6] => ([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,0,1,1,0,1,0,0,1,0,0,0] => [3,5,6,4,7,2,1] => [2,4,1,3,7,6,5] => ([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7) => 3

[1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [3,5,6,7,4,2,1] => [2,3,7,1,6,5,4] => ([(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 4

[1,1,1,0,1,1,0,1,1,0,0,0,0,0] => [3,5,7,6,4,2,1] => [7,2,6,1,5,4,3] => ([(0,1),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,1,1,0,1,1,1,0,0,0,0,0,1,0] => [3,6,5,4,2,1,7] => [6,5,1,4,3,2,7] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,1,1,0,1,1,1,0,0,0,0,1,0,0] => [3,6,5,4,2,7,1] => [5,4,1,3,2,7,6] => ([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,0,1,1,1,0,0,0,1,0,0,0] => [3,6,5,4,7,2,1] => [4,3,1,2,7,6,5] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [3,6,5,7,4,2,1] => [3,2,7,1,6,5,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 4

[1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [3,6,7,5,4,2,1] => [2,7,6,1,5,4,3] => ([(0,1),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [4,3,2,1,5,7,6] => [3,2,1,7,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7) => 3

[1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [4,3,2,1,6,5,7] => [3,2,1,6,4,5,7] => ([(1,6),(2,6),(3,4),(3,5),(4,5)],7) => 3

[1,1,1,1,0,0,0,0,1,1,0,1,0,0] => [4,3,2,1,6,7,5] => [3,2,1,5,7,4,6] => ([(0,6),(1,5),(2,3),(2,4),(3,4),(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] => [2,1,7,3,6,4,5] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,1,0,0,0,1,0,0,1,0,1,0] => [4,3,2,5,1,6,7] => [3,2,1,5,4,6,7] => ([(2,3),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,1,0,0,0,1,0,0,1,1,0,0] => [4,3,2,5,1,7,6] => [3,2,1,4,7,5,6] => ([(1,6),(2,6),(3,4),(3,5),(4,5)],7) => 3

[1,1,1,1,0,0,0,1,0,1,0,0,1,0] => [4,3,2,5,6,1,7] => [3,2,1,4,6,5,7] => ([(2,3),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [4,3,2,5,6,7,1] => [3,2,1,4,5,7,6] => ([(2,3),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,1,0,0,0,1,0,1,1,0,0,0] => [4,3,2,5,7,6,1] => [2,1,7,3,4,6,5] => ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,1,0,0,0,1,1,0,0,0,1,0] => [4,3,2,6,5,1,7] => [2,1,6,3,5,4,7] => ([(1,2),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,1,0,0,0,1,1,0,0,1,0,0] => [4,3,2,6,5,7,1] => [2,1,5,3,4,7,6] => ([(0,3),(1,2),(4,6),(5,6)],7) => 2

[1,1,1,1,0,0,0,1,1,0,1,0,0,0] => [4,3,2,6,7,5,1] => [2,1,4,7,3,6,5] => ([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,1,0,0,0,1,1,1,0,0,0,0] => [4,3,2,7,6,5,1] => [1,7,2,6,3,5,4] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,1,0,0,1,0,0,0,1,0,1,0] => [4,3,5,2,1,6,7] => [2,1,5,4,3,6,7] => ([(2,3),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,1,0,0,1,0,0,0,1,1,0,0] => [4,3,5,2,1,7,6] => [2,1,4,3,7,5,6] => ([(0,3),(1,2),(4,6),(5,6)],7) => 2

[1,1,1,1,0,0,1,0,0,1,0,0,1,0] => [4,3,5,2,6,1,7] => [2,1,4,3,6,5,7] => ([(1,6),(2,5),(3,4)],7) => 2

[1,1,1,1,0,0,1,0,0,1,0,1,0,0] => [4,3,5,2,6,7,1] => [2,1,4,3,5,7,6] => ([(1,6),(2,5),(3,4)],7) => 2

[1,1,1,1,0,0,1,0,0,1,1,0,0,0] => [4,3,5,2,7,6,1] => [2,1,3,7,4,6,5] => ([(1,2),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,1,0,0,1,0,1,0,0,0,1,0] => [4,3,5,6,2,1,7] => [2,1,3,6,5,4,7] => ([(2,3),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,1,0,0,1,0,1,0,0,1,0,0] => [4,3,5,6,2,7,1] => [2,1,3,5,4,7,6] => ([(1,6),(2,5),(3,4)],7) => 2

[1,1,1,1,0,0,1,0,1,0,1,0,0,0] => [4,3,5,6,7,2,1] => [2,1,3,4,7,6,5] => ([(2,3),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,1,0,0,1,0,1,1,0,0,0,0] => [4,3,5,7,6,2,1] => [1,7,2,3,6,5,4] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,1,0,0,1,1,0,0,0,0,1,0] => [4,3,6,5,2,1,7] => [1,6,2,5,4,3,7] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,1,0,0,1,1,0,0,0,1,0,0] => [4,3,6,5,2,7,1] => [1,5,2,4,3,7,6] => ([(1,2),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,1,0,0,1,1,0,0,1,0,0,0] => [4,3,6,5,7,2,1] => [1,4,2,3,7,6,5] => ([(1,6),(2,6),(3,4),(3,5),(4,5)],7) => 3

