Identifier
-
Mp00201:
Dyck paths
—Ringel⟶
Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ ℤ
Values
[1,0] => [2,1] => [1,2] => ([(0,1)],2) => 1
[1,0,1,0] => [3,1,2] => [2,1,3] => ([(0,2),(1,2)],3) => 1
[1,1,0,0] => [2,3,1] => [1,3,2] => ([(0,1),(0,2)],3) => 2
[1,0,1,0,1,0] => [4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4) => 1
[1,0,1,1,0,0] => [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4) => 2
[1,1,0,0,1,0] => [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4) => 2
[1,1,0,1,0,0] => [4,3,1,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4) => 1
[1,1,1,0,0,0] => [2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4) => 3
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5) => 1
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5) => 2
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5) => 1
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5) => 3
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5) => 2
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5) => 3
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5) => 1
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5) => 1
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5) => 2
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5) => 3
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5) => 2
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => 1
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5) => 4
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [5,4,3,2,1,6] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 1
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [4,6,3,2,1,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5)],6) => 2
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [5,3,6,2,1,4] => ([(0,5),(1,5),(2,4),(3,4),(3,5)],6) => 2
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [4,3,5,2,1,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 1
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [3,6,5,2,1,4] => ([(0,5),(1,5),(2,3),(2,4),(2,5)],6) => 3
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [5,4,2,6,1,3] => ([(0,5),(1,5),(2,4),(3,4),(3,5)],6) => 2
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [4,6,2,5,1,3] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6) => 3
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [5,3,2,4,1,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 1
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [4,3,2,5,1,6] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6) => 1
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [3,6,2,4,1,5] => ([(0,5),(1,4),(2,3),(2,4),(4,5)],6) => 2
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [5,2,6,4,1,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5)],6) => 3
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [4,2,5,6,1,3] => ([(0,5),(1,4),(2,4),(2,5),(5,3)],6) => 2
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [3,2,5,4,1,6] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 1
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [2,6,5,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 4
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [5,4,3,1,6,2] => ([(0,5),(1,5),(2,5),(3,4),(3,5)],6) => 2
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [4,6,3,1,5,2] => ([(0,5),(1,4),(1,5),(2,3),(2,5)],6) => 3
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [5,3,6,1,4,2] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6) => 3
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [4,3,5,1,6,2] => ([(0,4),(1,4),(2,3),(2,5),(4,5)],6) => 2
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [3,6,5,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6) => 4
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [5,4,2,1,3,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 1
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [4,6,2,1,3,5] => ([(0,4),(1,4),(2,3),(2,5),(4,5)],6) => 2
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [5,3,2,1,4,6] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6) => 1
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [4,3,2,1,6,5] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 2
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [3,6,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,5),(5,4)],6) => 2
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [5,2,6,1,3,4] => ([(0,5),(1,4),(2,4),(2,5),(5,3)],6) => 2
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [4,2,5,1,3,6] => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6) => 1
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [3,2,5,1,4,6] => ([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6) => 1
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [2,6,5,1,3,4] => ([(0,5),(1,2),(1,3),(1,5),(5,4)],6) => 3
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [5,4,1,6,3,2] => ([(0,5),(1,5),(2,3),(2,4),(2,5)],6) => 3
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [4,6,1,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5)],6) => 4
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [5,3,1,4,6,2] => ([(0,5),(1,4),(2,3),(2,4),(4,5)],6) => 2
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,5),(5,4)],6) => 2
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [3,6,1,4,5,2] => ([(0,4),(0,5),(1,3),(1,5),(5,2)],6) => 3
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [5,2,1,4,3,6] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 1
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [4,2,1,5,3,6] => ([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6) => 1
