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Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St001879
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
{{1},{2,3,4,5},{6}}
=> [1,3,4,5,2,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 8
{{1},{2,3,4},{5},{6}}
=> [1,3,4,2,5,6] => [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 7
{{1},{2,3,5},{4},{6}}
=> [1,3,5,4,2,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 8
{{1},{2,3},{4},{5},{6}}
=> [1,3,2,4,5,6] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
{{1},{2,4,5},{3},{6}}
=> [1,4,3,5,2,6] => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 7
{{1},{2,4},{3,5},{6}}
=> [1,4,5,2,3,6] => [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 7
{{1},{2,4},{3},{5},{6}}
=> [1,4,3,2,5,6] => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 7
{{1},{2,5},{3,4},{6}}
=> [1,5,4,3,2,6] => [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 10
{{1},{2},{3,4,5},{6}}
=> [1,2,4,5,3,6] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 7
{{1},{2},{3,4},{5},{6}}
=> [1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
{{1},{2,5},{3},{4},{6}}
=> [1,5,3,4,2,6] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 8
{{1},{2},{3,5},{4},{6}}
=> [1,2,5,4,3,6] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 7
{{1},{2},{3},{4,5},{6}}
=> [1,2,3,5,4,6] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
{{1},{2,3,4,5,6},{7}}
=> [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 10
{{1},{2,3,4,5},{6},{7}}
=> [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> 9
{{1},{2,3,4,6},{5},{7}}
=> [1,3,4,6,5,2,7] => [1,5,6,2,3,4,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=> 10
{{1},{2,3,4},{5},{6},{7}}
=> [1,3,4,2,5,6,7] => [1,4,2,3,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> 8
{{1},{2,3,5,6},{4},{7}}
=> [1,3,5,4,6,2,7] => [1,4,6,2,3,5,7] => ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
=> 9
{{1},{2,3,5},{4,6},{7}}
=> [1,3,5,6,2,4,7] => [1,5,2,3,6,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=> 9
{{1},{2,3,5},{4},{6},{7}}
=> [1,3,5,4,2,6,7] => [1,4,5,2,3,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> 9
{{1},{2,3,6},{4,5},{7}}
=> [1,3,6,5,4,2,7] => [1,5,4,6,2,3,7] => ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> 13
{{1},{2,3,6},{4},{5},{7}}
=> [1,3,6,4,5,2,7] => [1,4,5,6,2,3,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=> 10
{{1},{2,3},{4},{5,6},{7}}
=> [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 8
{{1},{2,3},{4},{5},{6},{7}}
=> [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 7
{{1},{2,4,5,6},{3},{7}}
=> [1,4,3,5,6,2,7] => [1,3,6,2,4,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> 9
{{1},{2,4,5},{3,6},{7}}
=> [1,4,6,5,2,3,7] => [1,5,2,4,6,3,7] => ([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=> 11
{{1},{2,4,5},{3},{6},{7}}
=> [1,4,3,5,2,6,7] => [1,3,5,2,4,6,7] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 8
{{1},{2,4,6},{3,5},{7}}
=> [1,4,5,6,3,2,7] => [1,5,3,6,2,4,7] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 11
{{1},{2,4},{3,5},{6},{7}}
=> [1,4,5,2,3,6,7] => [1,4,2,5,3,6,7] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 8
{{1},{2,4,6},{3},{5},{7}}
=> [1,4,3,6,5,2,7] => [1,3,5,6,2,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> 9
{{1},{2,4},{3,6},{5},{7}}
=> [1,4,6,2,5,3,7] => [1,4,2,5,6,3,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> 9
{{1},{2,4},{3},{5},{6},{7}}
=> [1,4,3,2,5,6,7] => [1,3,4,2,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> 8
{{1},{2,5},{3,4,6},{7}}
=> [1,5,4,6,2,3,7] => [1,5,2,6,3,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> 9
{{1},{2,5},{3,4},{6},{7}}
=> [1,5,4,3,2,6,7] => [1,4,3,5,2,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> 11
{{1},{2,6},{3,4,5},{7}}
=> [1,6,4,5,3,2,7] => [1,5,3,4,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=> 13
{{1},{2},{3,4,5,6},{7}}
=> [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> 9
{{1},{2},{3,4,5},{6},{7}}
=> [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> 8
{{1},{2,6},{3,4},{5},{7}}
=> [1,6,4,3,5,2,7] => [1,4,3,5,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> 12
{{1},{2},{3,4,6},{5},{7}}
=> [1,2,4,6,5,3,7] => [1,2,5,6,3,4,7] => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> 9
{{1},{2},{3,4},{5},{6},{7}}
=> [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 7
{{1},{2,5,6},{3},{4},{7}}
=> [1,5,3,4,6,2,7] => [1,3,4,6,2,5,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=> 9
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001846
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001846: Lattices ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 50%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001846: Lattices ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 50%
Values
{{1},{2},{3}}
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
{{1},{2,3},{4}}
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 4
{{1},{2},{3},{4}}
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 3
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> 6
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 5
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 6
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 5
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 4
{{1},{2,3,4,5},{6}}
=> [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 8
{{1},{2,3,4},{5},{6}}
=> [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> 7
{{1},{2,3,5},{4},{6}}
=> [1,3,5,4,2,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,6),(2,7),(3,8),(3,10),(4,8),(4,9),(5,3),(5,4),(5,7),(6,2),(6,5),(7,9),(7,10),(8,11),(9,11),(10,11),(11,1)],12)
=> ? = 8
{{1},{2,3},{4},{5},{6}}
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> 6
{{1},{2,4,5},{3},{6}}
=> [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(2,10),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(6,3),(6,4),(6,5),(7,11),(8,11),(9,2),(9,11),(10,1),(11,10)],12)
=> ? = 7
{{1},{2,4},{3,5},{6}}
=> [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
=> ? = 7
{{1},{2,4},{3},{5},{6}}
=> [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,6),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,5)],11)
=> ? = 7
{{1},{2,5},{3,4},{6}}
=> [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,7),(4,9),(4,11),(5,7),(5,8),(5,10),(6,2),(6,3),(6,4),(6,5),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(11,15),(11,16),(12,14),(12,15),(13,17),(14,17),(15,17),(16,17),(17,1)],18)
=> ? = 10
{{1},{2},{3,4,5},{6}}
=> [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> 7
{{1},{2},{3,4},{5},{6}}
=> [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> 6
{{1},{2,5},{3},{4},{6}}
=> [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,6),(2,10),(2,11),(3,7),(3,9),(4,7),(4,8),(5,2),(5,8),(5,9),(6,3),(6,4),(6,5),(7,12),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13),(13,1)],14)
=> ? = 8
{{1},{2},{3,5},{4},{6}}
=> [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,5),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,6),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,1)],11)
=> ? = 7
{{1},{2},{3},{4,5},{6}}
=> [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> 6
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 5
{{1},{2,3,4,5,6},{7}}
=> [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ([(0,7),(2,8),(3,11),(4,5),(4,8),(5,6),(5,10),(6,3),(6,9),(7,2),(7,4),(8,10),(9,11),(10,9),(11,1)],12)
=> ? = 10
{{1},{2,3,4,5},{6},{7}}
=> [1,3,4,5,2,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> ([(0,7),(2,8),(3,9),(4,5),(4,8),(5,3),(5,10),(6,1),(7,2),(7,4),(8,10),(9,6),(10,9)],11)
=> ? = 9
{{1},{2,3,4,6},{5},{7}}
=> [1,3,4,6,5,2,7] => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(0,7),(1,9),(3,8),(3,11),(4,8),(4,10),(5,6),(5,9),(6,3),(6,4),(6,12),(7,1),(7,5),(8,13),(9,12),(10,13),(11,13),(12,10),(12,11),(13,2)],14)
=> ? = 10
{{1},{2,3,4},{5},{6},{7}}
=> [1,3,4,2,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> ([(0,7),(1,9),(3,8),(4,2),(5,1),(5,8),(6,4),(7,3),(7,5),(8,9),(9,6)],10)
=> ? = 8
{{1},{2,3,5,6},{4},{7}}
=> [1,3,5,4,6,2,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(0,7),(2,8),(3,12),(4,10),(4,11),(5,9),(5,11),(6,4),(6,5),(6,8),(7,2),(7,6),(8,9),(8,10),(9,13),(10,13),(11,3),(11,13),(12,1),(13,12)],14)
=> ? = 9
{{1},{2,3,5},{4,6},{7}}
=> [1,3,5,6,2,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> ([(0,7),(2,11),(3,10),(4,9),(5,6),(5,11),(6,4),(6,8),(7,2),(7,5),(8,9),(8,10),(9,12),(10,12),(11,3),(11,8),(12,1)],13)
=> ? = 9
{{1},{2,3,5},{4},{6},{7}}
=> [1,3,5,4,2,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7)
=> ([(0,7),(2,8),(3,9),(3,11),(4,9),(4,10),(5,1),(6,3),(6,4),(6,8),(7,2),(7,6),(8,10),(8,11),(9,12),(10,12),(11,12),(12,5)],13)
=> ? = 9
{{1},{2,3,6},{4,5},{7}}
=> [1,3,6,5,4,2,7] => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,6),(2,8),(3,10),(3,11),(3,14),(4,9),(4,11),(4,13),(5,9),(5,10),(5,12),(6,2),(6,7),(7,3),(7,4),(7,5),(7,8),(8,12),(8,13),(8,14),(9,17),(9,18),(10,15),(10,18),(11,16),(11,18),(12,15),(12,17),(13,16),(13,17),(14,15),(14,16),(15,19),(16,19),(17,19),(18,19),(19,1)],20)
=> ? = 13
{{1},{2,3,6},{4},{5},{7}}
=> [1,3,6,4,5,2,7] => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(0,7),(1,9),(3,8),(3,12),(4,10),(4,11),(5,4),(5,8),(5,13),(6,3),(6,5),(6,9),(7,1),(7,6),(8,11),(8,14),(9,12),(9,13),(10,15),(11,15),(12,14),(13,10),(13,14),(14,15),(15,2)],16)
=> ? = 10
{{1},{2,3},{4},{5,6},{7}}
=> [1,3,2,4,6,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,7),(2,9),(3,9),(4,8),(5,8),(6,2),(6,3),(7,4),(7,5),(8,6),(9,1)],10)
=> ? = 8
{{1},{2,3},{4},{5},{6},{7}}
=> [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,7),(2,8),(3,8),(4,5),(5,1),(6,4),(7,2),(7,3),(8,6)],9)
=> 7
{{1},{2,4,5,6},{3},{7}}
=> [1,4,3,5,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(0,7),(2,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,11),(7,3),(7,4),(7,5),(8,13),(9,13),(10,6),(10,13),(11,12),(12,1),(13,11)],14)
=> ? = 9
{{1},{2,4,5},{3,6},{7}}
=> [1,4,6,5,2,3,7] => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7)
=> ([(0,7),(2,8),(2,12),(3,8),(3,11),(4,10),(5,4),(5,9),(6,2),(6,3),(6,9),(7,5),(7,6),(8,15),(9,10),(9,11),(9,12),(10,13),(10,14),(11,13),(11,15),(12,14),(12,15),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 11
{{1},{2,4,5},{3},{6},{7}}
=> [1,4,3,5,2,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,3),(7,4),(7,5),(8,12),(9,12),(10,6),(11,2),(11,12),(12,10)],13)
=> ? = 8
{{1},{2,4,6},{3,5},{7}}
=> [1,4,5,6,3,2,7] => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,7),(2,8),(2,10),(3,8),(3,9),(4,11),(4,12),(5,6),(5,9),(5,10),(6,4),(6,13),(6,14),(7,2),(7,3),(7,5),(8,15),(9,13),(9,15),(10,14),(10,15),(11,17),(12,17),(13,11),(13,16),(14,12),(14,16),(15,16),(16,17),(17,1)],18)
=> ? = 11
{{1},{2,4},{3,5},{6},{7}}
=> [1,4,5,2,3,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> ([(0,7),(2,10),(3,9),(4,3),(4,8),(5,2),(5,8),(6,1),(7,4),(7,5),(8,9),(8,10),(9,11),(10,11),(11,6)],12)
=> ? = 8
{{1},{2,4,6},{3},{5},{7}}
=> [1,4,3,6,5,2,7] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(2,10),(2,13),(3,9),(3,13),(4,8),(4,12),(5,8),(5,11),(6,9),(6,10),(7,2),(7,3),(7,6),(8,15),(9,14),(10,14),(11,15),(12,15),(13,4),(13,5),(13,14),(14,11),(14,12),(15,1)],16)
=> ? = 9
{{1},{2,4},{3,6},{5},{7}}
=> [1,4,6,2,5,3,7] => ([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(0,7),(1,10),(1,11),(3,9),(4,8),(5,4),(5,13),(6,3),(6,13),(7,5),(7,6),(8,10),(8,12),(9,11),(9,12),(10,14),(11,14),(12,14),(13,1),(13,8),(13,9),(14,2)],15)
=> ? = 9
{{1},{2,4},{3},{5},{6},{7}}
=> [1,4,3,2,5,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,7),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,1),(6,5),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,6)],12)
=> ? = 8
{{1},{2,5},{3,4,6},{7}}
=> [1,5,4,6,2,3,7] => ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(0,7),(2,11),(2,12),(3,8),(4,10),(4,13),(5,9),(5,13),(6,2),(6,9),(6,10),(7,4),(7,5),(7,6),(8,14),(9,11),(9,16),(10,12),(10,16),(11,15),(12,15),(13,3),(13,16),(14,1),(15,14),(16,8),(16,15)],17)
=> ? = 9
{{1},{2,5},{3,4},{6},{7}}
=> [1,5,4,3,2,6,7] => ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,1),(7,2),(7,3),(7,4),(7,5),(8,14),(8,17),(9,14),(9,15),(10,14),(10,16),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,18),(15,18),(16,18),(17,18),(18,6)],19)
=> ? = 11
{{1},{2,6},{3,4,5},{7}}
=> [1,6,4,5,3,2,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ?
