- FindStat

Identifier

Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000314: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 196 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

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

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

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

$F_{5} = 14\ q + 14\ q^{2} + 9\ q^{3} + 4\ q^{4} + q^{5}$

$F_{6} = 42\ q + 42\ q^{2} + 28\ q^{3} + 14\ q^{4} + 5\ q^{5} + q^{6}$

Description

The number of left-to-right-maxima of a permutation.
An integer

$\sigma_i$ in the one-line notation of a permutation

$\sigma$ is a left-to-right-maximum if there does not exist a

$j < i$ such that

$\sigma_j > \sigma_i$ .
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.

Map

inverse zeta map

Description

The inverse zeta map on Dyck paths.
See its inverse, the zeta map Mp00030zeta map, for the definition and details.

Map

reverse

Description

The reversal of a Dyck path.
This is the Dyck path obtained by reading the path backwards.

Map

to 132-avoiding permutation

Description

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