[1,1,1,1,0,0,1,1,0,1,0,0,0,0] => [4,3,6,7,5,2,1] => [1,3,7,2,6,5,4] => ([(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,1,0,0,1,1,1,0,0,0,0,0] => [4,3,7,6,5,2,1] => [7,1,6,2,5,4,3] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,1,1,1,0,1,0,0,0,0,1,0,1,0] => [4,5,3,2,1,6,7] => [1,5,4,3,2,6,7] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,1,0,1,0,0,0,0,1,1,0,0] => [4,5,3,2,1,7,6] => [1,4,3,2,7,5,6] => ([(1,6),(2,6),(3,4),(3,5),(4,5)],7) => 3

[1,1,1,1,0,1,0,0,0,1,0,0,1,0] => [4,5,3,2,6,1,7] => [1,4,3,2,6,5,7] => ([(2,3),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,1,0,1,0,0,0,1,0,1,0,0] => [4,5,3,2,6,7,1] => [1,4,3,2,5,7,6] => ([(2,3),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,1,0,1,0,0,0,1,1,0,0,0] => [4,5,3,2,7,6,1] => [1,3,2,7,4,6,5] => ([(1,2),(3,6),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,1,0,1,0,0,1,0,0,0,1,0] => [4,5,3,6,2,1,7] => [1,3,2,6,5,4,7] => ([(2,3),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,1,0,1,0,0,1,0,0,1,0,0] => [4,5,3,6,2,7,1] => [1,3,2,5,4,7,6] => ([(1,6),(2,5),(3,4)],7) => 2

[1,1,1,1,0,1,0,0,1,0,1,0,0,0] => [4,5,3,6,7,2,1] => [1,3,2,4,7,6,5] => ([(2,3),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,1,0,1,0,0,1,1,0,0,0,0] => [4,5,3,7,6,2,1] => [1,2,7,3,6,5,4] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,1,0,1,0,1,0,0,0,0,1,0] => [4,5,6,3,2,1,7] => [1,2,6,5,4,3,7] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,1,0,1,0,1,0,0,0,1,0,0] => [4,5,6,3,2,7,1] => [1,2,5,4,3,7,6] => ([(2,3),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,1,0,1,0,1,0,0,1,0,0,0] => [4,5,6,3,7,2,1] => [1,2,4,3,7,6,5] => ([(2,3),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [4,5,6,7,3,2,1] => [1,2,3,7,6,5,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [4,5,7,6,3,2,1] => [7,1,2,6,5,4,3] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [4,6,5,3,2,1,7] => [6,1,5,4,3,2,7] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,1,1,1,0,1,1,0,0,0,0,1,0,0] => [4,6,5,3,2,7,1] => [5,1,4,3,2,7,6] => ([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,1,0,1,1,0,0,0,1,0,0,0] => [4,6,5,3,7,2,1] => [4,1,3,2,7,6,5] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [4,6,5,7,3,2,1] => [3,1,2,7,6,5,4] => ([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [4,6,7,5,3,2,1] => [2,7,1,6,5,4,3] => ([(0,1),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [5,4,3,2,1,7,6] => [4,3,2,1,7,5,6] => ([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,1,1,0,0,0,0,1,0,0,1,0] => [5,4,3,2,6,1,7] => [4,3,2,1,6,5,7] => ([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,1,1,0,0,0,0,1,0,1,0,0] => [5,4,3,2,6,7,1] => [4,3,2,1,5,7,6] => ([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,1,1,0,0,0,0,1,1,0,0,0] => [5,4,3,2,7,6,1] => [3,2,1,7,4,6,5] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,1,1,0,0,0,1,0,0,0,1,0] => [5,4,3,6,2,1,7] => [3,2,1,6,5,4,7] => ([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7) => 3

[1,1,1,1,1,0,0,0,1,0,0,1,0,0] => [5,4,3,6,2,7,1] => [3,2,1,5,4,7,6] => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,1,1,0,0,0,1,0,1,0,0,0] => [5,4,3,6,7,2,1] => [3,2,1,4,7,6,5] => ([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7) => 3

[1,1,1,1,1,0,0,0,1,1,0,0,0,0] => [5,4,3,7,6,2,1] => [2,1,7,3,6,5,4] => ([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,1,1,0,0,1,0,0,0,0,1,0] => [5,4,6,3,2,1,7] => [2,1,6,5,4,3,7] => ([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,1,1,0,0,1,0,0,0,1,0,0] => [5,4,6,3,2,7,1] => [2,1,5,4,3,7,6] => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,1,1,0,0,1,0,0,1,0,0,0] => [5,4,6,3,7,2,1] => [2,1,4,3,7,6,5] => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7) => 3