=> ? = 13
{{1},{2},{3,4,5,6},{7}}
=> [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ([(0,6),(2,10),(3,8),(4,2),(4,9),(5,4),(5,8),(6,7),(7,3),(7,5),(8,9),(9,10),(10,1)],11)
=> ? = 9
{{1},{2},{3,4,5},{6},{7}}
=> [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,6),(2,9),(3,8),(4,2),(4,8),(5,1),(6,7),(7,3),(7,4),(8,9),(9,5)],10)
=> ? = 8
{{1},{2,6},{3,4},{5},{7}}
=> [1,6,4,3,5,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 12
{{1},{2},{3,4,6},{5},{7}}
=> [1,2,4,6,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,5),(2,9),(2,11),(3,9),(3,10),(4,8),(5,6),(6,4),(6,7),(7,2),(7,3),(7,8),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 9
{{1},{2},{3,4},{5},{6},{7}}
=> [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
=> ? = 7
{{1},{2,5,6},{3},{4},{7}}
=> [1,5,3,4,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=> ([(0,7),(1,12),(3,11),(3,13),(4,8),(4,9),(5,8),(5,10),(6,3),(6,9),(6,10),(7,4),(7,5),(7,6),(8,15),(9,13),(9,15),(10,11),(10,15),(11,14),(12,2),(13,1),(13,14),(14,12),(15,14)],16)
=> ? = 9
{{1},{2,5},{3,6},{4},{7}}
=> [1,5,6,4,2,3,7] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
=> ([(0,2),(2,3),(2,4),(2,5),(3,13),(3,14),(4,7),(4,14),(4,15),(5,6),(5,13),(5,15),(6,9),(6,11),(7,10),(7,12),(8,19),(9,17),(10,18),(11,8),(11,17),(12,8),(12,18),(13,9),(13,16),(14,10),(14,16),(15,11),(15,12),(15,16),(16,17),(16,18),(17,19),(18,19),(19,1)],20)
=> ? = 9
{{1},{2,5},{3},{4},{6},{7}}
=> [1,5,3,4,2,6,7] => ([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=> ([(0,7),(1,11),(1,12),(3,8),(3,10),(4,8),(4,9),(5,2),(6,1),(6,9),(6,10),(7,3),(7,4),(7,6),(8,14),(9,11),(9,14),(10,12),(10,14),(11,13),(12,13),(13,5),(14,13)],15)
=> ? = 9
{{1},{2,6},{3,5},{4},{7}}
=> [1,6,5,4,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 13
{{1},{2},{3,5,6},{4},{7}}
=> [1,2,5,4,6,3,7] => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(2,11),(3,9),(3,10),(4,8),(4,10),(5,8),(5,9),(6,7),(7,3),(7,4),(7,5),(8,12),(9,12),(10,2),(10,12),(11,1),(12,11)],13)
=> ? = 8
{{1},{2},{3,5},{4,6},{7}}
=> [1,2,5,6,3,4,7] => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(0,6),(2,10),(3,9),(4,3),(4,8),(5,2),(5,8),(6,7),(7,4),(7,5),(8,9),(8,10),(9,11),(10,11),(11,1)],12)
=> ? = 8
{{1},{2},{3,5},{4},{6},{7}}
=> [1,2,5,4,3,6,7] => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,1),(6,7),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,5)],12)
=> ? = 8
{{1},{2,6},{3},{4,5},{7}}
=> [1,6,3,5,4,2,7] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ?
=> ? = 11
{{1},{2},{3,6},{4,5},{7}}
=> [1,2,6,5,4,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,6),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,7),(7,2),(7,3),(7,4),(7,5),(8,14),(8,17),(9,14),(9,15),(10,14),(10,16),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,18),(15,18),(16,18),(17,18),(18,1)],19)
=> ? = 11
{{1},{2},{3},{4,5,6},{7}}
=> [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ([(0,6),(1,9),(3,8),(4,7),(5,1),(5,8),(6,4),(7,3),(7,5),(8,9),(9,2)],10)
=> ? = 8
{{1},{2},{3},{4,5},{6},{7}}
=> [1,2,3,5,4,6,7] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(2,8),(3,8),(4,7),(5,1),(6,4),(7,2),(7,3),(8,5)],9)
=> 7
{{1},{2,6},{3},{4},{5},{7}}
=> [1,6,3,4,5,2,7] => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,7),(2,8),(2,10),(3,8),(3,9),(4,11),(4,12),(5,6),(5,9),(5,10),(6,4),(6,13),(6,14),(7,2),(7,3),(7,5),(8,15),(9,13),(9,15),(10,14),(10,15),(11,17),(12,17),(13,11),(13,16),(14,12),(14,16),(15,16),(16,17),(17,1)],18)
=> ? = 10
{{1},{2},{3,6},{4},{5},{7}}
=> [1,2,6,4,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,6),(2,11),(2,12),(3,8),(3,10),(4,8),(4,9),(5,2),(5,9),(5,10),(6,7),(7,3),(7,4),(7,5),(8,13),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14),(14,1)],15)
=> ? = 9
{{1},{2},{3},{4,6},{5},{7}}
=> [1,2,3,6,5,4,7] => ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,6),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,7),(6,5),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,1)],12)
=> ? = 8
{{1},{2},{3},{4},{5,6},{7}}
=> [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
=> ? = 7
{{1},{2},{3},{4},{5},{6},{7}}
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 6
Description
The number of elements which do not have a complement in the lattice.
A complement of an element $x$ in a lattice is an element $y$ such that the meet of $x$ and $y$ is the bottom element and their join is the top element.
Matching statistic: St001616
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 50%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 50%
Values
{{1},{2},{3}}
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
{{1},{2,3},{4}}
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6 = 4 + 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> 8 = 6 + 2
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 7 = 5 + 2
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 6 + 2
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 7 = 5 + 2
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
{{1},{2,3,4,5},{6}}
=> [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 8 + 2
{{1},{2,3,4},{5},{6}}
=> [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> 9 = 7 + 2
{{1},{2,3,5},{4},{6}}
=> [1,3,5,4,2,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,6),(2,7),(3,8),(3,10),(4,8),(4,9),(5,3),(5,4),(5,7),(6,2),(6,5),(7,9),(7,10),(8,11),(9,11),(10,11),(11,1)],12)
=> ? = 8 + 2
{{1},{2,3},{4},{5},{6}}
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> 8 = 6 + 2
{{1},{2,4,5},{3},{6}}
=> [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(2,10),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(6,3),(6,4),(6,5),(7,11),(8,11),(9,2),(9,11),(10,1),(11,10)],12)
=> ? = 7 + 2
{{1},{2,4},{3,5},{6}}
=> [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
=> ? = 7 + 2
{{1},{2,4},{3},{5},{6}}
=> [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,6),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,5)],11)
=> ? = 7 + 2
{{1},{2,5},{3,4},{6}}
=> [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,7),(4,9),(4,11),(5,7),(5,8),(5,10),(6,2),(6,3),(6,4),(6,5),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(11,15),(11,16),(12,14),(12,15),(13,17),(14,17),(15,17),(16,17),(17,1)],18)
=> ? = 10 + 2
{{1},{2},{3,4,5},{6}}
=> [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> 9 = 7 + 2
{{1},{2},{3,4},{5},{6}}
=> [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> 8 = 6 + 2
{{1},{2,5},{3},{4},{6}}
=> [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,6),(2,10),(2,11),(3,7),(3,9),(4,7),(4,8),(5,2),(5,8),(5,9),(6,3),(6,4),(6,5),(7,12),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13),(13,1)],14)
=> ? = 8 + 2
{{1},{2},{3,5},{4},{6}}
=> [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,5),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,6),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,1)],11)
=> ? = 7 + 2
{{1},{2},{3},{4,5},{6}}
=> [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> 8 = 6 + 2
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 5 + 2
{{1},{2,3,4,5,6},{7}}
=> [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ([(0,7),(2,8),(3,11),(4,5),(4,8),(5,6),(5,10),(6,3),(6,9),(7,2),(7,4),(8,10),(9,11),(10,9),(11,1)],12)
=> ? = 10 + 2
{{1},{2,3,4,5},{6},{7}}
=> [1,3,4,5,2,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> ([(0,7),(2,8),(3,9),(4,5),(4,8),(5,3),(5,10),(6,1),(7,2),(7,4),(8,10),(9,6),(10,9)],11)
=> ? = 9 + 2
{{1},{2,3,4,6},{5},{7}}
=> [1,3,4,6,5,2,7] => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(0,7),(1,9),(3,8),(3,11),(4,8),(4,10),(5,6),(5,9),(6,3),(6,4),(6,12),(7,1),(7,5),(8,13),(9,12),(10,13),(11,13),(12,10),(12,11),(13,2)],14)
=> ? = 10 + 2
{{1},{2,3,4},{5},{6},{7}}
=> [1,3,4,2,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> ([(0,7),(1,9),(3,8),(4,2),(5,1),(5,8),(6,4),(7,3),(7,5),(8,9),(9,6)],10)
=> ? = 8 + 2
{{1},{2,3,5,6},{4},{7}}
=> [1,3,5,4,6,2,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(0,7),(2,8),(3,12),(4,10),(4,11),(5,9),(5,11),(6,4),(6,5),(6,8),(7,2),(7,6),(8,9),(8,10),(9,13),(10,13),(11,3),(11,13),(12,1),(13,12)],14)
=> ? = 9 + 2
{{1},{2,3,5},{4,6},{7}}
=> [1,3,5,6,2,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> ([(0,7),(2,11),(3,10),(4,9),(5,6),(5,11),(6,4),(6,8),(7,2),(7,5),(8,9),(8,10),(9,12),(10,12),(11,3),(11,8),(12,1)],13)
=> ? = 9 + 2
{{1},{2,3,5},{4},{6},{7}}
=> [1,3,5,4,2,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7)
=> ([(0,7),(2,8),(3,9),(3,11),(4,9),(4,10),(5,1),(6,3),(6,4),(6,8),(7,2),(7,6),(8,10),(8,11),(9,12),(10,12),(11,12),(12,5)],13)
=> ? = 9 + 2
{{1},{2,3,6},{4,5},{7}}
=> [1,3,6,5,4,2,7] => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,6),(2,8),(3,10),(3,11),(3,14),(4,9),(4,11),(4,13),(5,9),(5,10),(5,12),(6,2),(6,7),(7,3),(7,4),(7,5),(7,8),(8,12),(8,13),(8,14),(9,17),(9,18),(10,15),(10,18),(11,16),(11,18),(12,15),(12,17),(13,16),(13,17),(14,15),(14,16),(15,19),(16,19),(17,19),(18,19),(19,1)],20)
=> ? = 13 + 2
{{1},{2,3,6},{4},{5},{7}}
=> [1,3,6,4,5,2,7] => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(0,7),(1,9),(3,8),(3,12),(4,10),(4,11),(5,4),(5,8),(5,13),(6,3),(6,5),(6,9),(7,1),(7,6),(8,11),(8,14),(9,12),(9,13),(10,15),(11,15),(12,14),(13,10),(13,14),(14,15),(15,2)],16)
=> ? = 10 + 2
{{1},{2,3},{4},{5,6},{7}}
=> [1,3,2,4,6,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,7),(2,9),(3,9),(4,8),(5,8),(6,2),(6,3),(7,4),(7,5),(8,6),(9,1)],10)
=> ? = 8 + 2
{{1},{2,3},{4},{5},{6},{7}}
=> [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,7),(2,8),(3,8),(4,5),(5,1),(6,4),(7,2),(7,3),(8,6)],9)
=> 9 = 7 + 2
{{1},{2,4,5,6},{3},{7}}
=> [1,4,3,5,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(0,7),(2,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,11),(7,3),(7,4),(7,5),(8,13),(9,13),(10,6),(10,13),(11,12),(12,1),(13,11)],14)
=> ? = 9 + 2
{{1},{2,4,5},{3,6},{7}}
=> [1,4,6,5,2,3,7] => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7)
=> ([(0,7),(2,8),(2,12),(3,8),(3,11),(4,10),(5,4),(5,9),(6,2),(6,3),(6,9),(7,5),(7,6),(8,15),(9,10),(9,11),(9,12),(10,13),(10,14),(11,13),(11,15),(12,14),(12,15),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 11 + 2
{{1},{2,4,5},{3},{6},{7}}
=> [1,4,3,5,2,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,3),(7,4),(7,5),(8,12),(9,12),(10,6),(11,2),(11,12),(12,10)],13)
=> ? = 8 + 2
{{1},{2,4,6},{3,5},{7}}
=> [1,4,5,6,3,2,7] => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,7),(2,8),(2,10),(3,8),(3,9),(4,11),(4,12),(5,6),(5,9),(5,10),(6,4),(6,13),(6,14),(7,2),(7,3),(7,5),(8,15),(9,13),(9,15),(10,14),(10,15),(11,17),(12,17),(13,11),(13,16),(14,12),(14,16),(15,16),(16,17),(17,1)],18)
=> ? = 11 + 2
{{1},{2,4},{3,5},{6},{7}}
=> [1,4,5,2,3,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> ([(0,7),(2,10),(3,9),(4,3),(4,8),(5,2),(5,8),(6,1),(7,4),(7,5),(8,9),(8,10),(9,11),(10,11),(11,6)],12)
=> ? = 8 + 2
{{1},{2,4,6},{3},{5},{7}}
=> [1,4,3,6,5,2,7] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(2,10),(2,13),(3,9),(3,13),(4,8),(4,12),(5,8),(5,11),(6,9),(6,10),(7,2),(7,3),(7,6),(8,15),(9,14),(10,14),(11,15),(12,15),(13,4),(13,5),(13,14),(14,11),(14,12),(15,1)],16)
=> ? = 9 + 2
{{1},{2,4},{3,6},{5},{7}}
=> [1,4,6,2,5,3,7] => ([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(0,7),(1,10),(1,11),(3,9),(4,8),(5,4),(5,13),(6,3),(6,13),(7,5),(7,6),(8,10),(8,12),(9,11),(9,12),(10,14),(11,14),(12,14),(13,1),(13,8),(13,9),(14,2)],15)
=> ? = 9 + 2
{{1},{2,4},{3},{5},{6},{7}}
=> [1,4,3,2,5,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,7),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,1),(6,5),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,6)],12)
=> ? = 8 + 2
{{1},{2,5},{3,4,6},{7}}
=> [1,5,4,6,2,3,7] => ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(0,7),(2,11),(2,12),(3,8),(4,10),(4,13),(5,9),(5,13),(6,2),(6,9),(6,10),(7,4),(7,5),(7,6),(8,14),(9,11),(9,16),(10,12),(10,16),(11,15),(12,15),(13,3),(13,16),(14,1),(15,14),(16,8),(16,15)],17)
=> ? = 9 + 2
{{1},{2,5},{3,4},{6},{7}}
=> [1,5,4,3,2,6,7] => ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,1),(7,2),(7,3),(7,4),(7,5),(8,14),(8,17),(9,14),(9,15),(10,14),(10,16),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,18),(15,18),(16,18),(17,18),(18,6)],19)
=> ? = 11 + 2
{{1},{2,6},{3,4,5},{7}}
=> [1,6,4,5,3,2,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ?
