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Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St001879
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Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[3,2,1] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3
[5,2,3,4,1] => [5,2,4,3,1] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[5,2,4,3,1] => [5,2,4,3,1] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[5,3,2,4,1] => [5,3,2,4,1] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[5,3,4,2,1] => [5,3,4,2,1] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[5,4,2,3,1] => [5,4,2,3,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[6,2,3,4,5,1] => [6,2,5,4,3,1] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 8
[6,2,3,5,4,1] => [6,2,5,4,3,1] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 8
[6,2,4,3,5,1] => [6,2,5,4,3,1] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 8
[6,2,4,5,3,1] => [6,2,5,4,3,1] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 8
[6,2,5,3,4,1] => [6,2,5,4,3,1] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 8
[6,2,5,4,3,1] => [6,2,5,4,3,1] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 8
[6,3,2,4,5,1] => [6,3,2,5,4,1] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 8
[6,3,2,5,4,1] => [6,3,2,5,4,1] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 8
[6,3,4,2,5,1] => [6,3,5,2,4,1] => [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 7
[6,3,4,5,2,1] => [6,3,5,4,2,1] => [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 7
[6,3,5,2,4,1] => [6,3,5,2,4,1] => [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 7
[6,3,5,4,2,1] => [6,3,5,4,2,1] => [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 7
[6,4,2,3,5,1] => [6,4,2,5,3,1] => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 7
[6,4,2,5,3,1] => [6,4,2,5,3,1] => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 7
[6,4,3,2,5,1] => [6,4,3,2,5,1] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 8
[6,4,3,5,2,1] => [6,4,3,5,2,1] => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 7
[6,4,5,3,2,1] => [6,4,5,3,2,1] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
[6,5,2,3,4,1] => [6,5,2,4,3,1] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 7
[6,5,2,4,3,1] => [6,5,2,4,3,1] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 7
[6,5,3,2,4,1] => [6,5,3,2,4,1] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 7
[6,5,3,4,2,1] => [6,5,3,4,2,1] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
[6,5,4,2,3,1] => [6,5,4,2,3,1] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[7,2,3,4,5,6,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 10
[7,2,3,4,6,5,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 10
[7,2,3,5,4,6,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 10
[7,2,3,5,6,4,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 10
[7,2,3,6,4,5,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 10
[7,2,3,6,5,4,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 10
[7,2,4,3,5,6,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 10
[7,2,4,3,6,5,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 10
[7,2,4,5,3,6,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 10
[7,2,4,5,6,3,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 10
[7,2,4,6,3,5,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 10
[7,2,4,6,5,3,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 10
[7,2,5,3,4,6,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 10
[7,2,5,3,6,4,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 10
[7,2,5,4,3,6,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 10
[7,2,5,4,6,3,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 10
[7,2,5,6,3,4,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 10
[7,2,5,6,4,3,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 10
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: St000081
Values
[3,2,1] => ([],3)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[4,3,2,1] => ([],4)
=> ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7 = 6 + 1
[5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7 = 6 + 1
[5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7 = 6 + 1
[5,3,4,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[5,4,2,3,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[5,4,3,2,1] => ([],5)
=> ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 4 + 1
[6,2,3,4,5,1] => ([(2,3),(3,5),(5,4)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 8 + 1
[6,2,3,5,4,1] => ([(2,3),(3,4),(3,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 9 = 8 + 1
[6,2,4,3,5,1] => ([(2,3),(2,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 8 + 1
[6,2,4,5,3,1] => ([(2,3),(2,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 8 + 1
[6,2,5,3,4,1] => ([(2,3),(2,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 8 + 1
[6,2,5,4,3,1] => ([(2,3),(2,4),(2,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 9 = 8 + 1
[6,3,2,4,5,1] => ([(2,5),(3,5),(5,4)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 9 = 8 + 1
[6,3,2,5,4,1] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 8 + 1
