Identifier
-
Mp00201:
Dyck paths
—Ringel⟶
Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St001875: Lattices ⟶ ℤ
Values
[1,0] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[1,0,1,0] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,2),(2,1)],3) => 3
[1,1,0,0] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,2),(2,1)],3) => 3
[1,0,1,0,1,0] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,0,0] => [4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9) => ([(0,2),(2,1)],3) => 3
[1,1,1,0,0,0] => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9) => ([(0,2),(2,1)],3) => 3
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13) => ([(0,2),(2,1)],3) => 3
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12) => ([(0,2),(2,1)],3) => 3
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11) => ([(0,2),(2,1)],3) => 3
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13) => ([(0,2),(2,1)],3) => 3
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18) => ([(0,2),(2,1)],3) => 3
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(11,9),(12,10)],13) => ([(0,2),(2,1)],3) => 3
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,10),(6,11),(7,11),(8,12),(9,12),(11,8),(11,9),(12,10)],13) => ([(0,2),(2,1)],3) => 3
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(3,8),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13) => ([(0,2),(2,1)],3) => 3
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,11),(9,10),(11,9)],12) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(11,9),(12,10)],13) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(3,8),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10) => ([(0,2),(2,1)],3) => 3
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,11),(9,10),(11,9)],12) => ([(0,2),(2,1)],3) => 3
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10) => ([(0,2),(2,1)],3) => 3
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,13),(4,14),(5,11),(5,15),(6,12),(6,15),(8,11),(9,7),(10,7),(11,9),(12,10),(13,8),(14,8),(15,9),(15,10)],16) => ([(0,2),(2,1)],3) => 3
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,9),(4,12),(5,12),(6,10),(6,11),(8,7),(9,7),(10,13),(11,13),(12,8),(13,8),(13,9)],14) => ([(0,2),(2,1)],3) => 3
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18) => ([(0,2),(2,1)],3) => 3
[1,0,1,0,1,0,1,1,0,1,0,0] => [7,1,2,3,6,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,14),(2,15),(3,15),(4,13),(5,8),(6,14),(6,16),(7,12),(7,16),(9,11),(10,11),(11,8),(12,10),(13,12),(14,9),(15,13),(16,9),(16,10)],17) => ([(0,2),(2,1)],3) => 3
[1,0,1,0,1,1,0,1,0,0,1,0] => [7,1,2,5,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,13),(2,16),(3,14),(4,14),(5,12),(6,8),(7,13),(7,15),(9,11),(10,11),(11,8),(12,10),(13,9),(14,16),(15,9),(15,10),(16,12),(16,15)],17) => ([(0,2),(2,1)],3) => 3
[1,0,1,0,1,1,0,1,0,1,0,0] => [7,1,2,6,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,15),(2,14),(3,13),(4,12),(5,8),(6,14),(6,15),(7,11),(7,13),(9,12),(10,8),(11,10),(12,11),(13,10),(14,9),(15,9)],16) => ([(0,2),(2,1)],3) => 3
[1,0,1,0,1,1,1,0,1,0,0,0] => [7,1,2,5,6,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(2,10),(3,10),(4,9),(5,9),(6,8),(7,8),(7,12),(8,14),(9,13),(10,13),(12,14),(13,12),(14,11)],15) => ([(0,2),(2,1)],3) => 3
[1,0,1,1,0,1,0,0,1,0,1,0] => [7,1,4,2,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,13),(2,16),(3,14),(4,14),(5,12),(6,8),(7,13),(7,15),(9,11),(10,11),(11,8),(12,10),(13,9),(14,16),(15,9),(15,10),(16,12),(16,15)],17) => ([(0,2),(2,1)],3) => 3