=> ? = 13 + 2
{{1},{2},{3,4,5,6},{7}}
=> [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ([(0,6),(2,10),(3,8),(4,2),(4,9),(5,4),(5,8),(6,7),(7,3),(7,5),(8,9),(9,10),(10,1)],11)
=> ? = 9 + 2
{{1},{2},{3,4,5},{6},{7}}
=> [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,6),(2,9),(3,8),(4,2),(4,8),(5,1),(6,7),(7,3),(7,4),(8,9),(9,5)],10)
=> ? = 8 + 2
{{1},{2,6},{3,4},{5},{7}}
=> [1,6,4,3,5,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 12 + 2
{{1},{2},{3,4,6},{5},{7}}
=> [1,2,4,6,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,5),(2,9),(2,11),(3,9),(3,10),(4,8),(5,6),(6,4),(6,7),(7,2),(7,3),(7,8),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 9 + 2
{{1},{2},{3,4},{5},{6},{7}}
=> [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
=> ? = 7 + 2
{{1},{2,5,6},{3},{4},{7}}
=> [1,5,3,4,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=> ([(0,7),(1,12),(3,11),(3,13),(4,8),(4,9),(5,8),(5,10),(6,3),(6,9),(6,10),(7,4),(7,5),(7,6),(8,15),(9,13),(9,15),(10,11),(10,15),(11,14),(12,2),(13,1),(13,14),(14,12),(15,14)],16)
=> ? = 9 + 2
{{1},{2,5},{3,6},{4},{7}}
=> [1,5,6,4,2,3,7] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
=> ([(0,2),(2,3),(2,4),(2,5),(3,13),(3,14),(4,7),(4,14),(4,15),(5,6),(5,13),(5,15),(6,9),(6,11),(7,10),(7,12),(8,19),(9,17),(10,18),(11,8),(11,17),(12,8),(12,18),(13,9),(13,16),(14,10),(14,16),(15,11),(15,12),(15,16),(16,17),(16,18),(17,19),(18,19),(19,1)],20)
=> ? = 9 + 2
{{1},{2,5},{3},{4},{6},{7}}
=> [1,5,3,4,2,6,7] => ([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=> ([(0,7),(1,11),(1,12),(3,8),(3,10),(4,8),(4,9),(5,2),(6,1),(6,9),(6,10),(7,3),(7,4),(7,6),(8,14),(9,11),(9,14),(10,12),(10,14),(11,13),(12,13),(13,5),(14,13)],15)
=> ? = 9 + 2
{{1},{2,6},{3,5},{4},{7}}
=> [1,6,5,4,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 13 + 2
{{1},{2},{3,5,6},{4},{7}}
=> [1,2,5,4,6,3,7] => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(2,11),(3,9),(3,10),(4,8),(4,10),(5,8),(5,9),(6,7),(7,3),(7,4),(7,5),(8,12),(9,12),(10,2),(10,12),(11,1),(12,11)],13)
=> ? = 8 + 2
{{1},{2},{3,5},{4,6},{7}}
=> [1,2,5,6,3,4,7] => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(0,6),(2,10),(3,9),(4,3),(4,8),(5,2),(5,8),(6,7),(7,4),(7,5),(8,9),(8,10),(9,11),(10,11),(11,1)],12)
=> ? = 8 + 2
{{1},{2},{3,5},{4},{6},{7}}
=> [1,2,5,4,3,6,7] => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,1),(6,7),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,5)],12)
=> ? = 8 + 2
{{1},{2,6},{3},{4,5},{7}}
=> [1,6,3,5,4,2,7] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ?
=> ? = 11 + 2
{{1},{2},{3,6},{4,5},{7}}
=> [1,2,6,5,4,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,6),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,7),(7,2),(7,3),(7,4),(7,5),(8,14),(8,17),(9,14),(9,15),(10,14),(10,16),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,18),(15,18),(16,18),(17,18),(18,1)],19)
=> ? = 11 + 2
{{1},{2},{3},{4,5,6},{7}}
=> [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ([(0,6),(1,9),(3,8),(4,7),(5,1),(5,8),(6,4),(7,3),(7,5),(8,9),(9,2)],10)
=> ? = 8 + 2
{{1},{2},{3},{4,5},{6},{7}}
=> [1,2,3,5,4,6,7] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(2,8),(3,8),(4,7),(5,1),(6,4),(7,2),(7,3),(8,5)],9)
=> 9 = 7 + 2
{{1},{2,6},{3},{4},{5},{7}}
=> [1,6,3,4,5,2,7] => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,7),(2,8),(2,10),(3,8),(3,9),(4,11),(4,12),(5,6),(5,9),(5,10),(6,4),(6,13),(6,14),(7,2),(7,3),(7,5),(8,15),(9,13),(9,15),(10,14),(10,15),(11,17),(12,17),(13,11),(13,16),(14,12),(14,16),(15,16),(16,17),(17,1)],18)
=> ? = 10 + 2
{{1},{2},{3,6},{4},{5},{7}}
=> [1,2,6,4,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,6),(2,11),(2,12),(3,8),(3,10),(4,8),(4,9),(5,2),(5,9),(5,10),(6,7),(7,3),(7,4),(7,5),(8,13),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14),(14,1)],15)
=> ? = 9 + 2
{{1},{2},{3},{4,6},{5},{7}}
=> [1,2,3,6,5,4,7] => ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,6),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,7),(6,5),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,1)],12)
=> ? = 8 + 2
{{1},{2},{3},{4},{5,6},{7}}
=> [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
=> ? = 7 + 2
{{1},{2},{3},{4},{5},{6},{7}}
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 8 = 6 + 2
Description
The number of neutral elements in a lattice.
An element $e$ of the lattice $L$ is neutral if the sublattice generated by $e$, $x$ and $y$ is distributive for all $x, y \in L$.
Matching statistic: St000327
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000327: Posets ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 42%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000327: Posets ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 42%
Values
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 6
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 6
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
{{1},{2,3,4,5},{6}}
=> [1,3,4,5,2,6] => [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ? = 8
{{1},{2,3,4},{5},{6}}
=> [1,3,4,2,5,6] => [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ? = 7
{{1},{2,3,5},{4},{6}}
=> [1,3,5,4,2,6] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 8
{{1},{2,3},{4},{5},{6}}
=> [1,3,2,4,5,6] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 6
{{1},{2,4,5},{3},{6}}
=> [1,4,3,5,2,6] => [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ? = 7
{{1},{2,4},{3,5},{6}}
=> [1,4,5,2,3,6] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 7
{{1},{2,4},{3},{5},{6}}
=> [1,4,3,2,5,6] => [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ? = 7
{{1},{2,5},{3,4},{6}}
=> [1,5,4,3,2,6] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 10
{{1},{2},{3,4,5},{6}}
=> [1,2,4,5,3,6] => [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ? = 7
{{1},{2},{3,4},{5},{6}}
=> [1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 6
{{1},{2,5},{3},{4},{6}}
=> [1,5,3,4,2,6] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 8
{{1},{2},{3,5},{4},{6}}
=> [1,2,5,4,3,6] => [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ? = 7
{{1},{2},{3},{4,5},{6}}
=> [1,2,3,5,4,6] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 6
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 5
{{1},{2,3,4,5,6},{7}}
=> [1,3,4,5,6,2,7] => [1,6,3,4,5,2,7] => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ? = 10
{{1},{2,3,4,5},{6},{7}}
=> [1,3,4,5,2,6,7] => [1,5,3,4,2,6,7] => ([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=> ? = 9
{{1},{2,3,4,6},{5},{7}}
=> [1,3,4,6,5,2,7] => [1,6,3,5,4,2,7] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ? = 10
{{1},{2,3,4},{5},{6},{7}}
=> [1,3,4,2,5,6,7] => [1,4,3,2,5,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ? = 8
{{1},{2,3,5,6},{4},{7}}
=> [1,3,5,4,6,2,7] => [1,6,4,3,5,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ? = 9
{{1},{2,3,5},{4,6},{7}}
=> [1,3,5,6,2,4,7] => [1,5,6,4,2,3,7] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
=> ? = 9
{{1},{2,3,5},{4},{6},{7}}
=> [1,3,5,4,2,6,7] => [1,5,4,3,2,6,7] => ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ? = 9
{{1},{2,3,6},{4,5},{7}}
=> [1,3,6,5,4,2,7] => [1,6,5,4,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 13
{{1},{2,3,6},{4},{5},{7}}
=> [1,3,6,4,5,2,7] => [1,6,5,4,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 10
{{1},{2,3},{4},{5,6},{7}}
=> [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ? = 8
{{1},{2,3},{4},{5},{6},{7}}
=> [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 7
{{1},{2,4,5,6},{3},{7}}
=> [1,4,3,5,6,2,7] => [1,6,3,4,5,2,7] => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ? = 9
{{1},{2,4,5},{3,6},{7}}
=> [1,4,6,5,2,3,7] => [1,6,5,4,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 11
{{1},{2,4,5},{3},{6},{7}}
=> [1,4,3,5,2,6,7] => [1,5,3,4,2,6,7] => ([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=> ? = 8
{{1},{2,4,6},{3,5},{7}}
=> [1,4,5,6,3,2,7] => [1,6,5,4,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 11
{{1},{2,4},{3,5},{6},{7}}
=> [1,4,5,2,3,6,7] => [1,5,4,3,2,6,7] => ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ? = 8
{{1},{2,4,6},{3},{5},{7}}
=> [1,4,3,6,5,2,7] => [1,6,3,5,4,2,7] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ? = 9
{{1},{2,4},{3,6},{5},{7}}
=> [1,4,6,2,5,3,7] => [1,6,5,4,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 9
{{1},{2,4},{3},{5},{6},{7}}
=> [1,4,3,2,5,6,7] => [1,4,3,2,5,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ? = 8
{{1},{2,5},{3,4,6},{7}}
=> [1,5,4,6,2,3,7] => [1,6,5,4,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 9
{{1},{2,5},{3,4},{6},{7}}
=> [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ? = 11
{{1},{2,6},{3,4,5},{7}}
=> [1,6,4,5,3,2,7] => [1,6,5,4,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 13
{{1},{2},{3,4,5,6},{7}}
=> [1,2,4,5,6,3,7] => [1,2,6,4,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ? = 9
{{1},{2},{3,4,5},{6},{7}}
=> [1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ? = 8
{{1},{2,6},{3,4},{5},{7}}
=> [1,6,4,3,5,2,7] => [1,6,5,4,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 12
{{1},{2},{3,4,6},{5},{7}}
=> [1,2,4,6,5,3,7] => [1,2,6,5,4,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ? = 9
{{1},{2},{3,4},{5},{6},{7}}
=> [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 7
{{1},{2,5,6},{3},{4},{7}}
=> [1,5,3,4,6,2,7] => [1,6,4,3,5,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ? = 9
{{1},{2,5},{3,6},{4},{7}}
=> [1,5,6,4,2,3,7] => [1,6,5,4,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 9
{{1},{2,5},{3},{4},{6},{7}}
=> [1,5,3,4,2,6,7] => [1,5,4,3,2,6,7] => ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ? = 9
{{1},{2,6},{3,5},{4},{7}}
=> [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 13
{{1},{2},{3,5,6},{4},{7}}
=> [1,2,5,4,6,3,7] => [1,2,6,4,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ? = 8
{{1},{2},{3,5},{4,6},{7}}
=> [1,2,5,6,3,4,7] => [1,2,6,5,4,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ? = 8
{{1},{2},{3,5},{4},{6},{7}}
=> [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ? = 8
{{1},{2,6},{3},{4,5},{7}}
=> [1,6,3,5,4,2,7] => [1,6,5,4,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 11
{{1},{2},{3,6},{4,5},{7}}
=> [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ? = 11
Description
The number of cover relations in a poset.