[6,3,4,2,5,1] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7 + 1
[6,3,4,5,2,1] => ([(3,4),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 8 = 7 + 1
[6,3,5,2,4,1] => ([(2,5),(3,4),(3,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7 + 1
[6,3,5,4,2,1] => ([(3,4),(3,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 8 = 7 + 1
[6,4,2,3,5,1] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7 + 1
[6,4,2,5,3,1] => ([(2,5),(3,4),(3,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7 + 1
[6,4,3,2,5,1] => ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 9 = 8 + 1
[6,4,3,5,2,1] => ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 8 = 7 + 1
[6,4,5,3,2,1] => ([(4,5)],6)
=> ([(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[6,5,2,3,4,1] => ([(3,4),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 8 = 7 + 1
[6,5,2,4,3,1] => ([(3,4),(3,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 8 = 7 + 1
[6,5,3,2,4,1] => ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 8 = 7 + 1
[6,5,3,4,2,1] => ([(4,5)],6)
=> ([(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[6,5,4,2,3,1] => ([(4,5)],6)
=> ([(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[6,5,4,3,2,1] => ([],6)
=> ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6 = 5 + 1
[7,2,3,4,5,6,1] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,2,3,4,6,5,1] => ([(2,5),(5,6),(6,3),(6,4)],7)
=> ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,2,3,5,4,6,1] => ([(2,3),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,2,3,5,6,4,1] => ([(2,6),(5,4),(6,3),(6,5)],7)
=> ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,2,3,6,4,5,1] => ([(2,6),(5,4),(6,3),(6,5)],7)
=> ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,2,3,6,5,4,1] => ([(2,6),(6,3),(6,4),(6,5)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,2,4,3,5,6,1] => ([(2,3),(2,4),(3,6),(4,6),(6,5)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,2,4,3,6,5,1] => ([(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,2,4,5,3,6,1] => ([(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,2,4,5,6,3,1] => ([(2,4),(2,6),(5,3),(6,5)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,2,4,6,3,5,1] => ([(2,3),(2,4),(3,6),(4,5),(4,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,2,4,6,5,3,1] => ([(2,5),(2,6),(6,3),(6,4)],7)
=> ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,2,5,3,4,6,1] => ([(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,2,5,3,6,4,1] => ([(2,3),(2,4),(3,6),(4,5),(4,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,2,5,4,3,6,1] => ([(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,2,5,4,6,3,1] => ([(2,3),(2,4),(2,5),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,2,5,6,3,4,1] => ([(2,5),(2,6),(5,4),(6,3)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,2,5,6,4,3,1] => ([(2,4),(2,5),(2,6),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,2,6,3,4,5,1] => ([(2,4),(2,6),(5,3),(6,5)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,2,6,3,5,4,1] => ([(2,5),(2,6),(6,3),(6,4)],7)
=> ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,2,6,4,3,5,1] => ([(2,3),(2,4),(2,5),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,2,6,4,5,3,1] => ([(2,4),(2,5),(2,6),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,2,6,5,3,4,1] => ([(2,4),(2,5),(2,6),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,2,6,5,4,3,1] => ([(2,3),(2,4),(2,5),(2,6)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,3,2,4,5,6,1] => ([(2,6),(3,6),(4,5),(6,4)],7)
=> ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,3,2,4,6,5,1] => ([(2,6),(3,6),(6,4),(6,5)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,3,2,5,4,6,1] => ([(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,3,2,5,6,4,1] => ([(2,5),(2,6),(3,5),(3,6),(6,4)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,3,2,6,4,5,1] => ([(2,5),(2,6),(3,5),(3,6),(6,4)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,3,2,6,5,4,1] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 10 + 1
[7,3,4,2,5,6,1] => ([(2,6),(3,4),(4,6),(6,5)],7)
=> ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 9 + 1
[7,3,4,2,6,5,1] => ([(2,5),(2,6),(3,4),(4,5),(4,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 9 + 1
[7,3,4,5,2,6,1] => ([(2,6),(3,4),(4,5),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 9 + 1
[7,3,4,5,6,2,1] => ([(3,4),(4,6),(6,5)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 9 + 1
[7,3,4,6,2,5,1] => ([(2,6),(3,4),(4,5),(4,6)],7)
=> ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 9 + 1
[7,3,4,6,5,2,1] => ([(3,4),(4,5),(4,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 9 + 1
[7,3,5,2,4,6,1] => ([(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 9 + 1
[7,3,5,2,6,4,1] => ([(2,5),(2,6),(3,4),(3,5),(4,6)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 9 + 1
[7,3,5,4,2,6,1] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 9 + 1
[7,3,5,4,6,2,1] => ([(3,4),(3,5),(4,6),(5,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 9 + 1
[7,3,5,6,2,4,1] => ([(2,6),(3,4),(3,6),(4,5)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 9 + 1
Description
The number of edges of a graph.