[1,0,1,1,0,1,0,1,0,0,1,0] => [7,1,5,2,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,12),(2,11),(3,14),(4,13),(5,15),(6,8),(7,13),(7,14),(9,15),(10,8),(11,10),(12,10),(13,9),(14,9),(15,11),(15,12)],16) => ([(0,2),(2,1)],3) => 3
[1,0,1,1,0,1,1,0,0,1,0,0] => [7,1,4,2,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(2,10),(3,10),(4,10),(5,10),(6,10),(7,8),(8,9),(10,8)],11) => ([(0,2),(2,1)],3) => 3
[1,0,1,1,0,1,1,0,1,0,0,0] => [7,1,5,2,6,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,11),(3,11),(4,11),(5,8),(6,8),(7,9),(8,11),(9,10),(11,9)],12) => ([(0,2),(2,1)],3) => 3
[1,0,1,1,1,0,1,0,0,0,1,0] => [7,1,4,5,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(2,9),(3,8),(4,8),(5,11),(6,10),(7,12),(8,14),(9,14),(10,13),(11,13),(13,12),(14,10),(14,11)],15) => ([(0,2),(2,1)],3) => 3
[1,0,1,1,1,0,1,0,0,1,0,0] => [7,1,4,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,11),(3,11),(4,11),(5,8),(6,8),(7,9),(8,11),(9,10),(11,9)],12) => ([(0,2),(2,1)],3) => 3
[1,0,1,1,1,0,1,0,1,0,0,0] => [7,1,6,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,14),(2,13),(3,15),(4,12),(5,8),(6,13),(6,14),(7,11),(7,15),(9,11),(10,12),(11,10),(12,8),(13,9),(14,9),(15,10)],16) => ([(0,2),(2,1)],3) => 3
[1,0,1,1,1,1,0,1,0,0,0,0] => [7,1,4,5,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,12),(2,11),(3,10),(4,9),(5,8),(6,8),(7,9),(7,10),(8,14),(9,13),(10,13),(11,12),(13,14),(14,11)],15) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,0,0,1,0,1,0,1,0] => [7,3,1,2,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,14),(2,15),(3,15),(4,13),(5,8),(6,14),(6,16),(7,12),(7,16),(9,11),(10,11),(11,8),(12,10),(13,12),(14,9),(15,13),(16,9),(16,10)],17) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,0,0,1,1,0,1,0,0] => [7,3,1,2,6,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,9),(4,9),(5,12),(6,11),(7,10),(8,11),(9,12),(11,13),(12,13),(13,10)],14) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,0,1,0,0,1,0,1,0] => [7,4,1,2,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,15),(2,14),(3,13),(4,12),(5,8),(6,14),(6,15),(7,11),(7,13),(9,12),(10,8),(11,10),(12,11),(13,10),(14,9),(15,9)],16) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,0,1,1,0,0,1,0,0] => [7,4,1,2,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,10),(3,10),(4,10),(5,8),(6,8),(7,9),(8,10),(10,9)],11) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,1,0,0,1,0,0,1,0] => [7,3,1,5,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(2,10),(3,10),(4,10),(5,10),(6,10),(7,8),(8,9),(10,8)],11) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,1,0,0,1,0,1,0,0] => [7,3,1,6,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,10),(3,10),(4,10),(5,8),(6,8),(7,9),(8,10),(10,9)],11) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,1,0,1,0,0,0,1,0] => [7,4,1,5,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,11),(3,11),(4,11),(5,8),(6,8),(7,9),(8,11),(9,10),(11,9)],12) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,1,0,1,0,0,1,0,0] => [7,4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,10),(3,10),(4,10),(5,8),(6,8),(7,9),(8,10),(10,9)],11) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,1,1,0,0,1,0,0,0] => [7,3,1,5,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,10),(3,10),(4,10),(5,8),(6,8),(7,9),(8,10),(10,9)],11) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,1,1,0,1,0,0,0,0] => [7,4,1,5,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(2,11),(3,9),(4,9),(5,8),(6,8),(7,10),(8,11),(9,11),(11,10)],12) => ([(0,2),(2,1)],3) => 3