Equivalently, this is also the number of edges in the Hasse diagram [1].
Matching statistic: St000550
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St000550: Lattices ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St000550: Lattices ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%
Values
{{1},{2},{3}}
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
{{1},{2,3},{4}}
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6 = 4 + 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 6 + 2
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 7 = 5 + 2
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 6 + 2
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 7 = 5 + 2
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
{{1},{2,3,4,5},{6}}
=> [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 8 + 2
{{1},{2,3,4},{5},{6}}
=> [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 7 + 2
{{1},{2,3,5},{4},{6}}
=> [1,3,5,4,2,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,6),(2,7),(3,8),(3,10),(4,8),(4,9),(5,3),(5,4),(5,7),(6,2),(6,5),(7,9),(7,10),(8,11),(9,11),(10,11),(11,1)],12)
=> ? = 8 + 2
{{1},{2,3},{4},{5},{6}}
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 6 + 2
{{1},{2,4,5},{3},{6}}
=> [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(2,10),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(6,3),(6,4),(6,5),(7,11),(8,11),(9,2),(9,11),(10,1),(11,10)],12)
=> ? = 7 + 2
{{1},{2,4},{3,5},{6}}
=> [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
=> ? = 7 + 2
{{1},{2,4},{3},{5},{6}}
=> [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,6),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,5)],11)
=> ? = 7 + 2
{{1},{2,5},{3,4},{6}}
=> [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,7),(4,9),(4,11),(5,7),(5,8),(5,10),(6,2),(6,3),(6,4),(6,5),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(11,15),(11,16),(12,14),(12,15),(13,17),(14,17),(15,17),(16,17),(17,1)],18)
=> ? = 10 + 2
{{1},{2},{3,4,5},{6}}
=> [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 7 + 2
{{1},{2},{3,4},{5},{6}}
=> [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 6 + 2
{{1},{2,5},{3},{4},{6}}
=> [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,6),(2,10),(2,11),(3,7),(3,9),(4,7),(4,8),(5,2),(5,8),(5,9),(6,3),(6,4),(6,5),(7,12),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13),(13,1)],14)
=> ? = 8 + 2
{{1},{2},{3,5},{4},{6}}
=> [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,5),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,6),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,1)],11)
=> ? = 7 + 2
{{1},{2},{3},{4,5},{6}}
=> [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 6 + 2
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 5 + 2
{{1},{2,3,4,5,6},{7}}
=> [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ([(0,7),(2,8),(3,11),(4,5),(4,8),(5,6),(5,10),(6,3),(6,9),(7,2),(7,4),(8,10),(9,11),(10,9),(11,1)],12)
=> ? = 10 + 2
{{1},{2,3,4,5},{6},{7}}
=> [1,3,4,5,2,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> ([(0,7),(2,8),(3,9),(4,5),(4,8),(5,3),(5,10),(6,1),(7,2),(7,4),(8,10),(9,6),(10,9)],11)
=> ? = 9 + 2
{{1},{2,3,4,6},{5},{7}}
=> [1,3,4,6,5,2,7] => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(0,7),(1,9),(3,8),(3,11),(4,8),(4,10),(5,6),(5,9),(6,3),(6,4),(6,12),(7,1),(7,5),(8,13),(9,12),(10,13),(11,13),(12,10),(12,11),(13,2)],14)
=> ? = 10 + 2
{{1},{2,3,4},{5},{6},{7}}
=> [1,3,4,2,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> ([(0,7),(1,9),(3,8),(4,2),(5,1),(5,8),(6,4),(7,3),(7,5),(8,9),(9,6)],10)
=> ? = 8 + 2
{{1},{2,3,5,6},{4},{7}}
=> [1,3,5,4,6,2,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(0,7),(2,8),(3,12),(4,10),(4,11),(5,9),(5,11),(6,4),(6,5),(6,8),(7,2),(7,6),(8,9),(8,10),(9,13),(10,13),(11,3),(11,13),(12,1),(13,12)],14)
=> ? = 9 + 2
{{1},{2,3,5},{4,6},{7}}
=> [1,3,5,6,2,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> ([(0,7),(2,11),(3,10),(4,9),(5,6),(5,11),(6,4),(6,8),(7,2),(7,5),(8,9),(8,10),(9,12),(10,12),(11,3),(11,8),(12,1)],13)
=> ? = 9 + 2
{{1},{2,3,5},{4},{6},{7}}
=> [1,3,5,4,2,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7)
=> ([(0,7),(2,8),(3,9),(3,11),(4,9),(4,10),(5,1),(6,3),(6,4),(6,8),(7,2),(7,6),(8,10),(8,11),(9,12),(10,12),(11,12),(12,5)],13)
=> ? = 9 + 2
{{1},{2,3,6},{4,5},{7}}
=> [1,3,6,5,4,2,7] => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,6),(2,8),(3,10),(3,11),(3,14),(4,9),(4,11),(4,13),(5,9),(5,10),(5,12),(6,2),(6,7),(7,3),(7,4),(7,5),(7,8),(8,12),(8,13),(8,14),(9,17),(9,18),(10,15),(10,18),(11,16),(11,18),(12,15),(12,17),(13,16),(13,17),(14,15),(14,16),(15,19),(16,19),(17,19),(18,19),(19,1)],20)
=> ? = 13 + 2
{{1},{2,3,6},{4},{5},{7}}
=> [1,3,6,4,5,2,7] => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(0,7),(1,9),(3,8),(3,12),(4,10),(4,11),(5,4),(5,8),(5,13),(6,3),(6,5),(6,9),(7,1),(7,6),(8,11),(8,14),(9,12),(9,13),(10,15),(11,15),(12,14),(13,10),(13,14),(14,15),(15,2)],16)
=> ? = 10 + 2
{{1},{2,3},{4},{5,6},{7}}
=> [1,3,2,4,6,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,7),(2,9),(3,9),(4,8),(5,8),(6,2),(6,3),(7,4),(7,5),(8,6),(9,1)],10)
=> ? = 8 + 2
{{1},{2,3},{4},{5},{6},{7}}
=> [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,7),(2,8),(3,8),(4,5),(5,1),(6,4),(7,2),(7,3),(8,6)],9)
=> ? = 7 + 2
{{1},{2,4,5,6},{3},{7}}
=> [1,4,3,5,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(0,7),(2,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,11),(7,3),(7,4),(7,5),(8,13),(9,13),(10,6),(10,13),(11,12),(12,1),(13,11)],14)
=> ? = 9 + 2
{{1},{2,4,5},{3,6},{7}}
=> [1,4,6,5,2,3,7] => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7)
=> ([(0,7),(2,8),(2,12),(3,8),(3,11),(4,10),(5,4),(5,9),(6,2),(6,3),(6,9),(7,5),(7,6),(8,15),(9,10),(9,11),(9,12),(10,13),(10,14),(11,13),(11,15),(12,14),(12,15),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 11 + 2
{{1},{2,4,5},{3},{6},{7}}
=> [1,4,3,5,2,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,3),(7,4),(7,5),(8,12),(9,12),(10,6),(11,2),(11,12),(12,10)],13)
=> ? = 8 + 2
{{1},{2,4,6},{3,5},{7}}
=> [1,4,5,6,3,2,7] => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,7),(2,8),(2,10),(3,8),(3,9),(4,11),(4,12),(5,6),(5,9),(5,10),(6,4),(6,13),(6,14),(7,2),(7,3),(7,5),(8,15),(9,13),(9,15),(10,14),(10,15),(11,17),(12,17),(13,11),(13,16),(14,12),(14,16),(15,16),(16,17),(17,1)],18)
=> ? = 11 + 2
{{1},{2,4},{3,5},{6},{7}}
=> [1,4,5,2,3,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> ([(0,7),(2,10),(3,9),(4,3),(4,8),(5,2),(5,8),(6,1),(7,4),(7,5),(8,9),(8,10),(9,11),(10,11),(11,6)],12)
=> ? = 8 + 2
{{1},{2,4,6},{3},{5},{7}}
=> [1,4,3,6,5,2,7] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(2,10),(2,13),(3,9),(3,13),(4,8),(4,12),(5,8),(5,11),(6,9),(6,10),(7,2),(7,3),(7,6),(8,15),(9,14),(10,14),(11,15),(12,15),(13,4),(13,5),(13,14),(14,11),(14,12),(15,1)],16)
=> ? = 9 + 2
{{1},{2,4},{3,6},{5},{7}}
=> [1,4,6,2,5,3,7] => ([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(0,7),(1,10),(1,11),(3,9),(4,8),(5,4),(5,13),(6,3),(6,13),(7,5),(7,6),(8,10),(8,12),(9,11),(9,12),(10,14),(11,14),(12,14),(13,1),(13,8),(13,9),(14,2)],15)
=> ? = 9 + 2
{{1},{2,4},{3},{5},{6},{7}}
=> [1,4,3,2,5,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,7),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,1),(6,5),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,6)],12)
=> ? = 8 + 2
{{1},{2,5},{3,4,6},{7}}
=> [1,5,4,6,2,3,7] => ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(0,7),(2,11),(2,12),(3,8),(4,10),(4,13),(5,9),(5,13),(6,2),(6,9),(6,10),(7,4),(7,5),(7,6),(8,14),(9,11),(9,16),(10,12),(10,16),(11,15),(12,15),(13,3),(13,16),(14,1),(15,14),(16,8),(16,15)],17)
=> ? = 9 + 2
{{1},{2,5},{3,4},{6},{7}}
=> [1,5,4,3,2,6,7] => ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,1),(7,2),(7,3),(7,4),(7,5),(8,14),(8,17),(9,14),(9,15),(10,14),(10,16),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,18),(15,18),(16,18),(17,18),(18,6)],19)
=> ? = 11 + 2
{{1},{2,6},{3,4,5},{7}}
=> [1,6,4,5,3,2,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ?
