Identifier
-
Mp00032:
Dyck paths
—inverse zeta map⟶
Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001876: Lattices ⟶ ℤ (values match St001877Number of indecomposable injective modules with projective dimension 2.)
Values
[1,1,1,0,0,0] => [1,0,1,0,1,0] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 0
[1,1,1,0,0,0,1,0] => [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
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => ([(1,4),(2,3),(2,4)],5) => ([(0,2),(2,1)],3) => 0
[1,1,1,0,0,0,1,1,0,0] => [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
[1,1,1,0,0,1,0,1,0,0] => [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
[1,1,1,0,1,0,0,1,0,0] => [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
[1,1,1,1,0,0,0,0,1,0] => [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
[1,1,1,1,0,0,0,1,0,0] => [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
[1,1,1,1,1,0,0,0,0,0] => [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
[1,1,1,0,0,0,1,0,1,0,1,0] => [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
[1,1,1,0,0,0,1,0,1,1,0,0] => [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
[1,1,1,0,0,0,1,1,0,0,1,0] => [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
[1,1,1,0,0,0,1,1,0,1,0,0] => [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
[1,1,1,0,0,0,1,1,1,0,0,0] => [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
[1,1,1,0,0,1,0,1,0,0,1,0] => [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)],6) => ([(0,2),(2,1)],3) => 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] => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => ([(0,2),(2,1)],3) => 0
[1,1,1,0,0,1,1,0,1,0,0,0] => [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
[1,1,1,0,1,0,0,1,0,0,1,0] => [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)],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
[1,1,1,0,1,0,1,0,1,0,0,0] => [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
[1,1,1,0,1,1,0,0,1,0,0,0] => [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
[1,1,1,1,0,0,0,0,1,0,1,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => ([(0,3),(2,1),(3,2)],4) => 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] => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6) => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,1,1,0,0,0,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6) => ([(0,2),(2,1)],3) => 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] => ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6) => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,1,1,0,0,0,1,1,0,0,0] => [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
[1,1,1,1,0,0,1,0,0,1,0,0] => [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
[1,1,1,1,0,0,1,0,1,0,0,0] => [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
[1,1,1,1,0,1,0,0,0,1,0,0] => [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
[1,1,1,1,0,1,0,0,1,0,0,0] => [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
[1,1,1,1,1,0,0,0,0,0,1,0] => [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
[1,1,1,1,1,0,0,0,0,1,0,0] => [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
[1,1,1,1,1,0,0,0,1,0,0,0] => [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
[1,1,1,1,1,1,0,0,0,0,0,0] => [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
[1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [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
[1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [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
[1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [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
[1,1,1,0,0,0,1,0,1,1,0,1,0,0] => [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
[1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => ([(0,6),(1,5),(2,3),(2,5),(3,6),(5,6),(6,4)],7) => ([(0,2),(2,1)],3) => 0
[1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [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
[1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [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
[1,1,1,0,0,0,1,1,0,1,0,0,1,0] => [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
[1,1,1,0,0,0,1,1,0,1,0,1,0,0] => [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
[1,1,1,0,0,0,1,1,0,1,1,0,0,0] => [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
[1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [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
[1,1,1,0,0,0,1,1,1,0,0,1,0,0] => [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
[1,1,1,0,0,0,1,1,1,0,1,0,0,0] => [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] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => ([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7) => ([(0,2),(2,1)],3) => 0
[1,1,1,0,0,1,0,1,0,0,1,0,1,0] => [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
[1,1,1,0,0,1,0,1,0,0,1,1,0,0] => [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
[1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [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
[1,1,1,0,0,1,0,1,1,0,0,0,1,0] => [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
[1,1,1,0,0,1,0,1,1,0,0,1,0,0] => [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
[1,1,1,0,0,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,1,0,1,0,0] => ([(0,3),(1,5),(1,6),(3,5),(3,6),(4,2),(5,4),(6,4)],7) => ([(0,2),(2,1)],3) => 0
