- FindStat

Identifier

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000868: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 305 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,1,1,0,1,0,0,0,0,0] => [6,5,3,4,2,1] => [5,3,4,2,1,6] => [4,3,2,5,1,6] => 9

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

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

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

$F_{5} = 1 + 4\ q + 9\ q^{2} + 12\ q^{3} + 10\ q^{4} + 4\ q^{5} + 2\ q^{6}$

$F_{6} = 1 + 5\ q + 14\ q^{2} + 25\ q^{3} + 31\ q^{4} + 26\ q^{5} + 16\ q^{6} + 9\ q^{7} + 4\ q^{8} + q^{9}$

Description

The aid statistic in the sense of Shareshian-Wachs.
This is the number of admissible inversions St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. plus the number of descents St000021The number of descents of a permutation.. This statistic was introduced by John Shareshian and Michelle L. Wachs in [1]. Theorem 4.1 states that the aid statistic together with the descent statistic is Euler-Mahonian.

Map

Inverse Kreweras complement

Description

Sends the permutation

$\pi \in \mathfrak{S}_n$ to the permutation

$c\pi^{-1}$ where

$c = (1,\ldots,n)$ is the long cycle.

Map

to non-crossing permutation

Description

Sends a Dyck path

$D$ with valley at positions

$\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation

$\pi$ having descents

$\{i_1,\ldots,i_k\}$ and whose inverse has descents

$\{j_1,\ldots,j_k\}$ .
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to

$n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..

Map

Clarke-Steingrimsson-Zeng inverse

Description

The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map

$\Phi$ in [1, sec.3].