=> ? = 13 + 2
{{1},{2},{3,4,5,6},{7}}
=> [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ([(0,6),(2,10),(3,8),(4,2),(4,9),(5,4),(5,8),(6,7),(7,3),(7,5),(8,9),(9,10),(10,1)],11)
=> ? = 9 + 2
{{1},{2},{3,4,5},{6},{7}}
=> [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,6),(2,9),(3,8),(4,2),(4,8),(5,1),(6,7),(7,3),(7,4),(8,9),(9,5)],10)
=> ? = 8 + 2
{{1},{2,6},{3,4},{5},{7}}
=> [1,6,4,3,5,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 12 + 2
{{1},{2},{3,4,6},{5},{7}}
=> [1,2,4,6,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,5),(2,9),(2,11),(3,9),(3,10),(4,8),(5,6),(6,4),(6,7),(7,2),(7,3),(7,8),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 9 + 2
{{1},{2},{3,4},{5},{6},{7}}
=> [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
=> ? = 7 + 2
{{1},{2,5,6},{3},{4},{7}}
=> [1,5,3,4,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=> ([(0,7),(1,12),(3,11),(3,13),(4,8),(4,9),(5,8),(5,10),(6,3),(6,9),(6,10),(7,4),(7,5),(7,6),(8,15),(9,13),(9,15),(10,11),(10,15),(11,14),(12,2),(13,1),(13,14),(14,12),(15,14)],16)
=> ? = 9 + 2
{{1},{2,5},{3,6},{4},{7}}
=> [1,5,6,4,2,3,7] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
=> ([(0,2),(2,3),(2,4),(2,5),(3,13),(3,14),(4,7),(4,14),(4,15),(5,6),(5,13),(5,15),(6,9),(6,11),(7,10),(7,12),(8,19),(9,17),(10,18),(11,8),(11,17),(12,8),(12,18),(13,9),(13,16),(14,10),(14,16),(15,11),(15,12),(15,16),(16,17),(16,18),(17,19),(18,19),(19,1)],20)
=> ? = 9 + 2
{{1},{2,5},{3},{4},{6},{7}}
=> [1,5,3,4,2,6,7] => ([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=> ([(0,7),(1,11),(1,12),(3,8),(3,10),(4,8),(4,9),(5,2),(6,1),(6,9),(6,10),(7,3),(7,4),(7,6),(8,14),(9,11),(9,14),(10,12),(10,14),(11,13),(12,13),(13,5),(14,13)],15)
=> ? = 9 + 2
{{1},{2,6},{3,5},{4},{7}}
=> [1,6,5,4,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 13 + 2
{{1},{2},{3,5,6},{4},{7}}
=> [1,2,5,4,6,3,7] => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(2,11),(3,9),(3,10),(4,8),(4,10),(5,8),(5,9),(6,7),(7,3),(7,4),(7,5),(8,12),(9,12),(10,2),(10,12),(11,1),(12,11)],13)
=> ? = 8 + 2
{{1},{2},{3,5},{4,6},{7}}
=> [1,2,5,6,3,4,7] => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(0,6),(2,10),(3,9),(4,3),(4,8),(5,2),(5,8),(6,7),(7,4),(7,5),(8,9),(8,10),(9,11),(10,11),(11,1)],12)
=> ? = 8 + 2
{{1},{2},{3,5},{4},{6},{7}}
=> [1,2,5,4,3,6,7] => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,1),(6,7),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,5)],12)
=> ? = 8 + 2
{{1},{2,6},{3},{4,5},{7}}
=> [1,6,3,5,4,2,7] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ?
=> ? = 11 + 2
Description
The number of modular elements of a lattice.
A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$, and is modular if both $(x, y)$ and $(y, x)$ are modular pairs for every $y\in L$.
Matching statistic: St000551
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St000551: Lattices ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St000551: Lattices ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%
Values
{{1},{2},{3}}
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
{{1},{2,3},{4}}
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6 = 4 + 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 6 + 2
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 7 = 5 + 2
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 6 + 2
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 7 = 5 + 2
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
{{1},{2,3,4,5},{6}}
=> [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 8 + 2
{{1},{2,3,4},{5},{6}}
=> [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 7 + 2
{{1},{2,3,5},{4},{6}}
=> [1,3,5,4,2,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,6),(2,7),(3,8),(3,10),(4,8),(4,9),(5,3),(5,4),(5,7),(6,2),(6,5),(7,9),(7,10),(8,11),(9,11),(10,11),(11,1)],12)
=> ? = 8 + 2
{{1},{2,3},{4},{5},{6}}
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 6 + 2
{{1},{2,4,5},{3},{6}}
=> [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(2,10),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(6,3),(6,4),(6,5),(7,11),(8,11),(9,2),(9,11),(10,1),(11,10)],12)
=> ? = 7 + 2
{{1},{2,4},{3,5},{6}}
=> [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
=> ? = 7 + 2
{{1},{2,4},{3},{5},{6}}
=> [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,6),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,5)],11)
=> ? = 7 + 2
{{1},{2,5},{3,4},{6}}
=> [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,7),(4,9),(4,11),(5,7),(5,8),(5,10),(6,2),(6,3),(6,4),(6,5),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(11,15),(11,16),(12,14),(12,15),(13,17),(14,17),(15,17),(16,17),(17,1)],18)
=> ? = 10 + 2
{{1},{2},{3,4,5},{6}}
=> [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 7 + 2
{{1},{2},{3,4},{5},{6}}
=> [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 6 + 2
{{1},{2,5},{3},{4},{6}}
=> [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,6),(2,10),(2,11),(3,7),(3,9),(4,7),(4,8),(5,2),(5,8),(5,9),(6,3),(6,4),(6,5),(7,12),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13),(13,1)],14)
=> ? = 8 + 2
{{1},{2},{3,5},{4},{6}}
=> [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,5),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,6),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,1)],11)
=> ? = 7 + 2
{{1},{2},{3},{4,5},{6}}
=> [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 6 + 2
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 5 + 2
{{1},{2,3,4,5,6},{7}}
=> [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ([(0,7),(2,8),(3,11),(4,5),(4,8),(5,6),(5,10),(6,3),(6,9),(7,2),(7,4),(8,10),(9,11),(10,9),(11,1)],12)
=> ? = 10 + 2
{{1},{2,3,4,5},{6},{7}}
=> [1,3,4,5,2,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> ([(0,7),(2,8),(3,9),(4,5),(4,8),(5,3),(5,10),(6,1),(7,2),(7,4),(8,10),(9,6),(10,9)],11)
=> ? = 9 + 2
{{1},{2,3,4,6},{5},{7}}
=> [1,3,4,6,5,2,7] => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(0,7),(1,9),(3,8),(3,11),(4,8),(4,10),(5,6),(5,9),(6,3),(6,4),(6,12),(7,1),(7,5),(8,13),(9,12),(10,13),(11,13),(12,10),(12,11),(13,2)],14)
=> ? = 10 + 2
{{1},{2,3,4},{5},{6},{7}}
=> [1,3,4,2,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> ([(0,7),(1,9),(3,8),(4,2),(5,1),(5,8),(6,4),(7,3),(7,5),(8,9),(9,6)],10)
=> ? = 8 + 2
{{1},{2,3,5,6},{4},{7}}
=> [1,3,5,4,6,2,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(0,7),(2,8),(3,12),(4,10),(4,11),(5,9),(5,11),(6,4),(6,5),(6,8),(7,2),(7,6),(8,9),(8,10),(9,13),(10,13),(11,3),(11,13),(12,1),(13,12)],14)
=> ? = 9 + 2
{{1},{2,3,5},{4,6},{7}}
=> [1,3,5,6,2,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> ([(0,7),(2,11),(3,10),(4,9),(5,6),(5,11),(6,4),(6,8),(7,2),(7,5),(8,9),(8,10),(9,12),(10,12),(11,3),(11,8),(12,1)],13)
=> ? = 9 + 2
{{1},{2,3,5},{4},{6},{7}}
=> [1,3,5,4,2,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7)
=> ([(0,7),(2,8),(3,9),(3,11),(4,9),(4,10),(5,1),(6,3),(6,4),(6,8),(7,2),(7,6),(8,10),(8,11),(9,12),(10,12),(11,12),(12,5)],13)
=> ? = 9 + 2
{{1},{2,3,6},{4,5},{7}}
=> [1,3,6,5,4,2,7] => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,6),(2,8),(3,10),(3,11),(3,14),(4,9),(4,11),(4,13),(5,9),(5,10),(5,12),(6,2),(6,7),(7,3),(7,4),(7,5),(7,8),(8,12),(8,13),(8,14),(9,17),(9,18),(10,15),(10,18),(11,16),(11,18),(12,15),(12,17),(13,16),(13,17),(14,15),(14,16),(15,19),(16,19),(17,19),(18,19),(19,1)],20)
=> ? = 13 + 2
{{1},{2,3,6},{4},{5},{7}}
=> [1,3,6,4,5,2,7] => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(0,7),(1,9),(3,8),(3,12),(4,10),(4,11),(5,4),(5,8),(5,13),(6,3),(6,5),(6,9),(7,1),(7,6),(8,11),(8,14),(9,12),(9,13),(10,15),(11,15),(12,14),(13,10),(13,14),(14,15),(15,2)],16)
=> ? = 10 + 2
{{1},{2,3},{4},{5,6},{7}}
=> [1,3,2,4,6,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,7),(2,9),(3,9),(4,8),(5,8),(6,2),(6,3),(7,4),(7,5),(8,6),(9,1)],10)
=> ? = 8 + 2
{{1},{2,3},{4},{5},{6},{7}}
=> [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,7),(2,8),(3,8),(4,5),(5,1),(6,4),(7,2),(7,3),(8,6)],9)
=> ? = 7 + 2
{{1},{2,4,5,6},{3},{7}}
=> [1,4,3,5,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(0,7),(2,12),(3,8),(3,9),(4,9),(4,10),(5,8),(5,10),(6,2),(6,11),(7,3),(7,4),(7,5),(8,13),(9,13),(10,6),(10,13),(11,12),(12,1),(13,11)],14)
=> ? = 9 + 2
{{1},{2,4,5},{3,6},{7}}
=> [1,4,6,5,2,3,7] => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7)
=> ([(0,7),(2,8),(2,12),(3,8),(3,11),(4,10),(5,4),(5,9),(6,2),(6,3),(6,9),(7,5),(7,6),(8,15),(9,10),(9,11),(9,12),(10,13),(10,14),(11,13),(11,15),(12,14),(12,15),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 11 + 2
{{1},{2,4,5},{3},{6},{7}}
=> [1,4,3,5,2,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,10),(3,9),(3,11),(4,8),(4,11),(5,8),(5,9),(6,1),(7,3),(7,4),(7,5),(8,12),(9,12),(10,6),(11,2),(11,12),(12,10)],13)
=> ? = 8 + 2
{{1},{2,4,6},{3,5},{7}}
=> [1,4,5,6,3,2,7] => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,7),(2,8),(2,10),(3,8),(3,9),(4,11),(4,12),(5,6),(5,9),(5,10),(6,4),(6,13),(6,14),(7,2),(7,3),(7,5),(8,15),(9,13),(9,15),(10,14),(10,15),(11,17),(12,17),(13,11),(13,16),(14,12),(14,16),(15,16),(16,17),(17,1)],18)
=> ? = 11 + 2
{{1},{2,4},{3,5},{6},{7}}
=> [1,4,5,2,3,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> ([(0,7),(2,10),(3,9),(4,3),(4,8),(5,2),(5,8),(6,1),(7,4),(7,5),(8,9),(8,10),(9,11),(10,11),(11,6)],12)
=> ? = 8 + 2
{{1},{2,4,6},{3},{5},{7}}
=> [1,4,3,6,5,2,7] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(2,10),(2,13),(3,9),(3,13),(4,8),(4,12),(5,8),(5,11),(6,9),(6,10),(7,2),(7,3),(7,6),(8,15),(9,14),(10,14),(11,15),(12,15),(13,4),(13,5),(13,14),(14,11),(14,12),(15,1)],16)
=> ? = 9 + 2
{{1},{2,4},{3,6},{5},{7}}
=> [1,4,6,2,5,3,7] => ([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(0,7),(1,10),(1,11),(3,9),(4,8),(5,4),(5,13),(6,3),(6,13),(7,5),(7,6),(8,10),(8,12),(9,11),(9,12),(10,14),(11,14),(12,14),(13,1),(13,8),(13,9),(14,2)],15)
=> ? = 9 + 2
{{1},{2,4},{3},{5},{6},{7}}
=> [1,4,3,2,5,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,7),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,1),(6,5),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,6)],12)
=> ? = 8 + 2
{{1},{2,5},{3,4,6},{7}}
=> [1,5,4,6,2,3,7] => ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(0,7),(2,11),(2,12),(3,8),(4,10),(4,13),(5,9),(5,13),(6,2),(6,9),(6,10),(7,4),(7,5),(7,6),(8,14),(9,11),(9,16),(10,12),(10,16),(11,15),(12,15),(13,3),(13,16),(14,1),(15,14),(16,8),(16,15)],17)
=> ? = 9 + 2
{{1},{2,5},{3,4},{6},{7}}
=> [1,5,4,3,2,6,7] => ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,1),(7,2),(7,3),(7,4),(7,5),(8,14),(8,17),(9,14),(9,15),(10,14),(10,16),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,18),(15,18),(16,18),(17,18),(18,6)],19)
=> ? = 11 + 2
{{1},{2,6},{3,4,5},{7}}
=> [1,6,4,5,3,2,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ?
