Identifier
-
Mp00129:
Dyck paths
—to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶
Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤ
Values
[1,0,1,0,1,0] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[1,1,0,1,0,0] => [3,1,2] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[1,0,1,0,1,0,1,0] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[1,0,1,1,0,1,0,0] => [2,4,1,3] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[1,1,0,0,1,1,0,0] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[1,1,0,1,0,1,0,0] => [3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,1,0,1,0,0,0] => [4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 5
[1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 5
[1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 5
[1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 5
[1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,0,1,0,1,0,1,1,0,1,0,0] => [2,3,4,6,1,5] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 4
[1,0,1,0,1,1,0,0,1,1,0,0] => [2,3,1,5,4,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 4
[1,0,1,0,1,1,0,1,0,1,0,0] => [2,3,5,6,1,4] => [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 5
[1,0,1,0,1,1,1,0,1,0,0,0] => [2,3,6,1,4,5] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => 4
[1,0,1,1,0,0,1,0,1,1,0,0] => [2,1,4,5,3,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => 4
[1,0,1,1,0,0,1,1,1,0,0,0] => [2,1,4,3,5,6] => [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 5
[1,0,1,1,0,1,0,0,1,1,0,0] => [2,4,1,5,3,6] => [1,5,2,4,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6) => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => [2,4,5,6,1,3] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 6
[1,0,1,1,0,1,1,0,1,0,0,0] => [2,4,6,1,3,5] => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 6
[1,0,1,1,1,0,0,0,1,1,0,0] => [2,1,3,5,4,6] => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 6
[1,0,1,1,1,0,0,1,1,0,0,0] => [2,1,5,3,4,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 4
[1,0,1,1,1,0,1,0,1,0,0,0] => [2,5,6,1,3,4] => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 5
[1,0,1,1,1,1,0,1,0,0,0,0] => [2,6,1,3,4,5] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 4
[1,1,0,0,1,0,1,0,1,1,0,0] => [1,3,4,5,2,6] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 4
[1,1,0,0,1,0,1,1,1,0,0,0] => [1,3,4,2,5,6] => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 5
[1,1,0,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4,6] => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 6
[1,1,0,0,1,1,1,1,0,0,0,0] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 6
[1,1,0,1,0,1,0,1,0,1,0,0] => [3,4,5,6,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,0,1,0,1,1,0,1,0,0,0] => [3,4,6,1,2,5] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 5
[1,1,0,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4,6] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 5
[1,1,0,1,1,0,1,0,1,0,0,0] => [3,5,6,1,2,4] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 6
[1,1,0,1,1,1,0,1,0,0,0,0] => [3,6,1,2,4,5] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 5
[1,1,1,0,0,0,1,0,1,1,0,0] => [1,2,4,5,3,6] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 5
[1,1,1,0,0,0,1,1,1,0,0,0] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 6
[1,1,1,0,0,1,0,0,1,1,0,0] => [1,4,2,5,3,6] => [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 6
[1,1,1,0,0,1,0,1,1,0,0,0] => [1,4,5,2,3,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => 4
[1,1,1,0,0,1,1,1,0,0,0,0] => [1,4,2,3,5,6] => [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 5
[1,1,1,0,1,0,1,0,1,0,0,0] => [4,5,6,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,1,1,0,1,0,0,0,0] => [4,6,1,2,3,5] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 6
[1,1,1,1,0,0,0,0,1,1,0,0] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 6
[1,1,1,1,0,0,0,1,1,0,0,0] => [1,2,5,3,4,6] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 5
[1,1,1,1,0,0,1,1,0,0,0,0] => [1,5,2,3,4,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 4
[1,1,1,1,0,1,0,1,0,0,0,0] => [5,6,1,2,3,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,1,0,1,0,0,0,0,0] => [6,1,2,3,4,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [2,3,4,5,7,1,6] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 4
[1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [2,3,4,1,6,5,7] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 4
[1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [2,3,4,6,7,1,5] => [1,5,2,3,4,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => 5
[1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [2,3,4,7,1,5,6] => [1,5,6,2,3,4,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7) => 4
[1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [2,3,1,5,6,4,7] => [1,5,6,2,3,4,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7) => 4
[1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [2,3,1,5,4,6,7] => [1,5,2,3,4,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => 5
[1,0,1,0,1,1,0,1,0,0,1,1,0,0] => [2,3,5,1,6,4,7] => [1,6,2,3,5,4,7] => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7) => 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [2,3,5,6,7,1,4] => [1,4,2,3,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => 6
