Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001875
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1},{2},{3}}
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
{{1,2,3,4}}
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
{{1,3},{2,4}}
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
{{1},{2},{3},{4}}
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
{{1,2,3,4,5}}
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 4
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 3
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 4
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 3
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
{{1},{2,4},{3,5}}
 => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
{{1,2,3,4,5},{6}}
 => [2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
{{1,2,3,4},{5,6}}
 => [2,3,4,1,6,5] => ([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
{{1,2,3,4},{5},{6}}
 => [2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
{{1,2,3,5},{4,6}}
 => [2,3,5,6,1,4] => ([(0,5),(1,4),(3,2),(4,3),(4,5)],6)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 5
{{1,2,3},{4,5,6}}
 => [2,3,1,5,6,4] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
{{1,2,3},{4,5},{6}}
 => [2,3,1,5,4,6] => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
{{1,2,3},{4},{5,6}}
 => [2,3,1,4,6,5] => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3
{{1,2,4,6},{3,5}}
 => [2,4,5,6,3,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3
{{1,2,4},{3,5,6}}
 => [2,4,5,1,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(3,5)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 5
{{1,2,4},{3,5},{6}}
 => [2,4,5,1,3,6] => ([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 4
{{1,2},{3,4,5,6}}
 => [2,1,4,5,6,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
{{1,2},{3,4,5},{6}}
 => [2,1,4,5,3,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
{{1,2},{3,4},{5,6}}
 => [2,1,4,3,6,5] => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
{{1,2,5},{3,6},{4}}
 => [2,5,6,4,1,3] => ([(0,5),(1,3),(1,4),(1,5),(4,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
{{1,2},{3,5},{4,6}}
 => [2,1,5,6,3,4] => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 4
{{1,2},{3},{4,5,6}}
 => [2,1,3,5,6,4] => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3
{{1,2,6},{3},{4},{5}}
 => [2,6,3,4,5,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3
{{1,3,4,5},{2,6}}
 => [3,6,4,5,1,2] => ([(0,4),(1,3),(1,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
{{1,3,4},{2,6},{5}}
 => [3,6,4,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
{{1,3,5,6},{2,4}}
 => [3,4,5,2,6,1] => ([(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
{{1,3,5},{2,4},{6}}
 => [3,4,5,2,1,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
{{1,3},{2,4,5,6}}
 => [3,4,1,5,6,2] => ([(0,4),(1,3),(1,5),(4,5),(5,2)],6)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 5
{{1,3},{2,4,5},{6}}
 => [3,4,1,5,2,6] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 4
{{1,3,6},{2,4},{5}}
 => [3,4,6,2,5,1] => ([(1,5),(2,3),(3,4),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
{{1,3},{2,4},{5,6}}
 => [3,4,1,2,6,5] => ([(0,3),(1,2),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 4
{{1,3},{2,4},{5},{6}}
 => [3,4,1,2,5,6] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
{{1,3},{2,5},{4,6}}
 => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,4),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 5
{{1,3,6},{2},{4},{5}}
 => [3,2,6,4,5,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
{{1,4,5},{2,3,6}}
 => [4,3,6,5,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
{{1,4},{2,3,6},{5}}
 => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
{{1,5},{2,3,4,6}}
 => [5,3,4,6,1,2] => ([(0,5),(1,3),(2,4),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
{{1,6},{2,3,4,5}}
 => [6,3,4,5,2,1] => ([(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St001615
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001615: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1},{2},{3}}
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,3,4}}
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,3},{2,4}}
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3 = 4 - 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3 = 4 - 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1},{2,4},{3,5}}
 => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
{{1,2,3,4,5},{6}}
 => [2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
{{1,2,3,4},{5,6}}
 => [2,3,4,1,6,5] => ([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
{{1,2,3,4},{5},{6}}
 => [2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,3,5},{4,6}}
 => [2,3,5,6,1,4] => ([(0,5),(1,4),(3,2),(4,3),(4,5)],6)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 4 = 5 - 1
{{1,2,3},{4,5,6}}
 => [2,3,1,5,6,4] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
{{1,2,3},{4,5},{6}}
 => [2,3,1,5,4,6] => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,3},{4},{5,6}}
 => [2,3,1,4,6,5] => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,4,6},{3,5}}
 => [2,4,5,6,3,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,4},{3,5,6}}
 => [2,4,5,1,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(3,5)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 4 = 5 - 1
{{1,2,4},{3,5},{6}}
 => [2,4,5,1,3,6] => ([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3 = 4 - 1
{{1,2},{3,4,5,6}}
 => [2,1,4,5,6,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
{{1,2},{3,4,5},{6}}
 => [2,1,4,5,3,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2},{3,4},{5,6}}
 => [2,1,4,3,6,5] => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,5},{3,6},{4}}
 => [2,5,6,4,1,3] => ([(0,5),(1,3),(1,4),(1,5),(4,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1,2},{3,5},{4,6}}
 => [2,1,5,6,3,4] => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3 = 4 - 1
{{1,2},{3},{4,5,6}}
 => [2,1,3,5,6,4] => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,6},{3},{4},{5}}
 => [2,6,3,4,5,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,3,4,5},{2,6}}