=> ? = 13 + 2
{{1},{2},{3,4,5,6},{7}}
=> [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ([(0,6),(2,10),(3,8),(4,2),(4,9),(5,4),(5,8),(6,7),(7,3),(7,5),(8,9),(9,10),(10,1)],11)
=> ? = 9 + 2
{{1},{2},{3,4,5},{6},{7}}
=> [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,6),(2,9),(3,8),(4,2),(4,8),(5,1),(6,7),(7,3),(7,4),(8,9),(9,5)],10)
=> ? = 8 + 2
{{1},{2,6},{3,4},{5},{7}}
=> [1,6,4,3,5,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 12 + 2
{{1},{2},{3,4,6},{5},{7}}
=> [1,2,4,6,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,5),(2,9),(2,11),(3,9),(3,10),(4,8),(5,6),(6,4),(6,7),(7,2),(7,3),(7,8),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 9 + 2
{{1},{2},{3,4},{5},{6},{7}}
=> [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
=> ? = 7 + 2
{{1},{2,5,6},{3},{4},{7}}
=> [1,5,3,4,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=> ([(0,7),(1,12),(3,11),(3,13),(4,8),(4,9),(5,8),(5,10),(6,3),(6,9),(6,10),(7,4),(7,5),(7,6),(8,15),(9,13),(9,15),(10,11),(10,15),(11,14),(12,2),(13,1),(13,14),(14,12),(15,14)],16)
=> ? = 9 + 2
{{1},{2,5},{3,6},{4},{7}}
=> [1,5,6,4,2,3,7] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
=> ([(0,2),(2,3),(2,4),(2,5),(3,13),(3,14),(4,7),(4,14),(4,15),(5,6),(5,13),(5,15),(6,9),(6,11),(7,10),(7,12),(8,19),(9,17),(10,18),(11,8),(11,17),(12,8),(12,18),(13,9),(13,16),(14,10),(14,16),(15,11),(15,12),(15,16),(16,17),(16,18),(17,19),(18,19),(19,1)],20)
=> ? = 9 + 2
{{1},{2,5},{3},{4},{6},{7}}
=> [1,5,3,4,2,6,7] => ([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=> ([(0,7),(1,11),(1,12),(3,8),(3,10),(4,8),(4,9),(5,2),(6,1),(6,9),(6,10),(7,3),(7,4),(7,6),(8,14),(9,11),(9,14),(10,12),(10,14),(11,13),(12,13),(13,5),(14,13)],15)
=> ? = 9 + 2
{{1},{2,6},{3,5},{4},{7}}
=> [1,6,5,4,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 13 + 2
{{1},{2},{3,5,6},{4},{7}}
=> [1,2,5,4,6,3,7] => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(2,11),(3,9),(3,10),(4,8),(4,10),(5,8),(5,9),(6,7),(7,3),(7,4),(7,5),(8,12),(9,12),(10,2),(10,12),(11,1),(12,11)],13)
=> ? = 8 + 2
{{1},{2},{3,5},{4,6},{7}}
=> [1,2,5,6,3,4,7] => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(0,6),(2,10),(3,9),(4,3),(4,8),(5,2),(5,8),(6,7),(7,4),(7,5),(8,9),(8,10),(9,11),(10,11),(11,1)],12)
=> ? = 8 + 2
{{1},{2},{3,5},{4},{6},{7}}
=> [1,2,5,4,3,6,7] => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(2,9),(2,10),(3,8),(3,10),(4,8),(4,9),(5,1),(6,7),(7,2),(7,3),(7,4),(8,11),(9,11),(10,11),(11,5)],12)
=> ? = 8 + 2
{{1},{2,6},{3},{4,5},{7}}
=> [1,6,3,5,4,2,7] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ?
=> ? = 11 + 2
Description
The number of left modular elements of a lattice.
A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$.
Matching statistic: St000912
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000912: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 42%
Mp00209: Permutations —pattern poset⟶ Posets
St000912: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 42%
Values
{{1},{2},{3}}
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3 = 2 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 5 = 4 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 6 + 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5 + 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 6 + 1
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5 + 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
{{1},{2,3,4,5},{6}}
=> [1,3,4,5,2,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 8 + 1
{{1},{2,3,4},{5},{6}}
=> [1,3,4,2,5,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 7 + 1
{{1},{2,3,5},{4},{6}}
=> [1,3,5,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,17),(2,10),(2,11),(2,17),(3,8),(3,9),(3,13),(3,17),(4,7),(4,9),(4,10),(4,17),(5,7),(5,8),(5,11),(5,12),(7,15),(7,20),(7,21),(8,14),(8,15),(8,20),(9,15),(9,16),(9,21),(10,21),(11,20),(11,21),(12,14),(12,20),(13,14),(13,16),(14,19),(15,18),(15,19),(16,18),(16,19),(17,16),(17,20),(17,21),(18,6),(19,6),(20,18),(20,19),(21,18)],22)
=> ? = 8 + 1
{{1},{2,3},{4},{5},{6}}
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 6 + 1
{{1},{2,4,5},{3},{6}}
=> [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,17),(2,10),(2,11),(2,17),(3,8),(3,9),(3,13),(3,17),(4,7),(4,9),(4,10),(4,17),(5,7),(5,8),(5,11),(5,12),(7,15),(7,20),(7,21),(8,14),(8,15),(8,20),(9,15),(9,16),(9,21),(10,21),(11,20),(11,21),(12,14),(12,20),(13,14),(13,16),(14,19),(15,18),(15,19),(16,18),(16,19),(17,16),(17,20),(17,21),(18,6),(19,6),(20,18),(20,19),(21,18)],22)
=> ? = 7 + 1
{{1},{2,4},{3,5},{6}}
=> [1,4,5,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,5),(3,7),(3,9),(3,11),(4,5),(4,7),(4,8),(4,10),(5,19),(7,13),(7,14),(7,19),(8,13),(8,15),(8,19),(9,13),(9,16),(9,19),(10,14),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(13,17),(13,18),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,18)],20)
=> ? = 7 + 1
{{1},{2,4},{3},{5},{6}}
=> [1,4,3,2,5,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 7 + 1
{{1},{2,5},{3,4},{6}}
=> [1,5,4,3,2,6] => ([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ? = 10 + 1
{{1},{2},{3,4,5},{6}}
=> [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 7 + 1
{{1},{2},{3,4},{5},{6}}
=> [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 6 + 1
{{1},{2,5},{3},{4},{6}}
=> [1,5,3,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(1,8),(1,18),(2,12),(2,13),(2,14),(2,18),(3,10),(3,11),(3,14),(3,18),(4,8),(4,9),(4,11),(4,13),(5,7),(5,9),(5,10),(5,12),(7,15),(7,19),(8,15),(8,20),(9,15),(9,16),(9,17),(10,16),(10,19),(10,23),(11,16),(11,20),(11,23),(12,17),(12,19),(12,23),(13,17),(13,20),(13,23),(14,23),(15,22),(16,21),(16,22),(17,21),(17,22),(18,19),(18,20),(18,23),(19,21),(19,22),(20,21),(20,22),(21,6),(22,6),(23,21)],24)
=> ? = 8 + 1
{{1},{2},{3,5},{4},{6}}
=> [1,2,5,4,3,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 7 + 1
{{1},{2},{3},{4,5},{6}}
=> [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 6 + 1
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
{{1},{2,3,4,5,6},{7}}
=> [1,3,4,5,6,2,7] => ([(0,1),(0,4),(0,5),(0,6),(1,18),(2,8),(2,9),(2,21),(3,2),(3,11),(3,12),(3,20),(4,10),(4,14),(4,18),(5,10),(5,13),(5,18),(6,3),(6,13),(6,14),(6,18),(8,17),(8,19),(9,17),(9,19),(10,15),(10,20),(11,8),(11,16),(11,21),(12,9),(12,16),(12,21),(13,11),(13,15),(13,20),(14,12),(14,15),(14,20),(15,16),(15,21),(16,17),(16,19),(17,7),(18,20),(19,7),(20,21),(21,19)],22)
=> ? = 10 + 1
{{1},{2,3,4,5},{6},{7}}
=> [1,3,4,5,2,6,7] => ([(0,1),(0,4),(0,5),(0,6),(1,20),(2,10),(2,12),(2,23),(3,9),(3,11),(3,23),(4,13),(4,14),(4,20),(5,3),(5,13),(5,15),(5,20),(6,2),(6,14),(6,15),(6,20),(8,19),(8,21),(9,17),(9,22),(10,18),(10,22),(11,8),(11,17),(11,22),(12,8),(12,18),(12,22),(13,9),(13,16),(13,23),(14,10),(14,16),(14,23),(15,11),(15,12),(15,16),(15,23),(16,17),(16,18),(16,22),(17,19),(17,21),(18,19),(18,21),(19,7),(20,23),(21,7),(22,21),(23,22)],24)
=> ? = 9 + 1
{{1},{2,3,4,6},{5},{7}}
=> [1,3,4,6,5,2,7] => ?
=> ? = 10 + 1
{{1},{2,3,4},{5},{6},{7}}
=> [1,3,4,2,5,6,7] => ([(0,1),(0,4),(0,5),(0,6),(1,18),(2,8),(2,9),(2,21),(3,2),(3,11),(3,12),(3,20),(4,10),(4,14),(4,18),(5,10),(5,13),(5,18),(6,3),(6,13),(6,14),(6,18),(8,17),(8,19),(9,17),(9,19),(10,15),(10,20),(11,8),(11,16),(11,21),(12,9),(12,16),(12,21),(13,11),(13,15),(13,20),(14,12),(14,15),(14,20),(15,16),(15,21),(16,17),(16,19),(17,7),(18,20),(19,7),(20,21),(21,19)],22)
=> ? = 8 + 1
{{1},{2,3,5,6},{4},{7}}
=> [1,3,5,4,6,2,7] => ?
=> ? = 9 + 1
{{1},{2,3,5},{4,6},{7}}
=> [1,3,5,6,2,4,7] => ?
=> ? = 9 + 1
{{1},{2,3,5},{4},{6},{7}}
=> [1,3,5,4,2,6,7] => ?
=> ? = 9 + 1
{{1},{2,3,6},{4,5},{7}}
=> [1,3,6,5,4,2,7] => ?
=> ? = 13 + 1
{{1},{2,3,6},{4},{5},{7}}
=> [1,3,6,4,5,2,7] => ?
=> ? = 10 + 1
{{1},{2,3},{4},{5,6},{7}}
=> [1,3,2,4,6,5,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(1,17),(1,18),(2,11),(2,12),(2,18),(3,10),(3,12),(3,17),(4,7),(4,8),(4,10),(4,18),(5,7),(5,9),(5,11),(5,17),(7,14),(7,21),(7,22),(8,14),(8,21),(8,23),(9,14),(9,22),(9,23),(10,21),(10,24),(11,22),(11,24),(12,24),(13,6),(14,15),(14,16),(15,13),(15,19),(16,13),(16,19),(17,21),(17,23),(17,24),(18,22),(18,23),(18,24),(19,6),(20,19),(21,15),(21,20),(22,16),(22,20),(23,15),(23,16),(23,20),(24,20)],25)
=> ? = 8 + 1
{{1},{2,3},{4},{5},{6},{7}}
=> [1,3,2,4,5,6,7] => ([(0,2),(0,3),(0,6),(1,10),(1,15),(2,12),(3,7),(3,12),(4,5),(4,11),(4,13),(5,1),(5,9),(5,14),(6,4),(6,7),(6,12),(7,11),(7,13),(9,10),(9,15),(10,8),(11,9),(11,14),(12,13),(13,14),(14,15),(15,8)],16)
=> ? = 7 + 1
{{1},{2,4,5,6},{3},{7}}
=> [1,4,3,5,6,2,7] => ?
=> ? = 9 + 1
{{1},{2,4,5},{3,6},{7}}
=> [1,4,6,5,2,3,7] => ?
=> ? = 11 + 1
{{1},{2,4,5},{3},{6},{7}}
=> [1,4,3,5,2,6,7] => ?
=> ? = 8 + 1
{{1},{2,4,6},{3,5},{7}}
=> [1,4,5,6,3,2,7] => ([(0,3),(0,4),(0,5),(0,6),(1,24),(2,8),(2,9),(2,10),(3,11),(3,13),(3,15),(4,11),(4,12),(4,14),(5,2),(5,14),(5,15),(5,16),(6,1),(6,12),(6,13),(6,16),(8,21),(8,22),(9,21),(9,23),(10,22),(10,23),(10,28),(11,17),(11,18),(12,18),(12,19),(12,24),(13,18),(13,20),(13,24),(14,8),(14,17),(14,19),(15,9),(15,17),(15,20),(16,10),(16,19),(16,20),(16,24),(17,21),(17,25),(18,25),(18,28),(19,22),(19,25),(19,28),(20,23),(20,25),(20,28),(21,26),(22,26),(22,27),(23,26),(23,27),(24,28),(25,26),(25,27),(26,7),(27,7),(28,27)],29)
=> ? = 11 + 1
{{1},{2,4},{3,5},{6},{7}}
=> [1,4,5,2,3,6,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(1,8),(2,12),(2,13),(2,16),(3,1),(3,14),(3,15),(3,16),(4,9),(4,11),(4,13),(4,15),(5,9),(5,11),(5,12),(5,14),(6,22),(6,23),(7,21),(7,22),(7,28),(8,21),(8,23),(8,28),(9,27),(11,17),(11,20),(11,27),(12,17),(12,18),(12,27),(13,17),(13,19),(13,27),(14,7),(14,18),(14,20),(14,27),(15,8),(15,19),(15,20),(15,27),(16,6),(16,18),(16,19),(17,24),(17,28),(18,22),(18,24),(18,28),(19,23),(19,24),(19,28),(20,21),(20,24),(20,28),(21,25),(21,26),(22,25),(22,26),(23,25),(23,26),(24,25),(24,26),(25,10),(26,10),(27,28),(28,26)],29)
=> ? = 8 + 1
{{1},{2,4,6},{3},{5},{7}}
=> [1,4,3,6,5,2,7] => ?
