- FindStat

Identifier

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000830: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => [4,3,2,1] => 8

[1,0,1,0,1,1,0,0] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => [4,3,1,2] => 8

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

[1,0,1,1,0,1,0,0] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => [4,3,1,2] => 8

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => [5,4,3,2,1] => 12

[1,0,1,0,1,0,1,1,0,0] => [1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => [5,4,3,1,2] => 12

[1,0,1,0,1,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => [5,4,1,3,2] => 12

[1,0,1,0,1,1,0,1,0,0] => [1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => [5,4,3,1,2] => 12

[1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [5,4,1,2,3] => 12

[1,0,1,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => [5,1,4,3,2] => 10

[1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [5,1,4,2,3] => 10

[1,0,1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => [5,4,1,3,2] => 12

[1,0,1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [5,1,4,2,3] => 10

[1,0,1,1,0,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [5,4,1,2,3] => 12

[1,0,1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [5,1,2,4,3] => 8

[1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [5,1,4,2,3] => 10

[1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [5,4,1,2,3] => 12

[1,0,1,1,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => 8

[1,1,0,0,1,0,1,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => [1,5,4,3,2] => 8

[1,1,0,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => [1,5,4,2,3] => 8

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

[1,1,0,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => [1,5,4,2,3] => 8

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

[1,1,0,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => [5,1,4,3,2] => 10

[1,1,0,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [5,1,4,2,3] => 10

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

[1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [5,1,4,2,3] => 10

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

[1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [5,1,2,4,3] => 8

[1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [5,1,4,2,3] => 10

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

[1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => 8

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

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

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

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

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

[1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [5,1,2,4,3] => 8

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

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

[1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => 8

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

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

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

[1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => 8

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

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 18

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => [6,5,4,3,1,2] => 18

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => [6,5,4,1,3,2] => 18

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => [6,5,4,3,1,2] => 18

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => [6,5,4,1,2,3] => 18

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => [6,5,1,4,3,2] => 16

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => [6,5,1,4,2,3] => 16

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => [6,5,4,1,3,2] => 18

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => [6,5,1,4,2,3] => 16

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => [6,5,4,1,2,3] => 18

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => [6,5,1,2,4,3] => 16

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => [6,5,1,4,2,3] => 16

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => [6,5,4,1,2,3] => 18

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => [6,5,1,2,3,4] => 16

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => [6,1,5,4,3,2] => 14

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => [6,1,5,4,2,3] => 14

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => [6,1,5,2,4,3] => 14

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => [6,1,5,4,2,3] => 14

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [6,1,5,2,3,4] => 14

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => [6,5,1,4,3,2] => 16

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => [6,5,1,4,2,3] => 16

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => [6,1,5,2,4,3] => 14

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => [6,5,1,4,2,3] => 16

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [6,1,5,2,3,4] => 14

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => [6,5,1,2,4,3] => 16

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => [6,5,1,4,2,3] => 16

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [6,1,5,2,3,4] => 14

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => [6,5,1,2,3,4] => 16

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => [6,1,2,5,4,3] => 12

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [6,1,2,5,3,4] => 12

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => [6,1,5,2,4,3] => 14

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [6,1,2,5,3,4] => 12

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [6,1,5,2,3,4] => 14

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => [6,5,1,2,4,3] => 16

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [6,1,2,5,3,4] => 12

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [6,1,5,2,3,4] => 14

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => [6,5,1,2,3,4] => 16

[1,0,1,1,1,1,0,0,0,0,1,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => [6,1,2,3,5,4] => 10

>>> Load all 281 entries. <<<

[1,0,1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [6,1,2,5,3,4] => 12

[1,0,1,1,1,1,0,0,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [6,1,5,2,3,4] => 14

[1,0,1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => [6,5,1,2,3,4] => 16

[1,0,1,1,1,1,1,0,0,0,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => 10

[1,1,0,0,1,0,1,0,1,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => [6,5,4,3,1,2] => [1,6,5,4,3,2] => 12

[1,1,0,0,1,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => [5,6,4,3,1,2] => [1,6,5,4,2,3] => 12

[1,1,0,0,1,0,1,1,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0,1,0] => [6,4,5,3,1,2] => [1,6,5,2,4,3] => 12

[1,1,0,0,1,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => [5,6,4,3,1,2] => [1,6,5,4,2,3] => 12

[1,1,0,0,1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,1,1,0,0,0] => [4,5,6,3,1,2] => [1,6,5,2,3,4] => 12

[1,1,0,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => [6,5,3,4,1,2] => [1,6,2,5,4,3] => 10

[1,1,0,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => [5,6,3,4,1,2] => [1,6,2,5,3,4] => 10