[1,0,1,0,1,1,0,1,1,0,1,0,0,0] => [2,3,5,7,1,4,6] => [1,4,6,2,3,5,7] => ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7) => 6
[1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [2,3,1,4,6,5,7] => [1,4,6,2,3,5,7] => ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7) => 6
[1,0,1,0,1,1,1,0,0,1,1,0,0,0] => [2,3,1,6,4,5,7] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 4
[1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [2,3,6,7,1,4,5] => [1,4,5,2,3,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7) => 5
[1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [2,3,7,1,4,5,6] => [1,4,5,6,2,3,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7) => 4
[1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [2,1,4,5,6,3,7] => [1,4,5,6,2,3,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7) => 4
[1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,4,5,3,6,7] => [1,4,5,2,3,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7) => 5
[1,0,1,1,0,0,1,1,0,1,1,0,0,0] => [2,1,4,6,3,5,7] => [1,4,6,2,3,5,7] => ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7) => 6
[1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,4,3,5,6,7] => [1,4,2,3,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => 6
[1,0,1,1,0,1,0,0,1,0,1,1,0,0] => [2,4,1,5,6,3,7] => [1,5,6,2,4,3,7] => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7) => 2
[1,0,1,1,0,1,0,0,1,1,1,0,0,0] => [2,4,1,5,3,6,7] => [1,5,2,4,3,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7) => 3
[1,0,1,1,0,1,0,1,0,0,1,1,0,0] => [2,4,5,1,6,3,7] => [1,6,2,4,5,3,7] => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7) => 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [2,4,5,6,7,1,3] => [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7) => 7
[1,0,1,1,0,1,0,1,1,0,1,0,0,0] => [2,4,5,7,1,3,6] => [1,3,6,2,4,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7) => 6
[1,0,1,1,0,1,1,0,0,0,1,1,0,0] => [2,4,1,3,6,5,7] => [1,3,6,2,4,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7) => 6
[1,0,1,1,0,1,1,0,0,1,1,0,0,0] => [2,4,1,6,3,5,7] => [1,6,2,4,3,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7) => 4
[1,0,1,1,0,1,1,0,1,0,1,0,0,0] => [2,4,6,7,1,3,5] => [1,3,5,2,4,6,7] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 7
[1,0,1,1,0,1,1,1,0,1,0,0,0,0] => [2,4,7,1,3,5,6] => [1,3,5,6,2,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7) => 6
[1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [2,1,3,5,6,4,7] => [1,3,5,6,2,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7) => 6
[1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [2,1,3,5,4,6,7] => [1,3,5,2,4,6,7] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 7
[1,0,1,1,1,0,0,1,0,0,1,1,0,0] => [2,1,5,3,6,4,7] => [1,5,2,3,6,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7) => 6
[1,0,1,1,1,0,0,1,0,1,1,0,0,0] => [2,1,5,6,3,4,7] => [1,5,6,2,3,4,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7) => 4
[1,0,1,1,1,0,0,1,1,1,0,0,0,0] => [2,1,5,3,4,6,7] => [1,5,2,3,4,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => 5
[1,0,1,1,1,0,1,0,0,1,1,0,0,0] => [2,5,1,6,3,4,7] => [1,6,2,5,3,4,7] => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7) => 2
[1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [2,5,6,7,1,3,4] => [1,3,4,2,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => 6
[1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [2,5,7,1,3,4,6] => [1,3,4,6,2,5,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7) => 6
[1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [2,1,3,4,6,5,7] => [1,3,4,6,2,5,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7) => 6
[1,0,1,1,1,1,0,0,0,1,1,0,0,0] => [2,1,3,6,4,5,7] => [1,3,6,2,4,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7) => 6
[1,0,1,1,1,1,0,0,1,1,0,0,0,0] => [2,1,6,3,4,5,7] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 4
[1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [2,6,7,1,3,4,5] => [1,3,4,5,2,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => 5
[1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [2,7,1,3,4,5,6] => [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 4
[1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [1,3,4,5,6,2,7] => [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 4
[1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [1,3,4,5,2,6,7] => [1,3,4,5,2,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => 5
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Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
Map
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.
Map
permutation poset
Description
Sends a permutation to its permutation poset.
For a permutation π of length n, this poset has vertices
{(i,π(i)) : 1≤i≤n}
and the cover relation is given by (w,x)≤(y,z) if w≤y and x≤z.
For example, the permutation [3,1,5,4,2] is mapped to the poset with cover relations
{(2,1)≺(5,2), (2,1)≺(4,4), (2,1)≺(3,5), (1,3)≺(4,4), (1,3)≺(3,5)}.
For a permutation π of length n, this poset has vertices
{(i,π(i)) : 1≤i≤n}
and the cover relation is given by (w,x)≤(y,z) if w≤y and x≤z.
For example, the permutation [3,1,5,4,2] is mapped to the poset with cover relations
{(2,1)≺(5,2), (2,1)≺(4,4), (2,1)≺(3,5), (1,3)≺(4,4), (1,3)≺(3,5)}.
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