Identifier
-
Mp00101:
Dyck paths
—decomposition reverse⟶
Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001877: Lattices ⟶ ℤ (values match St001876The number of 2-regular simple modules in the incidence algebra of the lattice.)
Values
[1,1,1,0,0,0] => [1,0,1,0,1,0] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 0
[1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => ([(0,3),(1,2),(1,3)],4) => ([(0,2),(2,1)],3) => 0
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => ([(0,3),(2,1),(3,2)],4) => 0
[1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => ([(1,4),(2,3),(2,4)],5) => ([(0,2),(2,1)],3) => 0
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,0,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => ([(0,2),(2,1)],3) => 0
[1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => ([(0,2),(2,1)],3) => 0
[1,1,0,1,1,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => ([(0,2),(2,1)],3) => 0
[1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,2),(2,1)],3) => 0
[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,4),(2,3),(3,1),(4,2)],5) => 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] => ([(2,5),(3,4),(3,5)],6) => ([(0,2),(2,1)],3) => 0
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => ([(0,3),(2,1),(3,2)],4) => 0
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6) => ([(0,2),(2,1)],3) => 0
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => ([(0,2),(2,1)],3) => 0
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => ([(0,2),(2,1)],3) => 0
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6) => ([(0,2),(2,1)],3) => 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] => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6) => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,1,0,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0,1,0] => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6) => ([(0,2),(2,1)],3) => 0
[1,1,0,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6) => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,0,1,0,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,1,0,0] => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6) => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,0,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,0,1,0,1,0,0,0] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,2),(2,1)],3) => 0
[1,1,0,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,1,0,0] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6) => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,0,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,1,0,0] => ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6) => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,0,1,1,0,1,0,1,0,1,0,0] => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6) => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,1,0,0,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6) => ([(0,2),(2,1)],3) => 0
[1,1,1,0,0,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0,1,0] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6) => ([(0,2),(2,1)],3) => 0
[1,1,1,0,0,1,1,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0,1,0] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6) => ([(0,2),(2,1)],3) => 0
[1,1,1,0,0,1,1,1,0,0,0,0] => [1,0,1,1,0,1,0,1,0,0,1,0] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,2),(2,1)],3) => 0
[1,1,1,0,1,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0,1,1,0,0] => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6) => ([(0,2),(2,1)],3) => 0
[1,1,1,0,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6) => ([(0,2),(2,1)],3) => 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] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6) => ([(0,2),(2,1)],3) => 0
[1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,1,0,0] => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6) => ([(0,2),(2,1)],3) => 0
[1,1,1,0,1,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,1,0,0] => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => ([(0,2),(2,1)],3) => 0
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,0,1,0,1,1,0,1,0,1,0,0] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6) => ([(0,2),(2,1)],3) => 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] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 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] => ([(3,6),(4,5),(4,6)],7) => ([(0,2),(2,1)],3) => 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => ([(0,3),(2,1),(3,2)],4) => 0
[1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7) => ([(0,2),(2,1)],3) => 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,1,1,0,1,0,0,1,1,0,0,0,0] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => ([(0,2),(2,1)],3) => 0
[1,0,1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,1,1,0,0,1,1,0,1,0,0,0,0] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => ([(0,2),(2,1)],3) => 0
[1,0,1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,0,1,0,1,0,0,0,0] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => ([(0,2),(2,1)],3) => 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] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,6)],7) => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(4,5)],7) => ([(0,2),(2,1)],3) => 0
[1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(2,6),(3,5),(3,6)],7) => ([(0,3),(2,1),(3,2)],4) => 0
[1,0,1,1,0,1,0,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,1,1,0,0,0] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(4,5)],7) => ([(0,3),(2,1),(3,2)],4) => 0
[1,0,1,1,0,1,0,1,1,1,0,0,0,0] => [1,1,0,1,1,1,0,1,0,1,0,0,0,0] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(4,6)],7) => ([(0,2),(2,1)],3) => 0
