Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St000672
(load all 4 compositions to match this statistic)
Mapping
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 1
[2,1] => 0
[1,2,3] => 2
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 0
[1,2,3,4] => 3
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 1
[2,1,3,4] => 2
[2,1,4,3] => 1
[2,3,1,4] => 2
[2,3,4,1] => 2
[2,4,1,3] => 2
[2,4,3,1] => 1
[3,1,2,4] => 2
[3,1,4,2] => 2
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 2
[3,4,2,1] => 1
[4,1,2,3] => 2
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 0
[1,2,3,4,5] => 4
[1,2,3,5,4] => 3
[1,2,4,3,5] => 3
[1,2,4,5,3] => 3
[1,2,5,3,4] => 3
[1,2,5,4,3] => 2
[1,3,2,4,5] => 3
[1,3,2,5,4] => 2
[1,3,4,2,5] => 3
[1,3,4,5,2] => 3
[1,3,5,2,4] => 3
[1,3,5,4,2] => 2
[1,4,2,3,5] => 3
[1,4,2,5,3] => 3
[1,4,3,2,5] => 2
[1,4,3,5,2] => 2
[1,4,5,2,3] => 3
Description
The number of minimal elements in Bruhat order not less than the permutation. The minimal elements in question are biGrassmannian, that is $$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$ for some $(r,a,b)$. This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Matching statistic: St001876
(load all 2 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00282: Posets —Dedekind-MacNeille completion⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 45%
Values
[1] => [1] => ([],1)
 => ([],1)
 => ? = 0
[1,2] => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,1] => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 0
[1,2,3] => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ? = 2
[1,3,2] => [2,3,1] => ([(1,2)],3)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ? = 1
[2,1,3] => [3,1,2] => ([(1,2)],3)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ? = 1
[2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0
[1,2,3,4] => [4,3,2,1] => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 3
[1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => ? = 2
[1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => ? = 2
[1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => ? = 2
[1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 2
[1,4,3,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ? = 1
[2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => ? = 2
[2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ? = 1
[2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => ? = 2
[2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ? = 2
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[2,4,3,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ? = 1
[3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 2
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ? = 1
[3,2,4,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ? = 1
[3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 2
[3,4,2,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ? = 2
[4,1,3,2] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ? = 1
[4,2,1,3] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ? = 1
[4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[4,3,1,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,2,3,4,5] => [5,4,3,2,1] => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 4
[1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => ? = 3
[1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => ? = 3
[1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => ? = 3
[1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ? = 3
[1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
 => ? = 2
[1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => ? = 3
[1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
 => ? = 2
[1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => ? = 3
[1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
 => ? = 3
[1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
 => ? = 3
[1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
 => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
 => ? = 2
[1,4,2,3,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ? = 3
[1,4,2,5,3] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
 => ? = 3
[1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
