- FindStat

Identifier

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001640: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 306 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [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] => 6

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$F_{1} = 1$

$F_{2} = 1 + q$

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

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

$F_{5} = 9 + 16\ q + 12\ q^{2} + 4\ q^{3} + q^{4}$

$F_{6} = 21 + 45\ q + 40\ q^{2} + 20\ q^{3} + 5\ q^{4} + q^{5}$

Description

The number of ascent tops in the permutation such that all smaller elements appear before.

Map

decomposition reverse

Description

This map is recursively defined as follows.
The unique empty path of semilength

$0$ is sent to itself.
Let

$D$ be a Dyck path of semilength

$n > 0$ and decompose it into

$1 D_1 0 D_2$ with Dyck paths

$D_1, D_2$ of respective semilengths

$n_1$ and

$n_2$ such that

$n_1$ is minimal. One then has

$n_1+n_2 = n-1$ .
Now let

$\tilde D_1$ and

$\tilde D_2$ be the recursively defined respective images of

$D_1$ and

$D_2$ under this map. The image of

$D$ is then defined as

$1 \tilde D_2 0 \tilde D_1$ .

Map

inverse

Description

Sends a permutation to its inverse.

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.