- FindStat

Identifier

Mp00263: Lattices —join irreducibles⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001876: Lattices ⟶ ℤ (values match St001877Number of indecomposable injective modules with projective dimension 2.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

maximal antichains

Description

The lattice of maximal antichains in a poset.
An antichain

$A$ in a poset is maximal if there is no antichain of larger cardinality which contains all elements of

$A$ .
The set of maximal antichains 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

$A$ .

Map

join irreducibles

Description

The poset of join irreducibles of a lattice.