[1,0,1,1,0,1,1,0,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,1,0,1,0,0,0] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,6)],7) => ([(0,3),(2,1),(3,2)],4) => 0
[1,0,1,1,0,1,1,1,0,0,0,0,1,0] => [1,1,1,0,0,1,1,0,1,0,1,0,0,0] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6)],7) => ([(0,3),(2,1),(3,2)],4) => 0
[1,0,1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,0,1,1,0,1,0,1,0,1,0,0,0] => ([(0,5),(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6)],7) => ([(0,3),(2,1),(3,2)],4) => 0
[1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0] => ([(0,2),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(6,3)],7) => ([(0,2),(2,1)],3) => 0
[1,0,1,1,1,0,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,1,0,0,1,0,0] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(2,6),(3,4),(3,5),(3,6)],7) => ([(0,2),(2,1)],3) => 0
[1,0,1,1,1,0,0,1,1,0,0,0,1,0] => [1,1,1,0,0,1,1,0,1,0,0,1,0,0] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6)],7) => ([(0,2),(2,1)],3) => 0
[1,0,1,1,1,0,0,1,1,1,0,0,0,0] => [1,1,0,1,1,0,1,0,1,0,0,1,0,0] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,6),(3,5),(3,6),(4,6)],7) => ([(0,2),(2,1)],3) => 0
[1,0,1,1,1,0,1,0,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,0,1,1,0,0,0] => ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4)],7) => ([(0,2),(2,1)],3) => 0
[1,0,1,1,1,0,1,0,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,1,1,0,0,0] => ([(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(6,3),(6,4)],7) => ([(0,2),(2,1)],3) => 0
[1,0,1,1,1,0,1,0,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,0,1,1,0,0,0] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6)],7) => ([(0,2),(2,1)],3) => 0
[1,0,1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,1,0,0,1,0,1,1,0,1,0,0,0] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6)],7) => ([(0,2),(2,1)],3) => 0
[1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [1,1,0,1,1,0,0,1,1,0,1,0,0,0] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(6,4)],7) => ([(0,2),(2,1)],3) => 0
[1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,0,1,0,1,1,0,1,0,1,0,0,0] => ([(0,5),(0,6),(1,4),(1,5),(2,3),(2,4),(3,5),(3,6),(4,6)],7) => ([(0,2),(2,1)],3) => 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] => ([(0,2),(0,6),(1,5),(1,6),(2,4),(2,5),(5,3),(6,3),(6,4)],7) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0] => ([(0,2),(0,3),(0,4),(0,5),(4,6),(5,1),(5,6)],7) => ([(0,2),(2,1)],3) => 0
[1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0,1,0] => ([(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,1),(4,5),(4,6)],7) => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,1,0,0,1,0] => ([(0,3),(0,4),(2,5),(3,5),(3,6),(4,1),(4,2),(4,6)],7) => ([(0,2),(2,1)],3) => 0
[1,1,0,0,1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => ([(0,2),(2,1)],3) => 0
[1,1,0,0,1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,0,1,1,0,1,0,0,0,1,0] => ([(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,1),(4,5),(4,6)],7) => ([(0,2),(2,1)],3) => 0
[1,1,0,0,1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0,1,0] => ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(3,6),(4,1),(4,5)],7) => ([(0,2),(2,1)],3) => 0
[1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0] => ([(0,2),(0,3),(1,4),(1,6),(2,4),(2,5),(3,1),(3,5),(5,6)],7) => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,1,0,1,0,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0,1,1,0,0] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(5,3),(6,2),(6,3)],7) => ([(0,2),(2,1)],3) => 0
[1,1,0,1,0,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0,1,1,0,0] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(5,2),(5,3),(6,3),(6,4)],7) => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,1,0,1,0,1,0,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0,1,1,1,0,0,0] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(6,3)],7) => ([(0,2),(2,1)],3) => 0
[1,1,0,1,0,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => ([(0,6),(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => ([(0,2),(2,1)],3) => 0
[1,1,0,1,0,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,1,0,1,0,0,0] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7) => ([(0,2),(2,1)],3) => 0
[1,1,0,1,0,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,1,1,0,1,0,1,0,0,0] => ([(0,5),(0,6),(1,4),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,2),(2,1)],3) => 0
[1,1,0,1,0,1,1,1,1,0,0,0,0,0] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7) => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,0,1,1,0,0,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,1,0,1,0,0] => ([(0,3),(1,5),(1,6),(3,5),(3,6),(5,4),(6,2),(6,4)],7) => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,1,0,1,1,0,0,1,1,1,0,0,0,0] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,5),(3,6),(4,6),(5,6)],7) => ([(0,2),(2,1)],3) => 0
[1,1,0,1,1,0,1,0,0,1,1,0,0,0] => [1,0,1,1,1,0,1,0,0,1,1,0,0,0] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7) => ([(0,2),(2,1)],3) => 0
[1,1,0,1,1,0,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,0,1,1,0,1,0,0,0] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7) => ([(0,2),(2,1)],3) => 0
[1,1,0,1,1,0,1,1,1,0,0,0,0,0] => [1,0,1,1,0,1,1,0,1,0,1,0,0,0] => ([(0,6),(1,4),(1,6),(2,3),(2,4),(3,6),(4,5),(6,5)],7) => ([(0,2),(2,1)],3) => 0
[1,1,0,1,1,1,0,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,1,0,1,0,0] => ([(0,6),(1,3),(1,6),(3,4),(3,5),(5,2),(6,4),(6,5)],7) => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,1,0,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,1,0,1,0,0] => ([(0,2),(0,6),(1,5),(1,6),(2,5),(5,3),(5,4),(6,3),(6,4)],7) => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => ([(0,2),(0,6),(1,5),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4)],7) => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0,1,0,1,0] => ([(0,5),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4)],7) => ([(0,2),(2,1)],3) => 0