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [3,2,1,4,5,6] => ([(0,5),(1,5),(2,5),(3,4),(5,3)],6) => 1
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [2,6,1,4,3,5] => ([(0,4),(0,5),(1,2),(1,4),(1,5),(4,3),(5,3)],6) => 2
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 4
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [4,1,5,6,3,2] => ([(0,5),(1,2),(1,3),(1,5),(5,4)],6) => 3
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [3,1,5,4,6,2] => ([(0,4),(0,5),(1,2),(1,4),(1,5),(4,3),(5,3)],6) => 2
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [2,1,5,4,3,6] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6) => 1
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6) => 5
[1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [6,5,4,3,2,1,7] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => [5,7,4,3,2,1,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6)],7) => 2
[1,0,1,0,1,0,1,1,0,0,1,0] => [5,1,2,3,7,4,6] => [6,4,7,3,2,1,5] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6)],7) => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [7,1,2,3,6,4,5] => [5,4,6,3,2,1,7] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => [4,7,6,3,2,1,5] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6)],7) => 3
[1,0,1,0,1,1,0,0,1,0,1,0] => [4,1,2,7,3,5,6] => [6,5,3,7,2,1,4] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [4,1,2,6,3,7,5] => [5,7,3,6,2,1,4] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5)],7) => 3
[1,0,1,0,1,1,0,1,0,0,1,0] => [7,1,2,5,3,4,6] => [6,4,3,5,2,1,7] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [7,1,2,6,3,4,5] => [5,4,3,6,2,1,7] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7) => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [6,1,2,5,3,7,4] => [4,7,3,5,2,1,6] => ([(0,6),(1,6),(2,5),(3,4),(3,5),(5,6)],7) => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [4,1,2,5,7,3,6] => [6,3,7,5,2,1,4] => ([(0,6),(1,6),(2,5),(3,4),(3,5),(3,6)],7) => 3
[1,0,1,0,1,1,1,0,0,1,0,0] => [4,1,2,7,6,3,5] => [5,3,6,7,2,1,4] => ([(0,6),(1,6),(2,5),(3,5),(3,6),(5,4)],7) => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [7,1,2,5,6,3,4] => [4,3,6,5,2,1,7] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [4,1,2,5,6,7,3] => [3,7,6,5,2,1,4] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6)],7) => 4
[1,0,1,1,0,0,1,0,1,0,1,0] => [3,1,7,2,4,5,6] => [6,5,4,2,7,1,3] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6)],7) => 2
[1,0,1,1,0,0,1,0,1,1,0,0] => [3,1,6,2,4,7,5] => [5,7,4,2,6,1,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,6)],7) => 3
[1,0,1,1,0,0,1,1,0,0,1,0] => [3,1,5,2,7,4,6] => [6,4,7,2,5,1,3] => ([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6)],7) => 3
[1,0,1,1,0,0,1,1,0,1,0,0] => [3,1,7,2,6,4,5] => [5,4,6,2,7,1,3] => ([(0,4),(1,5),(2,5),(3,4),(3,6),(5,6)],7) => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [7,1,4,2,3,5,6] => [6,5,3,2,4,1,7] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [6,1,4,2,3,7,5] => [5,7,3,2,4,1,6] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(5,6)],7) => 2
[1,0,1,1,0,1,0,1,0,0,1,0] => [7,1,5,2,3,4,6] => [6,4,3,2,5,1,7] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7) => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [6,1,7,2,3,4,5] => [5,4,3,2,7,1,6] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [6,1,5,2,3,7,4] => [4,7,3,2,5,1,6] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(6,5)],7) => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [5,1,4,2,7,3,6] => [6,3,7,2,4,1,5] => ([(0,6),(1,4),(2,5),(3,4),(3,5),(5,6)],7) => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [7,1,4,2,6,3,5] => [5,3,6,2,4,1,7] => ([(0,6),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7) => 1
[1,0,1,1,0,1,1,0,1,0,0,0] => [7,1,5,2,6,3,4] => [4,3,6,2,5,1,7] => ([(0,6),(1,5),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 1
[1,0,1,1,1,0,0,0,1,0,1,0] => [3,1,4,7,2,5,6] => [6,5,2,7,4,1,3] => ([(0,6),(1,6),(2,5),(3,4),(3,5),(3,6)],7) => 3
[1,0,1,1,1,0,0,1,0,0,1,0] => [3,1,7,5,2,4,6] => [6,4,2,5,7,1,3] => ([(0,6),(1,4),(2,5),(3,4),(3,5),(5,6)],7) => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [3,1,7,6,2,4,5] => [5,4,2,6,7,1,3] => ([(0,6),(1,6),(2,5),(3,5),(3,6),(6,4)],7) => 2
[1,0,1,1,1,0,1,0,0,0,1,0] => [7,1,4,5,2,3,6] => [6,3,2,5,4,1,7] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [7,1,4,6,2,3,5] => [5,3,2,6,4,1,7] => ([(0,6),(1,5),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 1
[1,0,1,1,1,0,1,0,1,0,0,0] => [7,1,6,5,2,3,4] => [4,3,2,5,6,1,7] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(6,4)],7) => 1
[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,7,5,6,2,4] => [4,2,6,5,7,1,3] => ([(0,4),(1,5),(1,6),(2,4),(2,5),(2,6),(5,3),(6,3)],7) => 2
[1,0,1,1,1,1,1,0,0,0,0,0] => [3,1,4,5,6,7,2] => [2,7,6,5,4,1,3] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6)],7) => 5
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => [6,5,4,3,1,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6)],7) => 2
[1,1,0,0,1,0,1,0,1,1,0,0] => [2,6,1,3,4,7,5] => [5,7,4,3,1,6,2] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6)],7) => 3
[1,1,0,0,1,0,1,1,0,0,1,0] => [2,5,1,3,7,4,6] => [6,4,7,3,1,5,2] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,6)],7) => 3
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Description
The number of maximal elements of a poset.
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reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
permutation poset
Description
Sends a permutation to its permutation poset.
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
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