Matching statistic: St001846
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001846: Lattices ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 67%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001846: Lattices ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 67%
Values
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[4,2,3,1] => [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
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 3
[5,2,3,4,1] => [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
[5,2,4,3,1] => [1,4,2,3,5] => ([(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
[5,3,2,4,1] => [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
[5,3,4,2,1] => [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
[5,4,2,3,1] => [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
[5,4,3,2,1] => [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
[6,2,3,4,5,1] => [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)
=> ? = 8
[6,2,3,5,4,1] => [1,5,4,2,3,6] => ([(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
[6,2,4,3,5,1] => [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
[6,2,4,5,3,1] => [1,5,3,2,4,6] => ([(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)
=> ? = 8
[6,2,5,3,4,1] => [1,5,2,4,3,6] => ([(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
[6,2,5,4,3,1] => [1,5,2,3,4,6] => ([(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
[6,3,2,4,5,1] => [1,4,5,3,2,6] => ([(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
[6,3,2,5,4,1] => [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)
=> ? = 8
[6,3,4,2,5,1] => [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
[6,3,4,5,2,1] => [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
[6,3,5,2,4,1] => [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 7
[6,3,5,4,2,1] => [1,4,2,3,5,6] => ([(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
[6,4,2,3,5,1] => [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)
=> ? = 7
[6,4,2,5,3,1] => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 7
[6,4,3,2,5,1] => [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
[6,4,3,5,2,1] => [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
[6,4,5,3,2,1] => [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
[6,5,2,3,4,1] => [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
[6,5,2,4,3,1] => [1,2,5,3,4,6] => ([(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
[6,5,3,2,4,1] => [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
[6,5,3,4,2,1] => [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
[6,5,4,2,3,1] => [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
[6,5,4,3,2,1] => [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
[7,2,3,4,5,6,1] => [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
[7,2,3,4,6,5,1] => [1,6,5,4,2,3,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ?
=> ? = 10
[7,2,3,5,4,6,1] => [1,6,5,3,4,2,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ?
=> ? = 10
[7,2,3,5,6,4,1] => [1,6,5,3,2,4,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 10
[7,2,3,6,4,5,1] => [1,6,5,2,4,3,7] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ?
=> ? = 10
[7,2,3,6,5,4,1] => [1,6,5,2,3,4,7] => ([(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
[7,2,4,3,5,6,1] => [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)
=> ?
=> ? = 10
[7,2,4,3,6,5,1] => [1,6,4,5,2,3,7] => ([(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)
=> ? = 10
[7,2,4,5,3,6,1] => [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)
=> ?
=> ? = 10
[7,2,4,5,6,3,1] => [1,6,4,3,2,5,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,7),(1,14),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,11),(6,12),(6,13),(7,3),(7,4),(7,5),(7,6),(8,17),(8,18),(9,15),(9,18),(10,16),(10,18),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,2),(15,19),(16,19),(17,19),(18,1),(18,19),(19,14)],20)
=> ? = 10
[7,2,4,6,3,5,1] => [1,6,4,2,5,3,7] => ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(0,7),(2,9),(2,12),(3,8),(3,14),(4,11),(4,13),(5,8),(5,10),(6,4),(6,10),(6,14),(7,3),(7,5),(7,6),(8,16),(9,17),(10,13),(10,16),(11,12),(11,15),(12,17),(13,15),(14,2),(14,11),(14,16),(15,17),(16,9),(16,15),(17,1)],18)
=> ? = 10
[7,2,4,6,5,3,1] => [1,6,4,2,3,5,7] => ([(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)
=> ? = 10
[7,2,5,3,4,6,1] => [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
[7,2,5,3,6,4,1] => [1,6,3,5,2,4,7] => ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(0,7),(2,9),(2,12),(3,8),(3,14),(4,11),(4,13),(5,8),(5,10),(6,4),(6,10),(6,14),(7,3),(7,5),(7,6),(8,16),(9,17),(10,13),(10,16),(11,12),(11,15),(12,17),(13,15),(14,2),(14,11),(14,16),(15,17),(16,9),(16,15),(17,1)],18)
=> ? = 10
[7,2,5,4,3,6,1] => [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
[7,2,5,4,6,3,1] => [1,6,3,4,2,5,7] => ([(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)
=> ? = 10
[7,2,5,6,3,4,1] => [1,6,3,2,5,4,7] => ([(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)
=> ? = 10
[7,2,5,6,4,3,1] => [1,6,3,2,4,5,7] => ([(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)
=> ? = 10
[7,2,6,3,4,5,1] => [1,6,2,5,4,3,7] => ([(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)
=> ? = 10
[7,2,6,3,5,4,1] => [1,6,2,5,3,4,7] => ([(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
[7,2,6,4,3,5,1] => [1,6,2,4,5,3,7] => ([(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
[7,2,6,4,5,3,1] => [1,6,2,4,3,5,7] => ([(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)
=> ? = 10
[7,2,6,5,3,4,1] => [1,6,2,3,5,4,7] => ([(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
[7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(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
[7,3,2,4,5,6,1] => [1,5,6,4,3,2,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ?