[1,1,0,0,1,1,0,1,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0,1,0] => [6,4,5,3,1,2] => [1,6,5,2,4,3] => 12

[1,1,0,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => [5,6,3,4,1,2] => [1,6,2,5,3,4] => 10

[1,1,0,0,1,1,0,1,1,0,0,0] => [1,1,0,0,1,0,1,1,1,0,0,0] => [4,5,6,3,1,2] => [1,6,5,2,3,4] => 12

[1,1,0,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => [6,3,4,5,1,2] => [1,6,2,3,5,4] => 8

[1,1,0,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => [5,6,3,4,1,2] => [1,6,2,5,3,4] => 10

[1,1,0,0,1,1,1,0,1,0,0,0] => [1,1,0,0,1,0,1,1,1,0,0,0] => [4,5,6,3,1,2] => [1,6,5,2,3,4] => 12

[1,1,0,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => [1,6,2,3,4,5] => 8

[1,1,0,1,0,0,1,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => [6,1,5,4,3,2] => 14

[1,1,0,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => [6,1,5,4,2,3] => 14

[1,1,0,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => [6,1,5,2,4,3] => 14

[1,1,0,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => [6,1,5,4,2,3] => 14

[1,1,0,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [6,1,5,2,3,4] => 14

[1,1,0,1,0,1,0,0,1,0,1,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => [6,5,3,4,1,2] => [1,6,2,5,4,3] => 10

[1,1,0,1,0,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => [5,6,3,4,1,2] => [1,6,2,5,3,4] => 10

[1,1,0,1,0,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => [6,1,5,2,4,3] => 14

[1,1,0,1,0,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => [5,6,3,4,1,2] => [1,6,2,5,3,4] => 10

[1,1,0,1,0,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [6,1,5,2,3,4] => 14

[1,1,0,1,0,1,1,0,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => [6,3,4,5,1,2] => [1,6,2,3,5,4] => 8

[1,1,0,1,0,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => [5,6,3,4,1,2] => [1,6,2,5,3,4] => 10

[1,1,0,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [6,1,5,2,3,4] => 14

[1,1,0,1,0,1,1,1,0,0,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => [1,6,2,3,4,5] => 8

[1,1,0,1,1,0,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => [6,1,2,5,4,3] => 12

[1,1,0,1,1,0,0,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [6,1,2,5,3,4] => 12

[1,1,0,1,1,0,0,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => [6,1,5,2,4,3] => 14

[1,1,0,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [6,1,2,5,3,4] => 12

[1,1,0,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [6,1,5,2,3,4] => 14

[1,1,0,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => [6,3,4,5,1,2] => [1,6,2,3,5,4] => 8

[1,1,0,1,1,0,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [6,1,2,5,3,4] => 12

[1,1,0,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [6,1,5,2,3,4] => 14

[1,1,0,1,1,0,1,1,0,0,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => [1,6,2,3,4,5] => 8

[1,1,0,1,1,1,0,0,0,0,1,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => [6,1,2,3,5,4] => 10

[1,1,0,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [6,1,2,5,3,4] => 12

[1,1,0,1,1,1,0,0,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [6,1,5,2,3,4] => 14

[1,1,0,1,1,1,0,1,0,0,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => [1,6,2,3,4,5] => 8

[1,1,0,1,1,1,1,0,0,0,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => 10

[1,1,1,0,0,0,1,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => [6,5,4,1,2,3] => [1,2,6,5,4,3] => 8

[1,1,1,0,0,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0,1,1,0,0] => [5,6,4,1,2,3] => [1,2,6,5,3,4] => 8

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

[1,1,1,0,0,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,1,0,0] => [5,6,4,1,2,3] => [1,2,6,5,3,4] => 8

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

[1,1,1,0,0,1,0,0,1,0,1,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => [6,5,3,4,1,2] => [1,6,2,5,4,3] => 10

[1,1,1,0,0,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => [5,6,3,4,1,2] => [1,6,2,5,3,4] => 10

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

[1,1,1,0,0,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => [5,6,3,4,1,2] => [1,6,2,5,3,4] => 10

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

[1,1,1,0,0,1,1,0,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => [6,3,4,5,1,2] => [1,6,2,3,5,4] => 8

[1,1,1,0,0,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => [5,6,3,4,1,2] => [1,6,2,5,3,4] => 10

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

[1,1,1,0,0,1,1,1,0,0,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => [1,6,2,3,4,5] => 8

[1,1,1,0,1,0,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => [6,1,2,5,4,3] => 12

[1,1,1,0,1,0,0,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [6,1,2,5,3,4] => 12

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

[1,1,1,0,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [6,1,2,5,3,4] => 12

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

[1,1,1,0,1,0,1,0,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => [6,3,4,5,1,2] => [1,6,2,3,5,4] => 8

[1,1,1,0,1,0,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [6,1,2,5,3,4] => 12