=> ? = 9 + 1
{{1},{2,4},{3,6},{5},{7}}
=> [1,4,6,2,5,3,7] => ?
=> ? = 9 + 1
{{1},{2,4},{3},{5},{6},{7}}
=> [1,4,3,2,5,6,7] => ([(0,4),(0,5),(0,6),(1,16),(2,8),(2,9),(3,2),(3,11),(3,12),(4,10),(4,13),(5,3),(5,13),(5,14),(6,1),(6,10),(6,14),(8,19),(9,19),(9,20),(10,15),(10,16),(11,9),(11,17),(11,18),(12,8),(12,17),(13,12),(13,15),(14,11),(14,15),(14,16),(15,17),(15,18),(16,18),(17,19),(17,20),(18,20),(19,7),(20,7)],21)
=> ? = 8 + 1
{{1},{2,5},{3,4,6},{7}}
=> [1,5,4,6,2,3,7] => ?
=> ? = 9 + 1
{{1},{2,5},{3,4},{6},{7}}
=> [1,5,4,3,2,6,7] => ([(0,4),(0,5),(0,6),(1,17),(2,7),(2,11),(3,1),(3,10),(3,12),(4,13),(4,14),(5,3),(5,14),(5,15),(6,2),(6,13),(6,15),(7,19),(9,20),(9,21),(10,17),(10,18),(11,9),(11,19),(12,9),(12,17),(12,18),(13,7),(13,16),(14,10),(14,16),(15,11),(15,12),(15,16),(16,18),(16,19),(17,21),(18,20),(18,21),(19,20),(20,8),(21,8)],22)
=> ? = 11 + 1
{{1},{2,6},{3,4,5},{7}}
=> [1,6,4,5,3,2,7] => ?
=> ? = 13 + 1
{{1},{2},{3,4,5,6},{7}}
=> [1,2,4,5,6,3,7] => ([(0,1),(0,4),(0,5),(0,6),(1,20),(2,10),(2,12),(2,23),(3,9),(3,11),(3,23),(4,13),(4,14),(4,20),(5,3),(5,13),(5,15),(5,20),(6,2),(6,14),(6,15),(6,20),(8,19),(8,21),(9,17),(9,22),(10,18),(10,22),(11,8),(11,17),(11,22),(12,8),(12,18),(12,22),(13,9),(13,16),(13,23),(14,10),(14,16),(14,23),(15,11),(15,12),(15,16),(15,23),(16,17),(16,18),(16,22),(17,19),(17,21),(18,19),(18,21),(19,7),(20,23),(21,7),(22,21),(23,22)],24)
=> ? = 9 + 1
{{1},{2},{3,4,5},{6},{7}}
=> [1,2,4,5,3,6,7] => ([(0,1),(0,4),(0,5),(0,6),(1,20),(2,10),(2,12),(2,23),(3,9),(3,11),(3,23),(4,13),(4,14),(4,20),(5,3),(5,13),(5,15),(5,20),(6,2),(6,14),(6,15),(6,20),(8,19),(8,21),(9,17),(9,22),(10,18),(10,22),(11,8),(11,17),(11,22),(12,8),(12,18),(12,22),(13,9),(13,16),(13,23),(14,10),(14,16),(14,23),(15,11),(15,12),(15,16),(15,23),(16,17),(16,18),(16,22),(17,19),(17,21),(18,19),(18,21),(19,7),(20,23),(21,7),(22,21),(23,22)],24)
=> ? = 8 + 1
{{1},{2,6},{3,4},{5},{7}}
=> [1,6,4,3,5,2,7] => ?
=> ? = 12 + 1
{{1},{2},{3,4,6},{5},{7}}
=> [1,2,4,6,5,3,7] => ?
=> ? = 9 + 1
{{1},{2},{3,4},{5},{6},{7}}
=> [1,2,4,3,5,6,7] => ([(0,1),(0,5),(0,6),(1,14),(2,11),(2,17),(3,4),(3,13),(3,17),(4,10),(4,16),(5,2),(5,12),(5,14),(6,3),(6,12),(6,14),(8,7),(9,8),(9,15),(10,8),(10,15),(11,9),(11,16),(12,11),(12,13),(12,17),(13,9),(13,10),(13,16),(14,17),(15,7),(16,15),(17,16)],18)
=> ? = 7 + 1
{{1},{2,5,6},{3},{4},{7}}
=> [1,5,3,4,6,2,7] => ?
=> ? = 9 + 1
{{1},{2,5},{3,6},{4},{7}}
=> [1,5,6,4,2,3,7] => ?
=> ? = 9 + 1
{{1},{2,5},{3},{4},{6},{7}}
=> [1,5,3,4,2,6,7] => ?
=> ? = 9 + 1
{{1},{2,6},{3,5},{4},{7}}
=> [1,6,5,4,3,2,7] => ([(0,4),(0,5),(0,6),(1,18),(2,1),(2,8),(2,9),(3,2),(3,11),(3,12),(4,10),(4,14),(5,10),(5,13),(6,3),(6,13),(6,14),(8,17),(8,18),(9,17),(9,18),(10,15),(11,8),(11,16),(12,9),(12,16),(13,11),(13,15),(14,12),(14,15),(15,16),(16,17),(17,7),(18,7)],19)
=> ? = 13 + 1
{{1},{2},{3,5,6},{4},{7}}
=> [1,2,5,4,6,3,7] => ?
=> ? = 8 + 1
{{1},{2},{3,5},{4,6},{7}}
=> [1,2,5,6,3,4,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(1,8),(2,12),(2,13),(2,16),(3,1),(3,14),(3,15),(3,16),(4,9),(4,11),(4,13),(4,15),(5,9),(5,11),(5,12),(5,14),(6,22),(6,23),(7,21),(7,22),(7,28),(8,21),(8,23),(8,28),(9,27),(11,17),(11,20),(11,27),(12,17),(12,18),(12,27),(13,17),(13,19),(13,27),(14,7),(14,18),(14,20),(14,27),(15,8),(15,19),(15,20),(15,27),(16,6),(16,18),(16,19),(17,24),(17,28),(18,22),(18,24),(18,28),(19,23),(19,24),(19,28),(20,21),(20,24),(20,28),(21,25),(21,26),(22,25),(22,26),(23,25),(23,26),(24,25),(24,26),(25,10),(26,10),(27,28),(28,26)],29)
=> ? = 8 + 1
{{1},{2},{3},{4},{5},{6},{7}}
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
Description
The number of maximal antichains in a poset.
Matching statistic: St001343
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00220: Set partitions —Yip⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001343: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 42%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001343: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 42%
Values
{{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3 = 2 + 1
{{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5 = 4 + 1
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
{{1},{2,3,4},{5}}
=> {{1,3,4},{2},{5}}
=> [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 6 + 1
{{1},{2,3},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 5 + 1
{{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 6 + 1
{{1},{2},{3,4},{5}}
=> {{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 5 + 1
{{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
{{1},{2,3,4,5},{6}}
=> {{1,3,4,5},{2},{6}}
=> [3,2,4,5,1,6] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? = 8 + 1
{{1},{2,3,4},{5},{6}}
=> {{1,3,4},{2},{5},{6}}
=> [3,2,4,1,5,6] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ? = 7 + 1
{{1},{2,3,5},{4},{6}}
=> {{1,3},{2,5},{4},{6}}
=> [3,5,1,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,17),(1,18),(2,12),(2,14),(2,18),(2,19),(3,11),(3,14),(3,17),(3,19),(4,10),(4,13),(4,17),(4,18),(4,19),(5,9),(5,13),(5,17),(5,18),(5,19),(6,8),(6,9),(6,10),(6,11),(6,12),(8,21),(8,22),(9,20),(9,21),(9,22),(9,25),(10,20),(10,21),(10,22),(10,25),(11,15),(11,20),(11,21),(12,15),(12,20),(12,22),(13,16),(13,25),(14,15),(14,25),(15,24),(16,23),(17,16),(17,21),(17,25),(18,16),(18,22),(18,25),(19,16),(19,20),(19,25),(20,23),(20,24),(21,23),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24)],26)
=> ? = 8 + 1
{{1},{2,3},{4},{5},{6}}
=> {{1,3},{2},{4},{5},{6}}
=> [3,2,1,4,5,6] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 6 + 1
{{1},{2,4,5},{3},{6}}
=> {{1},{2,4,5},{3},{6}}
=> [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,17),(2,10),(2,11),(2,17),(3,8),(3,9),(3,13),(3,17),(4,7),(4,9),(4,10),(4,17),(5,7),(5,8),(5,11),(5,12),(7,15),(7,20),(7,21),(8,14),(8,15),(8,20),(9,15),(9,16),(9,21),(10,21),(11,20),(11,21),(12,14),(12,20),(13,14),(13,16),(14,19),(15,18),(15,19),(16,18),(16,19),(17,16),(17,20),(17,21),(18,6),(19,6),(20,18),(20,19),(21,18)],22)
=> ? = 7 + 1
{{1},{2,4},{3,5},{6}}
=> {{1,5},{2,4},{3},{6}}
=> [5,4,3,2,1,6] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 7 + 1
{{1},{2,4},{3},{5},{6}}
=> {{1},{2,4},{3},{5},{6}}
=> [1,4,3,2,5,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 7 + 1
{{1},{2,5},{3,4},{6}}
=> {{1,4},{2,5},{3},{6}}
=> [4,5,3,1,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(1,9),(2,9),(2,10),(2,12),(3,8),(3,10),(3,11),(4,7),(4,11),(4,12),(5,17),(7,14),(7,15),(8,13),(8,14),(9,13),(9,15),(10,13),(10,16),(11,5),(11,14),(11,16),(12,5),(12,15),(12,16),(13,18),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
=> ? = 10 + 1
{{1},{2},{3,4,5},{6}}
=> {{1,4,5},{2},{3},{6}}
=> [4,2,3,5,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
=> ? = 7 + 1
{{1},{2},{3,4},{5},{6}}
=> {{1,4},{2},{3},{5},{6}}
=> [4,2,3,1,5,6] => ([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
=> ? = 6 + 1
{{1},{2,5},{3},{4},{6}}
=> {{1},{2},{3,5},{4},{6}}
=> [1,2,5,4,3,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 8 + 1
{{1},{2},{3,5},{4},{6}}
=> {{1},{2,5},{3},{4},{6}}
=> [1,5,3,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(1,8),(1,18),(2,12),(2,13),(2,14),(2,18),(3,10),(3,11),(3,14),(3,18),(4,8),(4,9),(4,11),(4,13),(5,7),(5,9),(5,10),(5,12),(7,15),(7,19),(8,15),(8,20),(9,15),(9,16),(9,17),(10,16),(10,19),(10,23),(11,16),(11,20),(11,23),(12,17),(12,19),(12,23),(13,17),(13,20),(13,23),(14,23),(15,22),(16,21),(16,22),(17,21),(17,22),(18,19),(18,20),(18,23),(19,21),(19,22),(20,21),(20,22),(21,6),(22,6),(23,21)],24)
=> ? = 7 + 1
{{1},{2},{3},{4,5},{6}}
=> {{1,5},{2},{3},{4},{6}}
=> [5,2,3,4,1,6] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,16),(2,8),(2,10),(2,12),(3,8),(3,9),(3,11),(4,9),(4,10),(4,13),(5,1),(5,11),(5,12),(5,13),(6,17),(8,19),(9,14),(9,19),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(12,19),(13,6),(13,14),(13,15),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,7),(18,7),(19,18)],20)
=> ? = 6 + 1
{{1},{2},{3},{4},{5},{6}}
=> {{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
{{1},{2,3,4,5,6},{7}}
=> {{1,3,4,5,6},{2},{7}}
=> [3,2,4,5,6,1,7] => ([(0,2),(0,4),(0,5),(0,6),(1,9),(1,22),(2,10),(2,18),(3,1),(3,7),(3,12),(3,16),(4,11),(4,14),(4,18),(5,3),(5,13),(5,14),(5,18),(6,10),(6,11),(6,13),(7,17),(7,22),(7,23),(9,19),(10,21),(11,15),(11,21),(12,9),(12,17),(12,22),(13,7),(13,15),(13,21),(14,12),(14,15),(14,16),(15,17),(15,23),(16,22),(16,23),(17,19),(17,20),(18,16),(18,21),(19,8),(20,8),(21,23),(22,19),(22,20),(23,20)],24)
=> ? = 10 + 1
{{1},{2,3,4,5},{6},{7}}
=> {{1,3,4,5},{2},{6},{7}}
=> [3,2,4,5,1,6,7] => ([(0,2),(0,4),(0,5),(0,6),(1,13),(1,24),(2,9),(2,18),(3,10),(3,12),(3,24),(4,3),(4,14),(4,15),(4,18),(5,1),(5,11),(5,15),(5,18),(6,9),(6,11),(6,14),(8,19),(9,21),(10,17),(10,22),(11,16),(11,21),(11,24),(12,8),(12,17),(12,23),(13,8),(13,23),(14,10),(14,16),(14,21),(15,12),(15,13),(15,16),(15,24),(16,17),(16,22),(16,23),(17,19),(17,20),(18,21),(18,24),(19,7),(20,7),(21,22),(22,20),(23,19),(23,20),(24,22),(24,23)],25)
=> ? = 9 + 1
{{1},{2,3,4,6},{5},{7}}
=> {{1,3,4},{2,6},{5},{7}}
=> [3,6,4,1,5,2,7] => ?