 => ? = 2
[1,4,3,5,2] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
 => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
 => ? = 2
[1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2)],8)
 => ? = 3
[1,4,5,3,2] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
 => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
 => ? = 2
[1,5,2,3,4] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ? = 3
[1,5,2,4,3] => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
 => ? = 2
[1,5,3,2,4] => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
 => ? = 2
[1,5,3,4,2] => [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
 => ? = 2
[1,5,4,2,3] => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
 => ? = 2
[1,5,4,3,2] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
 => ? = 1
[2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => ? = 3
[2,1,3,5,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
 => ? = 2
[2,1,4,3,5] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
 => ? = 2
[2,1,4,5,3] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
 => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7)
 => ? = 2
[3,5,1,4,2] => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => 3
[3,5,2,4,1] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,2,5,1,3] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => 3
[4,2,5,3,1] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,5,2,3,1] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 2
[4,5,3,1,2] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 2
[4,5,3,2,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 1
[5,2,4,1,3] => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[5,3,1,4,2] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[5,3,4,1,2] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 2
[5,3,4,2,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[5,4,2,3,1] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[5,4,3,1,2] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[3,6,2,5,1,4] => [4,1,5,2,6,3] => ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 3
[4,2,6,1,5,3] => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,4),(2,5),(3,4)],6)
 => ([(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)
 => 4
[4,6,2,5,3,1] => [1,3,5,2,6,4] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => 3
[4,6,3,1,5,2] => [2,5,1,3,6,4] => ([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => 3
[4,6,3,5,1,2] => [2,1,5,3,6,4] => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => 3
[4,6,3,5,2,1] => [1,2,5,3,6,4] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 2
[5,2,6,4,1,3] => [3,1,4,6,2,5] => ([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => 3
[5,3,1,6,4,2] => [2,4,6,1,3,5] => ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 3
[5,3,6,2,4,1] => [1,4,2,6,3,5] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => 3
[5,3,6,4,1,2] => [2,1,4,6,3,5] => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => 3
[5,3,6,4,2,1] => [1,2,4,6,3,5] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 2
[5,6,2,4,1,3] => [3,1,4,2,6,5] => ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
 => 3
[5,6,3,1,4,2] => [2,4,1,3,6,5] => ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
 => 3
[5,6,3,4,2,1] => [1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => 2
[5,6,4,2,3,1] => [1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 2
[5,6,4,3,1,2] => [2,1,3,4,6,5] => ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
 => 2
[5,6,4,3,2,1] => [1,2,3,4,6,5] => ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 1
[6,3,5,1,4,2] => [2,4,1,5,3,6] => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => 3
[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)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[6,4,2,5,1,3] => [3,1,5,2,4,6] => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => 3
[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)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[6,4,5,2,3,1] => [1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 2
[6,4,5,3,1,2] => [2,1,3,5,4,6] => ([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 2
[6,4,5,3,2,1] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 1