[1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [5,4,6,7,3,2,1] => [2,1,3,7,6,5,4] => ([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [5,4,7,6,3,2,1] => [1,7,2,6,5,4,3] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [5,6,4,3,2,1,7] => [1,6,5,4,3,2,7] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,1,1,1,1,0,1,0,0,0,0,1,0,0] => [5,6,4,3,2,7,1] => [1,5,4,3,2,7,6] => ([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,1,1,0,1,0,0,0,1,0,0,0] => [5,6,4,3,7,2,1] => [1,4,3,2,7,6,5] => ([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7) => 3

[1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [5,6,4,7,3,2,1] => [1,3,2,7,6,5,4] => ([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [5,6,7,4,3,2,1] => [1,2,7,6,5,4,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [6,5,4,3,2,7,1] => [5,4,3,2,1,7,6] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

[1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [6,5,4,3,7,2,1] => [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [6,5,4,7,3,2,1] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [6,5,7,4,3,2,1] => [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5

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$F_{1} = q$

$F_{2} = q + q^{2}$

$F_{3} = q + 3\ q^{2} + q^{3}$

$F_{4} = q + 7\ q^{2} + 5\ q^{3} + q^{4}$

$F_{5} = q + 15\ q^{2} + 18\ q^{3} + 7\ q^{4} + q^{5}$

$F_{6} = q + 31\ q^{2} + 56\ q^{3} + 34\ q^{4} + 9\ q^{5} + q^{6}$

Description

The tree-depth of a graph.
The tree-depth

$\operatorname{td}(G)$ of a graph

$G$ whose connected components are

$G_1,\ldots,G_p$ is recursively defined as

$\operatorname{td}(G)=\begin{cases} 1, & \text{if }|G|=1\\ 1 + \min_{v\in V} \operatorname{td}(G-v), & \text{if } p=1 \text{ and } |G| > 1\\ \max_{i=1}^p \operatorname{td}(G_i), & \text{otherwise} \end{cases}$
Nešetřil and Ossona de Mendez [2] proved that the tree-depth of a connected graph is equal to its minimum elimination tree height and its centered chromatic number (fewest colors needed for a vertex coloring where every connected induced subgraph has a color that appears exactly once).
Tree-depth is strictly greater than pathwidth. A longest path in

$G$ has at least

$\operatorname{td}(G)$ vertices [3].

Map

Lehmer-code to major-code bijection

Description

Sends a permutation to the unique permutation such that the Lehmer code is sent to the major code.
The Lehmer code encodes the inversions of a permutation and the major code encodes its major index. In particular, the number of inversions of a permutation equals the major index of its image under this map.
* The Lehmer code of a permutation

$\sigma$ is given by

$L(\sigma) = l_1 \ldots l_n$ with

$l_i = \# \{ j > i : \sigma_j < \sigma_i \}$ . In particular,

$l_i$ is the number of boxes in the

$i$ -th column of the Rothe diagram. For example, the Lehmer code of

$\sigma = [4,3,1,5,2]$ is

$32010$ . The Lehmer code

$L : \mathfrak{S}_n\ \tilde\longrightarrow\ S_n$ is a bijection between permutations of size

$n$ and sequences

$l_1\ldots l_n \in \mathbf{N}^n$ with

$l_i \leq i$ .
* The major code

$M(\sigma)$ of a permutation

$\sigma \in \mathfrak{S}_n$ is a way to encode a permutation as a sequence

$m_1 m_2 \ldots m_n$ with

$m_i \geq i$ . To define

$m_i$ , let

$\operatorname{del}_i(\sigma)$ be the normalized permutation obtained by removing all

$\sigma_j < i$ from the one-line notation of

$\sigma$ . The

$i$ -th index is then given by

$m_i = \operatorname{maj}(\operatorname{del}_i(\sigma)) - \operatorname{maj}(\operatorname{del}_{i-1}(\sigma)).$
For example, the permutation

$[9,3,5,7,2,1,4,6,8]$ has major code

$[5, 0, 1, 0, 1, 2, 0, 1, 0]$ since

$\operatorname{maj}([8,2,4,6,1,3,5,7]) = 5, \quad \operatorname{maj}([7,1,3,5,2,4,6]) = 5, \quad \operatorname{maj}([6,2,4,1,3,5]) = 4,$

$\operatorname{maj}([5,1,3,2,4]) = 4, \quad \operatorname{maj}([4,2,1,3]) = 3, \quad \operatorname{maj}([3,1,2]) = 1, \quad \operatorname{maj}([2,1]) = 1.$
Observe that the sum of the major code of

$\sigma$ equals the major index of

$\sigma$ .

Map

graph of inversions

Description

The graph of inversions of a permutation.
For a permutation of

$\{1,\dots,n\}$ , this is the graph with vertices

$\{1,\dots,n\}$ , where

$(i,j)$ is an edge if and only if it is an inversion of the permutation.

Map

to 312-avoiding permutation

Description

Sends a Dyck path to the 312-avoiding permutation according to Bandlow-Killpatrick.
This map is defined in [1] and sends the area (St000012The area of a Dyck path.) to the inversion number (St000018The number of inversions of a permutation.).