- FindStat

Identifier

Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001090: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

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

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

$F_{4} = 2\ q^{2} + 7\ q^{3} + 5\ q^{4}$

$F_{5} = 2\ q^{2} + 14\ q^{3} + 16\ q^{4} + 10\ q^{5}$

Description

The number of pop-stack-sorts needed to sort a permutation.
The pop-stack sorting operator is defined as follows. Process the permutation

$\pi$ from left to right. If the stack is empty or its top element is smaller than the current element, empty the stack completely and append its elements to the output in reverse order. Next, push the current element onto the stack. After having processed the last entry, append the stack to the output in reverse order.
A permutation is

$t$ -pop-stack sortable if it is sortable using

$t$ pop-stacks in series.

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

inverse Kreweras complement

Description

Return the inverse of the Kreweras complement of a Dyck path, regarded as a noncrossing set partition.
To identify Dyck paths and noncrossing set partitions, this maps uses the following classical bijection. The number of down steps after the

$i$ -th up step of the Dyck path is the size of the block of the set partition whose maximal element is

$i$ . If

$i$ is not a maximal element of a block, the

$(i+1)$ -st step is also an up step.

Map

Ringel

Description

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