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

[1,1,1,0,1,0,1,1,0,0,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => [1,6,2,3,4,5] => 8

[1,1,1,0,1,1,0,0,0,0,1,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => [6,1,2,3,5,4] => 10

[1,1,1,0,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [6,1,2,5,3,4] => 12

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

[1,1,1,0,1,1,0,1,0,0,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => [1,6,2,3,4,5] => 8

[1,1,1,0,1,1,1,0,0,0,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => 10

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

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

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

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

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

[1,1,1,1,0,0,1,0,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => [6,3,4,5,1,2] => [1,6,2,3,5,4] => 8

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

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

[1,1,1,1,0,0,1,1,0,0,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => [1,6,2,3,4,5] => 8

[1,1,1,1,0,1,0,0,0,0,1,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => [6,1,2,3,5,4] => 10

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

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

[1,1,1,1,0,1,0,1,0,0,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => [1,6,2,3,4,5] => 8

[1,1,1,1,0,1,1,0,0,0,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => 10

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

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

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

[1,1,1,1,1,0,0,1,0,0,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => [1,6,2,3,4,5] => 8

[1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => 10

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

[1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [7,6,5,4,1,2,3] => [1,2,7,6,5,4,3] => 12

[1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [6,7,5,4,1,2,3] => [1,2,7,6,5,3,4] => 12

[1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [7,5,6,4,1,2,3] => [1,2,7,6,3,5,4] => 12

[1,1,1,0,0,0,1,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [6,7,5,4,1,2,3] => [1,2,7,6,5,3,4] => 12

[1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [5,6,7,4,1,2,3] => [1,2,7,6,3,4,5] => 12

[1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [7,6,4,5,1,2,3] => [1,2,7,3,6,5,4] => 10

[1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [6,7,4,5,1,2,3] => [1,2,7,3,6,4,5] => 10

[1,1,1,0,0,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [7,5,6,4,1,2,3] => [1,2,7,6,3,5,4] => 12

[1,1,1,0,0,0,1,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [6,7,4,5,1,2,3] => [1,2,7,3,6,4,5] => 10

[1,1,1,0,0,0,1,1,0,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [5,6,7,4,1,2,3] => [1,2,7,6,3,4,5] => 12

[1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [7,4,5,6,1,2,3] => [1,2,7,3,4,6,5] => 8

[1,1,1,0,0,0,1,1,1,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [6,7,4,5,1,2,3] => [1,2,7,3,6,4,5] => 10

[1,1,1,0,0,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [5,6,7,4,1,2,3] => [1,2,7,6,3,4,5] => 12

[1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [4,5,6,7,1,2,3] => [1,2,7,3,4,5,6] => 8

[1,1,1,0,0,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [7,6,4,5,1,2,3] => [1,2,7,3,6,5,4] => 10

[1,1,1,0,0,1,0,1,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [6,7,4,5,1,2,3] => [1,2,7,3,6,4,5] => 10

[1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [6,7,4,5,1,2,3] => [1,2,7,3,6,4,5] => 10

[1,1,1,0,0,1,0,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [7,4,5,6,1,2,3] => [1,2,7,3,4,6,5] => 8

[1,1,1,0,0,1,0,1,1,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [6,7,4,5,1,2,3] => [1,2,7,3,6,4,5] => 10

[1,1,1,0,0,1,0,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [4,5,6,7,1,2,3] => [1,2,7,3,4,5,6] => 8

[1,1,1,0,0,1,1,0,1,0,0,0,1,0] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [7,4,5,6,1,2,3] => [1,2,7,3,4,6,5] => 8

[1,1,1,0,0,1,1,0,1,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [4,5,6,7,1,2,3] => [1,2,7,3,4,5,6] => 8

[1,1,1,0,0,1,1,1,0,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [4,5,6,7,1,2,3] => [1,2,7,3,4,5,6] => 8

[1,1,1,0,1,0,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [7,6,4,5,1,2,3] => [1,2,7,3,6,5,4] => 10

[1,1,1,0,1,0,0,1,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [6,7,4,5,1,2,3] => [1,2,7,3,6,4,5] => 10

[1,1,1,0,1,0,0,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [6,7,4,5,1,2,3] => [1,2,7,3,6,4,5] => 10

[1,1,1,0,1,0,0,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [7,4,5,6,1,2,3] => [1,2,7,3,4,6,5] => 8

[1,1,1,0,1,0,0,1,1,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [6,7,4,5,1,2,3] => [1,2,7,3,6,4,5] => 10

[1,1,1,0,1,0,0,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [4,5,6,7,1,2,3] => [1,2,7,3,4,5,6] => 8

[1,1,1,0,1,0,1,0,1,0,0,0,1,0] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [7,4,5,6,1,2,3] => [1,2,7,3,4,6,5] => 8