[1,1,1,0,0,1,1,0,1,0,0,0,1,0] => [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
[1,1,1,0,0,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,1,0,1,0,0] => ([(0,6),(1,2),(2,6),(3,5),(4,5),(6,3),(6,4)],7) => ([(0,2),(2,1)],3) => 0
[1,1,1,0,0,1,1,1,0,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,1,0,1,0,0] => ([(0,6),(1,4),(4,6),(5,2),(5,3),(6,5)],7) => ([(0,2),(2,1)],3) => 0
[1,1,1,0,1,0,0,1,0,0,1,0,1,0] => [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
[1,1,1,0,1,0,0,1,0,0,1,1,0,0] => [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
[1,1,1,0,1,0,0,1,0,1,0,1,0,0] => [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
[1,1,1,0,1,0,0,1,1,0,0,0,1,0] => [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
[1,1,1,0,1,0,0,1,1,0,0,1,0,0] => [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
[1,1,1,0,1,0,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,0,1,0,0,1,1,0,0] => ([(0,5),(0,6),(1,5),(1,6),(3,4),(4,2),(5,3),(6,4)],7) => ([(0,2),(2,1)],3) => 0
[1,1,1,0,1,0,1,0,1,0,0,0,1,0] => [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
[1,1,1,0,1,0,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => ([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7) => ([(0,2),(2,1)],3) => 0
[1,1,1,0,1,0,1,1,0,1,0,0,0,0] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => ([(0,5),(0,6),(1,5),(1,6),(4,2),(4,3),(5,4),(6,4)],7) => ([(0,2),(2,1)],3) => 0
[1,1,1,0,1,1,0,0,1,0,0,0,1,0] => [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
[1,1,1,0,1,1,0,0,1,1,0,0,0,0] => [1,0,1,1,0,1,0,0,1,0,1,1,0,0] => ([(0,6),(1,6),(2,5),(3,5),(4,3),(6,2),(6,4)],7) => ([(0,2),(2,1)],3) => 0
[1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(6,2),(6,3)],7) => ([(0,2),(2,1)],3) => 0
[1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [1,1,0,1,0,0,1,0,1,0,1,1,0,0] => ([(0,6),(1,6),(4,3),(5,2),(5,4),(6,5)],7) => ([(0,2),(2,1)],3) => 0
[1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [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
[1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [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
[1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [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
[1,1,1,1,0,0,0,0,1,1,0,1,0,0] => [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
[1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [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
[1,1,1,1,0,0,0,1,0,0,1,0,1,0] => [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
[1,1,1,1,0,0,0,1,0,0,1,1,0,0] => [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
[1,1,1,1,0,0,0,1,0,1,0,0,1,0] => [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
[1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [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
[1,1,1,1,0,0,0,1,0,1,1,0,0,0] => [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
[1,1,1,1,0,0,0,1,1,0,0,0,1,0] => [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
[1,1,1,1,0,0,0,1,1,0,0,1,0,0] => [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
[1,1,1,1,0,0,0,1,1,0,1,0,0,0] => [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
[1,1,1,1,0,0,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,0,1,0,1,0,0,1,0] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => ([(0,2),(2,1)],3) => 0
[1,1,1,1,0,0,1,0,0,1,0,0,1,0] => [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
[1,1,1,1,0,0,1,0,0,1,1,0,0,0] => [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
[1,1,1,1,0,0,1,0,1,0,0,0,1,0] => [1,1,1,0,0,1,1,0,1,0,0,1,0,0] => ([(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
[1,1,1,1,0,0,1,0,1,0,1,0,0,0] => [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
[1,1,1,1,0,0,1,0,1,1,0,0,0,0] => [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),(4,6),(5,6)],7) => ([(0,2),(2,1)],3) => 0
[1,1,1,1,0,0,1,1,0,0,1,0,0,0] => [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
[1,1,1,1,0,0,1,1,0,1,0,0,0,0] => [1,1,0,0,1,0,1,1,0,1,0,0,1,0] => ([(0,4),(0,5),(1,6),(4,6),(5,1),(6,2),(6,3)],7) => ([(0,2),(2,1)],3) => 0
[1,1,1,1,0,1,0,0,0,1,0,0,1,0] => [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
[1,1,1,1,0,1,0,0,0,1,1,0,0,0] => [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
[1,1,1,1,0,1,0,0,1,0,0,0,1,0] => [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
[1,1,1,1,0,1,0,0,1,0,1,0,0,0] => [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
[1,1,1,1,0,1,0,0,1,1,0,0,0,0] => [1,0,1,1,0,1,0,0,1,1,0,0,1,0] => ([(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1)],7) => ([(0,2),(2,1)],3) => 0
[1,1,1,1,0,1,0,1,0,0,1,0,0,0] => [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
[1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7) => ([(0,2),(2,1)],3) => 0
[1,1,1,1,0,1,1,0,0,0,1,0,0,0] => [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
[1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [1,1,0,1,0,0,1,0,1,1,0,0,1,0] => ([(0,3),(0,4),(3,6),(4,6),(5,1),(6,2),(6,5)],7) => ([(0,2),(2,1)],3) => 0
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Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
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.
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.
See its inverse, the zeta map Mp00030zeta map, for the definition and details.
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