[1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0] => ([(0,4),(1,5),(2,5),(2,6),(3,1),(3,6),(4,2),(4,3)],7) => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,1,0,0,1,0,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,1,0,0,1,0] => ([(0,2),(0,3),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1),(6,5)],7) => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,1,0,0,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,0,1,0,1,0,0,0,1,0] => ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7) => ([(0,2),(2,1)],3) => 0
[1,1,1,0,0,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,1,0,0,1,0] => ([(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(6,1)],7) => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,1,0,0,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,1,0,0,1,0] => ([(0,2),(0,3),(1,5),(1,6),(2,4),(3,1),(3,4),(4,5),(4,6)],7) => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,1,0,0,1,1,1,1,0,0,0,0,0] => [1,0,1,1,0,1,0,1,0,1,0,0,1,0] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5),(4,6),(5,6)],7) => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,1,0,1,0,0,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0,1,1,0,0] => ([(0,6),(1,6),(3,5),(4,2),(4,5),(6,3),(6,4)],7) => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,1,0,1,0,0,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0,1,1,0,0] => ([(0,5),(0,6),(1,5),(1,6),(4,2),(5,3),(5,4),(6,3),(6,4)],7) => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,1,0,1,0,0,1,1,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0,1,1,0,0] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(5,2),(6,3),(6,4)],7) => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,1,0,1,0,0,1,1,1,0,0,0,0] => [1,0,1,1,0,1,0,1,0,0,1,1,0,0] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(4,3),(5,4),(6,2),(6,4)],7) => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => ([(0,6),(1,5),(2,3),(2,5),(3,6),(5,6),(6,4)],7) => ([(0,2),(2,1)],3) => 0
[1,1,1,0,1,1,0,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0,1,1,0,1,0,0] => ([(0,6),(1,3),(3,6),(5,2),(6,4),(6,5)],7) => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,1,0,1,1,0,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,1,0,0] => ([(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7) => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,1,0,1,1,0,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,1,0,1,0,0] => ([(0,3),(1,4),(1,6),(2,5),(3,4),(3,6),(4,2),(6,5)],7) => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,1,0,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,1,0,1,0,0] => ([(0,5),(1,4),(1,5),(4,6),(5,6),(6,2),(6,3)],7) => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,1,0,1,1,1,0,0,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,1,0,1,0,0] => ([(0,6),(1,2),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4),(6,5)],7) => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => ([(0,3),(0,6),(1,5),(1,6),(3,5),(4,2),(5,4),(6,4)],7) => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7) => ([(0,2),(2,1)],3) => 0
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Description
Number of indecomposable injective modules with projective dimension 2.
Map
decomposition reverse
Description
This map is recursively defined as follows.
The unique empty path of semilength 0 is sent to itself.
Let D be a Dyck path of semilength n>0 and decompose it into 1D10D2 with Dyck paths D1,D2 of respective semilengths n1 and n2 such that n1 is minimal. One then has n1+n2=n−1.
Now let ˜D1 and ˜D2 be the recursively defined respective images of D1 and D2 under this map. The image of D is then defined as 1˜D20˜D1.
The unique empty path of semilength 0 is sent to itself.
Let D be a Dyck path of semilength n>0 and decompose it into 1D10D2 with Dyck paths D1,D2 of respective semilengths n1 and n2 such that n1 is minimal. One then has n1+n2=n−1.
Now let ˜D1 and ˜D2 be the recursively defined respective images of D1 and D2 under this map. The image of D is then defined as 1˜D20˜D1.
Map
Hessenberg poset
Description
The Hessenberg poset of a Dyck path.
Let D be a Dyck path of semilength n, regarded as a subdiagonal path from (0,0) to (n,n), and let \boldsymbol{m}_i be the x-coordinate of the i-th up step.
Then the Hessenberg poset (or natural unit interval order) corresponding to D has elements \{1,\dots,n\} with i < j if j < \boldsymbol{m}_i.
Let D be a Dyck path of semilength n, regarded as a subdiagonal path from (0,0) to (n,n), and let \boldsymbol{m}_i be the x-coordinate of the i-th up step.
Then the Hessenberg poset (or natural unit interval order) corresponding to D has elements \{1,\dots,n\} with i < j if j < \boldsymbol{m}_i.
Map
antichains of maximal size
Description
The lattice of antichains of maximal size in a poset.
The set of antichains of maximal size can be ordered by setting A \leq B \leftrightarrow \mathop{\downarrow} A \subseteq \mathop{\downarrow} B, where \mathop{\downarrow} A is the order ideal generated by A.
This is a sublattice of the lattice of all antichains with respect to the same order relation. In particular, it is distributive.
The set of antichains of maximal size can be ordered by setting A \leq B \leftrightarrow \mathop{\downarrow} A \subseteq \mathop{\downarrow} B, where \mathop{\downarrow} A is the order ideal generated by A.
This is a sublattice of the lattice of all antichains with respect to the same order relation. In particular, it is distributive.
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