 => [3,6,4,5,1,2] => ([(0,4),(1,3),(1,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1,3,4},{2,6},{5}}
 => [3,6,4,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,3,5,6},{2,4}}
 => [3,4,5,2,6,1] => ([(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,3,5},{2,4},{6}}
 => [3,4,5,2,1,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,3},{2,4,5,6}}
 => [3,4,1,5,6,2] => ([(0,4),(1,3),(1,5),(4,5),(5,2)],6)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 4 = 5 - 1
{{1,3},{2,4,5},{6}}
 => [3,4,1,5,2,6] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3 = 4 - 1
{{1,3,6},{2,4},{5}}
 => [3,4,6,2,5,1] => ([(1,5),(2,3),(3,4),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
{{1,3},{2,4},{5,6}}
 => [3,4,1,2,6,5] => ([(0,3),(1,2),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3 = 4 - 1
{{1,3},{2,4},{5},{6}}
 => [3,4,1,2,5,6] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1,3},{2,5},{4,6}}
 => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,4),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 4 = 5 - 1
{{1,3,6},{2},{4},{5}}
 => [3,2,6,4,5,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,4,5},{2,3,6}}
 => [4,3,6,5,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1,4},{2,3,6},{5}}
 => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,5},{2,3,4,6}}
 => [5,3,4,6,1,2] => ([(0,5),(1,3),(2,4),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1,6},{2,3,4,5}}
 => [6,3,4,5,2,1] => ([(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
Description
The number of join prime elements of a lattice. An element $x$ of a lattice $L$ is join-prime (or coprime) if $x \leq a \vee b$ implies $x \leq a$ or $x \leq b$ for every $a, b \in L$.
Matching statistic: St001617
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001617: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1},{2},{3}}
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,3,4}}
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,3},{2,4}}
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3 = 4 - 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3 = 4 - 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1},{2,4},{3,5}}
 => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
{{1,2,3,4,5},{6}}
 => [2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
{{1,2,3,4},{5,6}}
 => [2,3,4,1,6,5] => ([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
{{1,2,3,4},{5},{6}}
 => [2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,3,5},{4,6}}
 => [2,3,5,6,1,4] => ([(0,5),(1,4),(3,2),(4,3),(4,5)],6)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 4 = 5 - 1
{{1,2,3},{4,5,6}}
 => [2,3,1,5,6,4] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
{{1,2,3},{4,5},{6}}
 => [2,3,1,5,4,6] => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,3},{4},{5,6}}
 => [2,3,1,4,6,5] => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,4,6},{3,5}}
 => [2,4,5,6,3,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,4},{3,5,6}}
 => [2,4,5,1,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(3,5)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 4 = 5 - 1
{{1,2,4},{3,5},{6}}
 => [2,4,5,1,3,6] => ([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3 = 4 - 1
{{1,2},{3,4,5,6}}
 => [2,1,4,5,6,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
{{1,2},{3,4,5},{6}}
 => [2,1,4,5,3,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2},{3,4},{5,6}}
 => [2,1,4,3,6,5] => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,5},{3,6},{4}}
 => [2,5,6,4,1,3] => ([(0,5),(1,3),(1,4),(1,5),(4,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1,2},{3,5},{4,6}}
 => [2,1,5,6,3,4] => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3 = 4 - 1
{{1,2},{3},{4,5,6}}
 => [2,1,3,5,6,4] => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,6},{3},{4},{5}}
 => [2,6,3,4,5,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,3,4,5},{2,6}}
 => [3,6,4,5,1,2] => ([(0,4),(1,3),(1,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1,3,4},{2,6},{5}}
 => [3,6,4,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,3,5,6},{2,4}}
 => [3,4,5,2,6,1] => ([(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,3,5},{2,4},{6}}
 => [3,4,5,2,1,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,3},{2,4,5,6}}
 => [3,4,1,5,6,2] => ([(0,4),(1,3),(1,5),(4,5),(5,2)],6)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 4 = 5 - 1
{{1,3},{2,4,5},{6}}
 => [3,4,1,5,2,6] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3 = 4 - 1
{{1,3,6},{2,4},{5}}
 => [3,4,6,2,5,1] => ([(1,5),(2,3),(3,4),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
{{1,3},{2,4},{5,6}}
 => [3,4,1,2,6,5] => ([(0,3),(1,2),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3 = 4 - 1
{{1,3},{2,4},{5},{6}}
 => [3,4,1,2,5,6] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1,3},{2,5},{4,6}}
 => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,4),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 4 = 5 - 1
{{1,3,6},{2},{4},{5}}
 => [3,2,6,4,5,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,4,5},{2,3,6}}
 => [4,3,6,5,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1,4},{2,3,6},{5}}
 => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,5},{2,3,4,6}}
 => [5,3,4,6,1,2] => ([(0,5),(1,3),(2,4),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1,6},{2,3,4,5}}
 => [6,3,4,5,2,1] => ([(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
Description
The dimension of the space of valuations of a lattice. A valuation, or modular function, on a lattice $L$ is a function $v:L\mapsto\mathbb R$ satisfying $$ v(a\vee b) + v(a\wedge b) = v(a) + v(b). $$ It was shown by Birkhoff [1, thm. X.2], that a lattice with a positive valuation must be modular. This was sharpened by Fleischer and Traynor [2, thm. 1], which states that the modular functions on an arbitrary lattice are in bijection with the modular functions on its modular quotient [[Mp00196]]. Moreover, Birkhoff [1, thm. X.2] showed that the dimension of the space of modular functions equals the number of subsets of projective prime intervals.
