- FindStat

Identifier

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St001004: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 231 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [9,1,2,3,4,5,6,7,8] => [2,3,4,5,6,7,8,9,1] => [2,3,4,5,6,7,8,1] => 8

[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [8,1,2,3,4,5,6,9,7] => [2,3,4,5,6,7,9,1,8] => [2,3,4,5,6,7,1,8] => 8

[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [3,1,4,5,6,7,8,9,2] => [2,9,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => 8

[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0] => [9,3,1,2,4,5,6,7,8] => [3,4,2,5,6,7,8,9,1] => [3,4,2,5,6,7,8,1] => 7

[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [8,9,1,2,3,4,5,6,7] => [3,4,5,6,7,8,9,1,2] => [3,4,5,6,7,8,1,2] => 8

[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0] => [2,3,4,5,6,9,1,7,8] => [7,1,2,3,4,5,8,9,6] => [7,1,2,3,4,5,8,6] => 8

[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [2,3,4,5,6,7,9,1,8] => [8,1,2,3,4,5,6,9,7] => [8,1,2,3,4,5,6,7] => 8

[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => [9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 8

[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [2,3,4,5,6,7,8,10,1,9] => [9,1,2,3,4,5,6,7,10,8] => [9,1,2,3,4,5,6,7,8] => 9

[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [3,1,4,5,6,7,8,9,10,2] => [2,10,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => 9

[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => [2,3,4,5,6,7,8,9,11,1,10] => [10,1,2,3,4,5,6,7,8,11,9] => [10,1,2,3,4,5,6,7,8,9] => 10

[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0] => [2,3,4,5,6,7,10,1,8,9] => [8,1,2,3,4,5,6,9,10,7] => [8,1,2,3,4,5,6,9,7] => 9

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

[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,1] => [10,1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => 9

[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [10,1,2,3,4,5,6,7,8,9] => [2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,1] => 9

[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [9,1,2,3,4,5,6,7,10,8] => [2,3,4,5,6,7,8,10,1,9] => [2,3,4,5,6,7,8,1,9] => 9

[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [11,1,2,3,4,5,6,7,8,9,10] => [2,3,4,5,6,7,8,9,10,11,1] => [2,3,4,5,6,7,8,9,10,1] => 10

[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [10,1,2,3,4,5,6,7,8,11,9] => [2,3,4,5,6,7,8,9,11,1,10] => [2,3,4,5,6,7,8,9,1,10] => 10

[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => [11,1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => 10

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$F_{1} = q$

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

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

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

$F_{5} = q^{2} + 8\ q^{3} + 16\ q^{4} + 17\ q^{5}$

$F_{6} = q^{2} + 12\ q^{3} + 33\ q^{4} + 50\ q^{5} + 36\ q^{6}$

Description

The number of indices that are either left-to-right maxima or right-to-left minima.
The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a

$321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see St000371The number of mid points of decreasing subsequences of length 3 in a permutation. and St000372The number of mid points of increasing subsequences of length 3 in a permutation..

Map

Ringel

Description

The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.

Map

inverse

Description

Sends a permutation to its inverse.

Map

restriction

Description

The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.