[1,1,1,0,1,0,1,0,1,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [4,5,6,7,1,2,3] => [1,2,7,3,4,5,6] => 8

[1,1,1,0,1,0,1,1,0,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [4,5,6,7,1,2,3] => [1,2,7,3,4,5,6] => 8

[1,1,1,0,1,1,0,0,1,0,0,0,1,0] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [7,4,5,6,1,2,3] => [1,2,7,3,4,6,5] => 8

[1,1,1,0,1,1,0,0,1,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [4,5,6,7,1,2,3] => [1,2,7,3,4,5,6] => 8

[1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [4,5,6,7,1,2,3] => [1,2,7,3,4,5,6] => 8

[1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [4,5,6,7,1,2,3] => [1,2,7,3,4,5,6] => 8

[1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => 8

[1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [6,7,5,1,2,3,4] => [1,2,3,7,6,4,5] => 8

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

[1,1,1,1,0,0,0,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [6,7,5,1,2,3,4] => [1,2,3,7,6,4,5] => 8

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

[1,1,1,1,0,0,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [7,6,4,5,1,2,3] => [1,2,7,3,6,5,4] => 10

[1,1,1,1,0,0,0,1,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [6,7,4,5,1,2,3] => [1,2,7,3,6,4,5] => 10

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

[1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [6,7,4,5,1,2,3] => [1,2,7,3,6,4,5] => 10

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

[1,1,1,1,0,0,0,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [7,4,5,6,1,2,3] => [1,2,7,3,4,6,5] => 8

[1,1,1,1,0,0,0,1,1,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [6,7,4,5,1,2,3] => [1,2,7,3,6,4,5] => 10

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

[1,1,1,1,0,0,0,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [4,5,6,7,1,2,3] => [1,2,7,3,4,5,6] => 8

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

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

[1,1,1,1,0,0,1,0,1,0,0,0,1,0] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [7,4,5,6,1,2,3] => [1,2,7,3,4,6,5] => 8

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

[1,1,1,1,0,0,1,0,1,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [4,5,6,7,1,2,3] => [1,2,7,3,4,5,6] => 8

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

[1,1,1,1,0,0,1,1,0,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [4,5,6,7,1,2,3] => [1,2,7,3,4,5,6] => 8

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

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

[1,1,1,1,0,1,0,0,1,0,0,0,1,0] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [7,4,5,6,1,2,3] => [1,2,7,3,4,6,5] => 8

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

[1,1,1,1,0,1,0,0,1,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [4,5,6,7,1,2,3] => [1,2,7,3,4,5,6] => 8

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

[1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [4,5,6,7,1,2,3] => [1,2,7,3,4,5,6] => 8

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

[1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [4,5,6,7,1,2,3] => [1,2,7,3,4,5,6] => 8

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

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

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

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

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

[1,1,1,1,1,0,0,0,1,0,0,0,1,0] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [7,4,5,6,1,2,3] => [1,2,7,3,4,6,5] => 8

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

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

[1,1,1,1,1,0,0,0,1,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [4,5,6,7,1,2,3] => [1,2,7,3,4,5,6] => 8

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

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

[1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [4,5,6,7,1,2,3] => [1,2,7,3,4,5,6] => 8

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

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

[1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [4,5,6,7,1,2,3] => [1,2,7,3,4,5,6] => 8

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

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

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

[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [4,5,6,7,1,2,3] => [1,2,7,3,4,5,6] => 8

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

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$F_{2} = 1 + q^{2}$

$F_{3} = 1 + q^{2} + 3\ q^{4}$

$F_{4} = 1 + q^{2} + 4\ q^{4} + 5\ q^{6} + 3\ q^{8}$

$F_{5} = 1 + q^{2} + 5\ q^{4} + 9\ q^{6} + 10\ q^{8} + 8\ q^{10} + 8\ q^{12}$

$F_{6} = 1 + q^{2} + 6\ q^{4} + 14\ q^{6} + 19\ q^{8} + 21\ q^{10} + 23\ q^{12} + 24\ q^{14} + 15\ q^{16} + 8\ q^{18}$

Description

The total displacement of a permutation.
This is, for a permutation

$\pi$ of

$n$ , given by

$\sum_{i = 1}^n | \pi(i) - i |.$
This is twice the statistic St000029The depth of a permutation. and can be found in [3, Problem 5.1.1.28] and also in [1, 2].

Map

major-index to inversion-number bijection

Description

Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.

Map

bounce path

Description

Sends a Dyck path

$D$ of length

$2n$ to its bounce path.
This path is formed by starting at the endpoint

$(n,n)$ of

$D$ and travelling west until encountering the first vertical step of

$D$ , then south until hitting the diagonal, then west again to hit

$D$ , etc. until the point

$(0,0)$ is reached.
This map is the first part of the zeta map Mp00030zeta map.

Map

to 132-avoiding permutation

Description

Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].