[1,1,1,0,1,0,0,0,1,0,1,0] => [7,3,4,1,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(2,10),(3,10),(4,9),(5,9),(6,8),(7,8),(7,12),(8,14),(9,13),(10,13),(12,14),(13,12),(14,11)],15) => ([(0,2),(2,1)],3) => 3
[1,1,1,0,1,0,0,1,0,0,1,0] => [7,3,5,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,11),(3,11),(4,11),(5,8),(6,8),(7,9),(8,11),(9,10),(11,9)],12) => ([(0,2),(2,1)],3) => 3
[1,1,1,0,1,0,1,0,0,0,1,0] => [7,5,4,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,14),(2,13),(3,15),(4,12),(5,8),(6,13),(6,14),(7,11),(7,15),(9,11),(10,12),(11,10),(12,8),(13,9),(14,9),(15,10)],16) => ([(0,2),(2,1)],3) => 3
[1,1,1,0,1,1,0,0,0,1,0,0] => [7,3,4,1,6,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,10),(3,10),(4,10),(5,8),(6,8),(7,9),(8,10),(10,9)],11) => ([(0,2),(2,1)],3) => 3
[1,1,1,0,1,1,0,0,1,0,0,0] => [7,3,5,1,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,8),(9,8)],10) => ([(0,2),(2,1)],3) => 3
[1,1,1,0,1,1,0,1,0,0,0,0] => [7,5,4,1,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(2,11),(3,9),(4,9),(5,8),(6,8),(7,10),(8,11),(9,11),(11,10)],12) => ([(0,2),(2,1)],3) => 3
[1,1,1,1,0,1,0,0,0,0,1,0] => [7,3,4,5,1,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,12),(2,11),(3,10),(4,9),(5,8),(6,8),(7,9),(7,10),(8,14),(9,13),(10,13),(11,12),(13,14),(14,11)],15) => ([(0,2),(2,1)],3) => 3
[1,1,1,1,0,1,0,0,0,1,0,0] => [7,3,4,6,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(2,11),(3,9),(4,9),(5,8),(6,8),(7,10),(8,11),(9,11),(11,10)],12) => ([(0,2),(2,1)],3) => 3
[1,1,1,1,0,1,0,0,1,0,0,0] => [7,3,6,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(2,11),(3,9),(4,9),(5,8),(6,8),(7,10),(8,11),(9,11),(11,10)],12) => ([(0,2),(2,1)],3) => 3
[1,1,1,1,0,1,0,1,0,0,0,0] => [7,6,4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,17),(2,17),(3,14),(4,13),(5,15),(6,13),(6,14),(7,15),(7,16),(9,11),(10,9),(11,8),(12,8),(13,10),(14,10),(15,12),(16,11),(16,12),(17,9),(17,16)],18) => ([(0,2),(2,1)],3) => 3
[1,1,1,1,1,0,1,0,0,0,0,0] => [7,3,4,5,6,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,12),(2,15),(3,15),(4,14),(5,13),(6,13),(6,17),(7,14),(7,17),(9,16),(10,16),(11,8),(12,8),(13,9),(14,10),(15,11),(16,11),(16,12),(17,9),(17,10)],18) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,0,1,1,0,0,1,0,1,0,0] => [8,4,1,2,7,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,10),(5,10),(6,9),(7,9),(8,11),(9,12),(10,12),(12,11)],13) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,0,1,1,1,0,0,1,0,0,0] => [8,4,1,2,6,7,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,10),(5,10),(6,9),(7,9),(8,11),(9,12),(10,12),(12,11)],13) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,1,0,0,1,1,0,0,1,0,0] => [8,3,1,5,2,7,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,9),(10,9)],11) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,1,0,0,1,1,0,1,0,0,0] => [8,3,1,6,2,7,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,11),(3,11),(4,11),(5,11),(6,9),(7,9),(8,10),(9,11),(11,10)],12) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,1,0,1,1,0,0,0,1,0,0] => [8,4,1,5,2,7,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,9),(10,9)],11) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,1,0,1,1,0,0,1,0,0,0] => [8,4,1,6,2,7,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,9),(10,9)],11) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,1,1,0,0,1,0,0,1,0,0] => [8,3,1,5,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,9),(10,9)],11) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,1,1,0,0,1,0,1,0,0,0] => [8,3,1,7,6,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,10),(5,10),(6,9),(7,9),(8,11),(9,12),(10,12),(12,11)],13) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,1,1,0,1,0,0,0,1,0,0] => [8,4,1,5,7,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,11),(3,11),(4,11),(5,11),(6,9),(7,9),(8,10),(9,11),(11,10)],12) => ([(0,2),(2,1)],3) => 3