=> ? = 10 + 1
{{1},{2,3,4},{5},{6},{7}}
=> {{1,3,4},{2},{5},{6},{7}}
=> [3,2,4,1,5,6,7] => ([(0,1),(0,2),(0,5),(0,6),(1,10),(1,20),(2,11),(2,20),(3,4),(3,13),(3,14),(3,21),(4,7),(4,8),(4,19),(5,10),(5,12),(5,20),(6,3),(6,11),(6,12),(6,20),(7,17),(8,17),(8,18),(10,15),(11,14),(11,21),(12,13),(12,15),(12,21),(13,8),(13,16),(13,19),(14,7),(14,19),(15,16),(16,18),(17,9),(18,9),(19,17),(19,18),(20,15),(20,21),(21,16),(21,19)],22)
=> ? = 8 + 1
{{1},{2,3,5,6},{4},{7}}
=> {{1,3,6},{2,5},{4},{7}}
=> [3,5,6,4,2,1,7] => ?
=> ? = 9 + 1
{{1},{2,3,5},{4,6},{7}}
=> {{1,3},{2,5,6},{4},{7}}
=> [3,5,1,4,6,2,7] => ?
=> ? = 9 + 1
{{1},{2,3,5},{4},{6},{7}}
=> {{1,3},{2,5},{4},{6},{7}}
=> [3,5,1,4,2,6,7] => ?
=> ? = 9 + 1
{{1},{2,3,6},{4,5},{7}}
=> {{1,3,5},{2,6},{4},{7}}
=> [3,6,5,4,1,2,7] => ?
=> ? = 13 + 1
{{1},{2,3,6},{4},{5},{7}}
=> {{1,3},{2},{4,6},{5},{7}}
=> [3,2,1,6,5,4,7] => ([(0,3),(0,4),(0,5),(1,12),(1,17),(2,10),(2,11),(2,17),(3,9),(3,13),(4,2),(4,13),(4,14),(5,1),(5,9),(5,14),(6,18),(7,19),(9,15),(9,17),(10,7),(10,16),(11,6),(11,16),(11,20),(12,6),(12,20),(13,10),(13,15),(14,11),(14,12),(14,15),(15,16),(15,20),(16,18),(16,19),(17,7),(17,20),(18,8),(19,8),(20,18),(20,19)],21)
=> ? = 10 + 1
{{1},{2,3},{4},{5,6},{7}}
=> {{1,3,6},{2},{4},{5},{7}}
=> [3,2,6,4,5,1,7] => ?
=> ? = 8 + 1
{{1},{2,3},{4},{5},{6},{7}}
=> {{1,3},{2},{4},{5},{6},{7}}
=> [3,2,1,4,5,6,7] => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ? = 7 + 1
{{1},{2,4,5,6},{3},{7}}
=> {{1},{2,4,5,6},{3},{7}}
=> [1,4,3,5,6,2,7] => ?
=> ? = 9 + 1
{{1},{2,4,5},{3,6},{7}}
=> {{1,6},{2,4,5},{3},{7}}
=> [6,4,3,5,2,1,7] => ?
=> ? = 11 + 1
{{1},{2,4,5},{3},{6},{7}}
=> {{1},{2,4,5},{3},{6},{7}}
=> [1,4,3,5,2,6,7] => ?
=> ? = 8 + 1
{{1},{2,4,6},{3,5},{7}}
=> {{1,5},{2,4,6},{3},{7}}
=> [5,4,3,6,1,2,7] => ([(0,3),(0,4),(0,5),(0,6),(1,11),(1,27),(2,7),(2,9),(2,10),(3,13),(3,14),(3,16),(4,12),(4,13),(4,15),(5,2),(5,15),(5,16),(5,17),(6,1),(6,12),(6,14),(6,17),(7,22),(7,23),(9,18),(9,22),(9,28),(10,18),(10,23),(10,28),(11,18),(11,28),(12,19),(12,27),(13,19),(13,21),(14,19),(14,20),(14,27),(15,10),(15,21),(15,27),(16,7),(16,20),(16,21),(17,9),(17,11),(17,20),(17,27),(18,26),(19,24),(20,22),(20,24),(20,28),(21,23),(21,24),(22,25),(22,26),(23,25),(23,26),(24,25),(25,8),(26,8),(27,24),(27,28),(28,25),(28,26)],29)
=> ? = 11 + 1
{{1},{2,4},{3,5},{6},{7}}
=> {{1,5},{2,4},{3},{6},{7}}
=> [5,4,3,2,1,6,7] => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ? = 8 + 1
{{1},{2,4,6},{3},{5},{7}}
=> {{1,6},{2,4},{3},{5},{7}}
=> [6,4,3,2,5,1,7] => ?
=> ? = 9 + 1
{{1},{2,4},{3,6},{5},{7}}
=> {{1},{2,4},{3,6},{5},{7}}
=> [1,4,6,2,5,3,7] => ?
=> ? = 9 + 1
{{1},{2,4},{3},{5},{6},{7}}
=> {{1},{2,4},{3},{5},{6},{7}}
=> [1,4,3,2,5,6,7] => ([(0,4),(0,5),(0,6),(1,16),(2,8),(2,9),(3,2),(3,11),(3,12),(4,10),(4,13),(5,3),(5,13),(5,14),(6,1),(6,10),(6,14),(8,19),(9,19),(9,20),(10,15),(10,16),(11,9),(11,17),(11,18),(12,8),(12,17),(13,12),(13,15),(14,11),(14,15),(14,16),(15,17),(15,18),(16,18),(17,19),(17,20),(18,20),(19,7),(20,7)],21)
=> ? = 8 + 1
{{1},{2,5},{3,4,6},{7}}
=> {{1,4},{2,5,6},{3},{7}}
=> [4,5,3,1,6,2,7] => ?
=> ? = 9 + 1
{{1},{2,5},{3,4},{6},{7}}
=> {{1,4},{2,5},{3},{6},{7}}
=> [4,5,3,1,2,6,7] => ([(0,2),(0,3),(0,4),(0,5),(1,7),(1,8),(1,9),(2,11),(2,14),(2,16),(3,11),(3,13),(3,15),(4,12),(4,15),(4,16),(5,1),(5,12),(5,13),(5,14),(7,22),(7,23),(8,21),(8,22),(9,21),(9,23),(10,24),(11,17),(11,18),(12,7),(12,19),(12,20),(13,8),(13,17),(13,19),(14,9),(14,17),(14,20),(15,10),(15,18),(15,19),(16,10),(16,18),(16,20),(17,21),(17,25),(18,24),(18,25),(19,22),(19,24),(19,25),(20,23),(20,24),(20,25),(21,26),(22,26),(22,27),(23,26),(23,27),(24,27),(25,26),(25,27),(26,6),(27,6)],28)
=> ? = 11 + 1
{{1},{2,6},{3,4,5},{7}}
=> {{1,4,5},{2,6},{3},{7}}
=> [4,6,3,5,1,2,7] => ?
=> ? = 13 + 1
{{1},{2},{3,4,5,6},{7}}
=> {{1,4,5,6},{2},{3},{7}}
=> [4,2,3,5,6,1,7] => ?
=> ? = 9 + 1
{{1},{2},{3,4,5},{6},{7}}
=> {{1,4,5},{2},{3},{6},{7}}
=> [4,2,3,5,1,6,7] => ?
=> ? = 8 + 1
{{1},{2,6},{3,4},{5},{7}}
=> {{1,4},{2},{3,6},{5},{7}}
=> [4,2,6,1,5,3,7] => ?
=> ? = 12 + 1
{{1},{2},{3,4,6},{5},{7}}
=> {{1,4},{2,6},{3},{5},{7}}
=> [4,6,3,1,5,2,7] => ?
=> ? = 9 + 1
{{1},{2},{3,4},{5},{6},{7}}
=> {{1,4},{2},{3},{5},{6},{7}}
=> [4,2,3,1,5,6,7] => ([(0,1),(0,2),(0,3),(0,6),(1,14),(1,18),(2,10),(2,13),(2,18),(3,10),(3,12),(3,18),(4,5),(4,15),(4,16),(4,17),(5,7),(5,8),(5,9),(6,4),(6,12),(6,13),(6,14),(7,21),(8,21),(8,22),(9,21),(9,22),(10,23),(12,16),(12,23),(12,24),(13,17),(13,23),(13,24),(14,15),(14,24),(15,7),(15,19),(16,8),(16,19),(16,20),(17,9),(17,19),(17,20),(18,23),(18,24),(19,21),(19,22),(20,22),(21,11),(22,11),(23,20),(24,19),(24,20)],25)
=> ? = 7 + 1
{{1},{2,5,6},{3},{4},{7}}
=> {{1},{2},{3,5,6},{4},{7}}
=> [1,2,5,4,6,3,7] => ?
=> ? = 9 + 1
{{1},{2,5},{3,6},{4},{7}}
=> {{1},{2,6},{3,5},{4},{7}}
=> [1,6,5,4,3,2,7] => ([(0,4),(0,5),(0,6),(1,18),(2,1),(2,8),(2,9),(3,2),(3,11),(3,12),(4,10),(4,14),(5,10),(5,13),(6,3),(6,13),(6,14),(8,17),(8,18),(9,17),(9,18),(10,15),(11,8),(11,16),(12,9),(12,16),(13,11),(13,15),(14,12),(14,15),(15,16),(16,17),(17,7),(18,7)],19)
=> ? = 9 + 1
{{1},{2,5},{3},{4},{6},{7}}
=> {{1},{2},{3,5},{4},{6},{7}}
=> [1,2,5,4,3,6,7] => ([(0,4),(0,5),(0,6),(1,17),(2,10),(2,12),(3,9),(3,11),(4,3),(4,13),(4,15),(5,2),(5,14),(5,15),(6,1),(6,13),(6,14),(8,20),(9,18),(9,22),(10,19),(10,22),(11,8),(11,18),(12,8),(12,19),(13,9),(13,16),(13,17),(14,10),(14,16),(14,17),(15,11),(15,12),(15,16),(16,18),(16,19),(16,22),(17,22),(18,20),(18,21),(19,20),(19,21),(20,7),(21,7),(22,21)],23)
=> ? = 9 + 1
{{1},{2,6},{3,5},{4},{7}}
=> {{1},{2,5},{3,6},{4},{7}}
=> [1,5,6,4,2,3,7] => ?
=> ? = 13 + 1
{{1},{2},{3,5,6},{4},{7}}
=> {{1},{2,5,6},{3},{4},{7}}
=> [1,5,3,4,6,2,7] => ?
=> ? = 8 + 1
{{1},{2},{3,5},{4,6},{7}}
=> {{1,6},{2,5},{3},{4},{7}}
=> [6,5,3,4,2,1,7] => ([(0,1),(0,4),(0,5),(0,6),(1,9),(1,19),(2,11),(2,15),(2,23),(3,10),(3,14),(3,23),(4,3),(4,12),(4,16),(4,19),(5,2),(5,13),(5,16),(5,19),(6,9),(6,12),(6,13),(7,24),(9,22),(10,20),(10,26),(11,21),(11,26),(12,10),(12,18),(12,22),(13,11),(13,18),(13,22),(14,7),(14,17),(14,20),(15,7),(15,17),(15,21),(16,14),(16,15),(16,18),(16,23),(17,24),(17,25),(18,20),(18,21),(18,26),(19,22),(19,23),(20,24),(20,25),(21,24),(21,25),(22,26),(23,17),(23,26),(24,8),(25,8),(26,25)],27)
=> ? = 8 + 1
{{1},{2},{3},{4},{5},{6},{7}}
=> {{1},{2},{3},{4},{5},{6},{7}}
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
Description
The dimension of the reduced incidence algebra of a poset.
The reduced incidence algebra of a poset is the subalgebra of the incidence algebra consisting of the elements which assign the same value to any two intervals that are isomorphic to each other as posets.
Thus, this statistic returns the number of non-isomorphic intervals of the poset.
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