=> ? = 10
[7,3,2,4,6,5,1] => [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)
=> ? = 10
[7,3,2,5,4,6,1] => [1,5,6,3,4,2,7] => ([(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)
=> ? = 10
[7,3,2,5,6,4,1] => [1,5,6,3,2,4,7] => ([(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)
=> ? = 10
[7,3,2,6,4,5,1] => [1,5,6,2,4,3,7] => ([(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)
=> ? = 10
[7,3,2,6,5,4,1] => [1,5,6,2,3,4,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=> ([(0,7),(2,10),(3,9),(4,3),(4,8),(5,6),(5,8),(6,2),(6,11),(7,4),(7,5),(8,9),(8,11),(9,12),(10,13),(11,10),(11,12),(12,13),(13,1)],14)
=> ? = 10
[7,3,4,2,5,6,1] => [1,5,4,6,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 9
[7,3,4,2,6,5,1] => [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
[7,3,4,5,2,6,1] => [1,5,4,3,6,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,7),(1,14),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,11),(6,12),(6,13),(7,3),(7,4),(7,5),(7,6),(8,17),(8,18),(9,15),(9,18),(10,16),(10,18),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,2),(15,19),(16,19),(17,19),(18,1),(18,19),(19,14)],20)
=> ? = 9
[7,3,4,5,6,2,1] => [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)
=> ? = 9
[7,5,6,4,3,2,1] => [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
[7,6,5,3,4,2,1] => [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
[7,6,5,4,3,2,1] => [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
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 67%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 67%
Values
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
[4,2,3,1] => [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
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[5,2,3,4,1] => [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
[5,2,4,3,1] => [1,4,2,3,5] => ([(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
[5,3,2,4,1] => [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
[5,3,4,2,1] => [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
[5,4,2,3,1] => [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
[5,4,3,2,1] => [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
[6,2,3,4,5,1] => [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)
=> ? = 8 + 2
[6,2,3,5,4,1] => [1,5,4,2,3,6] => ([(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
[6,2,4,3,5,1] => [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
[6,2,4,5,3,1] => [1,5,3,2,4,6] => ([(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)
=> ? = 8 + 2
[6,2,5,3,4,1] => [1,5,2,4,3,6] => ([(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
[6,2,5,4,3,1] => [1,5,2,3,4,6] => ([(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
[6,3,2,4,5,1] => [1,4,5,3,2,6] => ([(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
[6,3,2,5,4,1] => [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)
=> ? = 8 + 2
[6,3,4,2,5,1] => [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
[6,3,4,5,2,1] => [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
[6,3,5,2,4,1] => [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 7 + 2
[6,3,5,4,2,1] => [1,4,2,3,5,6] => ([(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
[6,4,2,3,5,1] => [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)
=> ? = 7 + 2
[6,4,2,5,3,1] => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 7 + 2
[6,4,3,2,5,1] => [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
[6,4,3,5,2,1] => [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
[6,4,5,3,2,1] => [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
[6,5,2,3,4,1] => [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
[6,5,2,4,3,1] => [1,2,5,3,4,6] => ([(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
[6,5,3,2,4,1] => [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
[6,5,3,4,2,1] => [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
[6,5,4,2,3,1] => [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
[6,5,4,3,2,1] => [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
[7,2,3,4,5,6,1] => [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 + 2
[7,2,3,4,6,5,1] => [1,6,5,4,2,3,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ?
=> ? = 10 + 2
[7,2,3,5,4,6,1] => [1,6,5,3,4,2,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ?
=> ? = 10 + 2
[7,2,3,5,6,4,1] => [1,6,5,3,2,4,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 10 + 2
[7,2,3,6,4,5,1] => [1,6,5,2,4,3,7] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ?
=> ? = 10 + 2
[7,2,3,6,5,4,1] => [1,6,5,2,3,4,7] => ([(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
[7,2,4,3,5,6,1] => [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)
=> ?
