- FindStat

Identifier

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St001461: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 220 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

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

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

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

$F_{5} = 17\ q + 15\ q^{2} + 10\ q^{3}$

$F_{6} = 46\ q + 51\ q^{2} + 30\ q^{3} + 5\ q^{4}$

Description

The number of topologically connected components of the chord diagram of a permutation.
The chord diagram of a permutation

$\pi\in\mathfrak S_n$ is obtained by placing labels

$1,\dots,n$ in cyclic order on a cycle and drawing a (straight) arc from

$i$ to

$\pi(i)$ for every label

$i$ .
This statistic records the number of topologically connected components in the chord diagram. In particular, if two arcs cross, all four labels connected by the two arcs are in the same component.
The permutation

$\pi\in\mathfrak S_n$ stabilizes an interval

$I=\{a,a+1,\dots,b\}$ if

$\pi(I)=I$ . It is stabilized-interval-free, if the only interval

$\pi$ stablizes is

$\{1,\dots,n\}$ . Thus, this statistic is

$1$ if

$\pi$ is stabilized-interval-free.

Map

inverse

Description

Sends a permutation to its inverse.

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..