- FindStat

Identifier

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001962: Graphs ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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) => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 624 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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) => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 8 + 5\ q + q^{2}$

$F_{5} = 16 + 19\ q + 6\ q^{2} + q^{3}$

$F_{6} = 32 + 61\ q + 31\ q^{2} + 7\ q^{3} + q^{4}$

Description

The proper pathwidth of a graph.
The proper pathwidth

$\operatorname{ppw}(G)$ was introduced in [1] as the minimum width of a proper-path-decomposition. Barioli et al. [2] showed that if

$G$ has at least one edge, then

$\operatorname{ppw}(G)$ is the minimum

$k$ for which

$G$ is a minor of the Cartesian product

$K_k \square P$ of a complete graph on

$k$ vertices with a path; and further that

$\operatorname{ppw}(G)$ is the minor monotone floor

$\lfloor \operatorname{Z} \rfloor(G) := \min\{\operatorname{Z}(H) \mid G \preceq H\}$ of the zero forcing number

$\operatorname{Z}(G)$ . It can be shown [3, Corollary 9.130] that only the spanning supergraphs need to be considered for

$H$ in this definition, i.e.

$\lfloor \operatorname{Z} \rfloor(G) = \min\{\operatorname{Z}(H) \mid G \le H,\; V(H) = V(G)\}$ .
The minimum degree

$\delta$ , treewidth

$\operatorname{tw}$ , and pathwidth

$\operatorname{pw}$ satisfy

$\delta \le \operatorname{tw} \le \operatorname{pw} \le \operatorname{ppw} = \lfloor \operatorname{Z} \rfloor \le \operatorname{pw} + 1.$
Note that [4] uses a different notion of proper pathwidth, which is equal to bandwidth.

Map

cycle-as-one-line notation

Description

Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.

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 321-avoiding permutation (Krattenthaler)

Description

Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength

$n$ in an

$n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.