=> ? = 10 + 2
[7,2,4,3,6,5,1] => [1,6,4,5,2,3,7] => ([(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)
=> ? = 10 + 2
[7,2,4,5,3,6,1] => [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)
=> ?
=> ? = 10 + 2
[7,2,4,5,6,3,1] => [1,6,4,3,2,5,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,7),(1,14),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,11),(6,12),(6,13),(7,3),(7,4),(7,5),(7,6),(8,17),(8,18),(9,15),(9,18),(10,16),(10,18),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,2),(15,19),(16,19),(17,19),(18,1),(18,19),(19,14)],20)
=> ? = 10 + 2
[7,2,4,6,3,5,1] => [1,6,4,2,5,3,7] => ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(0,7),(2,9),(2,12),(3,8),(3,14),(4,11),(4,13),(5,8),(5,10),(6,4),(6,10),(6,14),(7,3),(7,5),(7,6),(8,16),(9,17),(10,13),(10,16),(11,12),(11,15),(12,17),(13,15),(14,2),(14,11),(14,16),(15,17),(16,9),(16,15),(17,1)],18)
=> ? = 10 + 2
[7,2,4,6,5,3,1] => [1,6,4,2,3,5,7] => ([(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)
=> ? = 10 + 2
[7,2,5,3,4,6,1] => [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 + 2
[7,2,5,3,6,4,1] => [1,6,3,5,2,4,7] => ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(0,7),(2,9),(2,12),(3,8),(3,14),(4,11),(4,13),(5,8),(5,10),(6,4),(6,10),(6,14),(7,3),(7,5),(7,6),(8,16),(9,17),(10,13),(10,16),(11,12),(11,15),(12,17),(13,15),(14,2),(14,11),(14,16),(15,17),(16,9),(16,15),(17,1)],18)
=> ? = 10 + 2
[7,2,5,4,3,6,1] => [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
[7,2,5,4,6,3,1] => [1,6,3,4,2,5,7] => ([(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)
=> ? = 10 + 2
[7,2,5,6,3,4,1] => [1,6,3,2,5,4,7] => ([(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)
=> ? = 10 + 2
[7,2,5,6,4,3,1] => [1,6,3,2,4,5,7] => ([(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)
=> ? = 10 + 2
[7,2,6,3,4,5,1] => [1,6,2,5,4,3,7] => ([(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)
=> ? = 10 + 2
[7,2,6,3,5,4,1] => [1,6,2,5,3,4,7] => ([(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
[7,2,6,4,3,5,1] => [1,6,2,4,5,3,7] => ([(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
[7,2,6,4,5,3,1] => [1,6,2,4,3,5,7] => ([(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)
=> ? = 10 + 2
[7,2,6,5,3,4,1] => [1,6,2,3,5,4,7] => ([(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
[7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(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
[7,3,2,4,5,6,1] => [1,5,6,4,3,2,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ?
=> ? = 10 + 2
[7,3,2,4,6,5,1] => [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)
=> ? = 10 + 2
[7,3,2,5,4,6,1] => [1,5,6,3,4,2,7] => ([(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)
=> ? = 10 + 2
[7,3,2,5,6,4,1] => [1,5,6,3,2,4,7] => ([(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)
=> ? = 10 + 2
[7,3,2,6,4,5,1] => [1,5,6,2,4,3,7] => ([(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)
=> ? = 10 + 2
[7,3,2,6,5,4,1] => [1,5,6,2,3,4,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=> ([(0,7),(2,10),(3,9),(4,3),(4,8),(5,6),(5,8),(6,2),(6,11),(7,4),(7,5),(8,9),(8,11),(9,12),(10,13),(11,10),(11,12),(12,13),(13,1)],14)
=> ? = 10 + 2
[7,3,4,2,5,6,1] => [1,5,4,6,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 9 + 2
[7,3,4,2,6,5,1] => [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
[7,3,4,5,2,6,1] => [1,5,4,3,6,2,7] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,7),(1,14),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,11),(6,12),(6,13),(7,3),(7,4),(7,5),(7,6),(8,17),(8,18),(9,15),(9,18),(10,16),(10,18),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,2),(15,19),(16,19),(17,19),(18,1),(18,19),(19,14)],20)
=> ? = 9 + 2
[7,3,4,5,6,2,1] => [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)
=> ? = 9 + 2
[7,5,6,4,3,2,1] => [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
[7,6,5,3,4,2,1] => [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
[7,6,5,4,3,2,1] => [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 $x, y \in L$.
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