Matching statistic: St001622
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001622: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1},{2},{3}}
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,3,4}}
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,3},{2,4}}
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3 = 4 - 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3 = 4 - 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1},{2,4},{3,5}}
 => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
{{1,2,3,4,5},{6}}
 => [2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
{{1,2,3,4},{5,6}}
 => [2,3,4,1,6,5] => ([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
{{1,2,3,4},{5},{6}}
 => [2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,3,5},{4,6}}
 => [2,3,5,6,1,4] => ([(0,5),(1,4),(3,2),(4,3),(4,5)],6)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 4 = 5 - 1
{{1,2,3},{4,5,6}}
 => [2,3,1,5,6,4] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
{{1,2,3},{4,5},{6}}
 => [2,3,1,5,4,6] => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,3},{4},{5,6}}
 => [2,3,1,4,6,5] => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,4,6},{3,5}}
 => [2,4,5,6,3,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,4},{3,5,6}}
 => [2,4,5,1,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(3,5)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 4 = 5 - 1
{{1,2,4},{3,5},{6}}
 => [2,4,5,1,3,6] => ([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3 = 4 - 1
{{1,2},{3,4,5,6}}
 => [2,1,4,5,6,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
{{1,2},{3,4,5},{6}}
 => [2,1,4,5,3,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2},{3,4},{5,6}}
 => [2,1,4,3,6,5] => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,5},{3,6},{4}}
 => [2,5,6,4,1,3] => ([(0,5),(1,3),(1,4),(1,5),(4,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1,2},{3,5},{4,6}}
 => [2,1,5,6,3,4] => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3 = 4 - 1
{{1,2},{3},{4,5,6}}
 => [2,1,3,5,6,4] => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,2,6},{3},{4},{5}}
 => [2,6,3,4,5,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,3,4,5},{2,6}}
 => [3,6,4,5,1,2] => ([(0,4),(1,3),(1,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1,3,4},{2,6},{5}}
 => [3,6,4,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,3,5,6},{2,4}}
 => [3,4,5,2,6,1] => ([(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,3,5},{2,4},{6}}
 => [3,4,5,2,1,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,3},{2,4,5,6}}
 => [3,4,1,5,6,2] => ([(0,4),(1,3),(1,5),(4,5),(5,2)],6)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 4 = 5 - 1
{{1,3},{2,4,5},{6}}
 => [3,4,1,5,2,6] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3 = 4 - 1
{{1,3,6},{2,4},{5}}
 => [3,4,6,2,5,1] => ([(1,5),(2,3),(3,4),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
{{1,3},{2,4},{5,6}}
 => [3,4,1,2,6,5] => ([(0,3),(1,2),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3 = 4 - 1
{{1,3},{2,4},{5},{6}}
 => [3,4,1,2,5,6] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1,3},{2,5},{4,6}}
 => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,4),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 4 = 5 - 1
{{1,3,6},{2},{4},{5}}
 => [3,2,6,4,5,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,4,5},{2,3,6}}
 => [4,3,6,5,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1,4},{2,3,6},{5}}
 => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
{{1,5},{2,3,4,6}}
 => [5,3,4,6,1,2] => ([(0,5),(1,3),(2,4),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1,6},{2,3,4,5}}
 => [6,3,4,5,2,1] => ([(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
Description
The number of join-irreducible elements of a lattice. An element $j$ of a lattice $L$ is '''join irreducible''' if it is not the least element and if $j=x\vee y$, then $j\in\{x,y\}$ for all $x,y\in L$.