[1,1,0,1,1,1,0,1,0,0,1,0,0,0] => [8,4,1,7,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,10),(5,10),(6,9),(7,9),(8,11),(9,12),(10,12),(12,11)],13) => ([(0,2),(2,1)],3) => 3
[1,1,1,0,1,0,0,1,1,0,0,1,0,0] => [8,3,5,1,2,7,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,11),(3,11),(4,11),(5,11),(6,9),(7,9),(8,10),(9,11),(11,10)],12) => ([(0,2),(2,1)],3) => 3
[1,1,1,0,1,0,1,1,0,0,0,1,0,0] => [8,5,4,1,2,7,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,10),(5,10),(6,9),(7,9),(8,11),(9,12),(10,12),(12,11)],13) => ([(0,2),(2,1)],3) => 3
[1,1,1,0,1,1,0,0,0,1,0,1,0,0] => [8,3,4,1,7,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,10),(5,10),(6,9),(7,9),(8,11),(9,12),(10,12),(12,11)],13) => ([(0,2),(2,1)],3) => 3
[1,1,1,0,1,1,0,0,1,0,0,1,0,0] => [8,3,5,1,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,9),(10,9)],11) => ([(0,2),(2,1)],3) => 3
[1,1,1,0,1,1,0,1,0,0,0,1,0,0] => [8,5,4,1,7,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,10),(5,10),(6,9),(7,9),(8,11),(9,12),(10,12),(12,11)],13) => ([(0,2),(2,1)],3) => 3
[1,1,1,0,1,1,1,0,0,0,1,0,0,0] => [8,3,4,1,6,7,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,10),(5,10),(6,9),(7,9),(8,11),(9,12),(10,12),(12,11)],13) => ([(0,2),(2,1)],3) => 3
[1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [8,3,5,1,6,7,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,11),(3,11),(4,11),(5,11),(6,9),(7,9),(8,10),(9,11),(11,10)],12) => ([(0,2),(2,1)],3) => 3
[1,1,1,1,0,1,1,0,0,0,1,0,0,0] => [8,3,4,6,1,7,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,11),(3,11),(4,11),(5,11),(6,9),(7,9),(8,10),(9,11),(11,10)],12) => ([(0,2),(2,1)],3) => 3
[1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [8,3,6,5,1,7,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,11),(3,11),(4,11),(5,11),(6,9),(7,9),(8,10),(9,11),(11,10)],12) => ([(0,2),(2,1)],3) => 3
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Description
The number of simple modules with projective dimension at most 1.
Map
lattice of intervals
Description
The lattice of intervals of a permutation.
An interval of a permutation $\pi$ is a possibly empty interval of values that appear in consecutive positions of $\pi$. The lattice of intervals of $\pi$ has as elements the intervals of $\pi$, ordered by set inclusion.
An interval of a permutation $\pi$ is a possibly empty interval of values that appear in consecutive positions of $\pi$. The lattice of intervals of $\pi$ has as elements the intervals of $\pi$, ordered by set inclusion.
Map
lattice of congruences
Description
The lattice of congruences of a lattice.
A congruence of a lattice is an equivalence relation such that $a_1 \cong a_2$ and $b_1 \cong b_2$ implies $a_1 \vee b_1 \cong a_2 \vee b_2$ and $a_1 \wedge b_1 \cong a_2 \wedge b_2$.
The set of congruences ordered by refinement forms a lattice.
A congruence of a lattice is an equivalence relation such that $a_1 \cong a_2$ and $b_1 \cong b_2$ implies $a_1 \vee b_1 \cong a_2 \vee b_2$ and $a_1 \wedge b_1 \cong a_2 \wedge b_2$.
The set of congruences ordered by refinement forms a lattice.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
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