[6,5,2,4,1,3] => [3,1,4,2,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St001773
(load all 10 compositions to match this statistic)
Mapping
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001773: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 36%
Values
[1] => [1] => 1 = 0 + 1
[1,2] => [1,2] => 2 = 1 + 1
[2,1] => [2,1] => 1 = 0 + 1
[1,2,3] => [1,2,3] => 3 = 2 + 1
[1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,3] => [2,1,3] => 2 = 1 + 1
[2,3,1] => [2,3,1] => 2 = 1 + 1
[3,1,2] => [3,1,2] => 2 = 1 + 1
[3,2,1] => [3,2,1] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[1,3,2,4] => [1,3,2,4] => 3 = 2 + 1
[1,3,4,2] => [1,3,4,2] => 3 = 2 + 1
[1,4,2,3] => [1,4,2,3] => 3 = 2 + 1
[1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => 3 = 2 + 1
[2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[2,3,1,4] => [2,3,1,4] => 3 = 2 + 1
[2,3,4,1] => [2,3,4,1] => 3 = 2 + 1
[2,4,1,3] => [2,4,1,3] => 3 = 2 + 1
[2,4,3,1] => [2,4,3,1] => 2 = 1 + 1
[3,1,2,4] => [3,1,2,4] => 3 = 2 + 1
[3,1,4,2] => [3,1,4,2] => 3 = 2 + 1
[3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[3,2,4,1] => [3,2,4,1] => 2 = 1 + 1
[3,4,1,2] => [3,4,1,2] => 3 = 2 + 1
[3,4,2,1] => [3,4,2,1] => 2 = 1 + 1
[4,1,2,3] => [4,1,2,3] => 3 = 2 + 1
[4,1,3,2] => [4,1,3,2] => 2 = 1 + 1
[4,2,1,3] => [4,2,1,3] => 2 = 1 + 1
[4,2,3,1] => [4,2,3,1] => 2 = 1 + 1
[4,3,1,2] => [4,3,1,2] => 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ? = 4 + 1
[1,2,3,5,4] => [1,2,3,5,4] => ? = 3 + 1
[1,2,4,3,5] => [1,2,4,3,5] => ? = 3 + 1
[1,2,4,5,3] => [1,2,4,5,3] => ? = 3 + 1
[1,2,5,3,4] => [1,2,5,3,4] => ? = 3 + 1
[1,2,5,4,3] => [1,2,5,4,3] => ? = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => ? = 3 + 1
[1,3,2,5,4] => [1,3,2,5,4] => ? = 2 + 1
[1,3,4,2,5] => [1,3,4,2,5] => ? = 3 + 1
[1,3,4,5,2] => [1,3,4,5,2] => ? = 3 + 1
[1,3,5,2,4] => [1,3,5,2,4] => ? = 3 + 1
[1,3,5,4,2] => [1,3,5,4,2] => ? = 2 + 1
[1,4,2,3,5] => [1,4,2,3,5] => ? = 3 + 1
[1,4,2,5,3] => [1,4,2,5,3] => ? = 3 + 1
[1,4,3,2,5] => [1,4,3,2,5] => ? = 2 + 1
[1,4,3,5,2] => [1,4,3,5,2] => ? = 2 + 1
[1,4,5,2,3] => [1,4,5,2,3] => ? = 3 + 1
[1,4,5,3,2] => [1,4,5,3,2] => ? = 2 + 1
[1,5,2,3,4] => [1,5,2,3,4] => ? = 3 + 1
[1,5,2,4,3] => [1,5,2,4,3] => ? = 2 + 1
[1,5,3,2,4] => [1,5,3,2,4] => ? = 2 + 1
[1,5,3,4,2] => [1,5,3,4,2] => ? = 2 + 1
[1,5,4,2,3] => [1,5,4,2,3] => ? = 2 + 1
[1,5,4,3,2] => [1,5,4,3,2] => ? = 1 + 1
[2,1,3,4,5] => [2,1,3,4,5] => ? = 3 + 1
[2,1,3,5,4] => [2,1,3,5,4] => ? = 2 + 1
[2,1,4,3,5] => [2,1,4,3,5] => ? = 2 + 1
[2,1,4,5,3] => [2,1,4,5,3] => ? = 2 + 1
[2,1,5,3,4] => [2,1,5,3,4] => ? = 2 + 1
[2,1,5,4,3] => [2,1,5,4,3] => ? = 1 + 1
[2,3,1,4,5] => [2,3,1,4,5] => ? = 3 + 1
[2,3,1,5,4] => [2,3,1,5,4] => ? = 2 + 1
[2,3,4,1,5] => [2,3,4,1,5] => ? = 3 + 1
[2,3,4,5,1] => [2,3,4,5,1] => ? = 3 + 1
[2,3,5,1,4] => [2,3,5,1,4] => ? = 3 + 1
[2,3,5,4,1] => [2,3,5,4,1] => ? = 2 + 1
[2,4,1,3,5] => [2,4,1,3,5] => ? = 3 + 1
[2,4,1,5,3] => [2,4,1,5,3] => ? = 3 + 1
[2,4,3,1,5] => [2,4,3,1,5] => ? = 2 + 1
[2,4,3,5,1] => [2,4,3,5,1] => ? = 2 + 1
[2,4,5,1,3] => [2,4,5,1,3] => ? = 3 + 1
[2,4,5,3,1] => [2,4,5,3,1] => ? = 2 + 1
[2,5,1,3,4] => [2,5,1,3,4] => ? = 3 + 1
[2,5,1,4,3] => [2,5,1,4,3] => ? = 2 + 1
[2,5,3,1,4] => [2,5,3,1,4] => ? = 2 + 1
[2,5,3,4,1] => [2,5,3,4,1] => ? = 2 + 1
[2,5,4,1,3] => [2,5,4,1,3] => ? = 2 + 1
[2,5,4,3,1] => [2,5,4,3,1] => ? = 1 + 1
[3,1,2,4,5] => [3,1,2,4,5] => ? = 3 + 1
[3,1,2,5,4] => [3,1,2,5,4] => ? = 2 + 1
Description
The number of minimal elements in Bruhat order not less than the signed permutation. The minimal elements in question are biGrassmannian, that is both the element and its inverse have at most one descent. This is the size of the essential set of the signed permutation, see [1].
Matching statistic: St001626
(load all 2 compositions to match this statistic)
Mapping
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001626: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 55%
Values
[1] => [1] => ([],1)
 => ([(0,1)],2)
 => 2 = 0 + 2
[1,2] => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[2,1] => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[1,3,2] => [3,1,2] => ([(1,2)],3)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3 = 1 + 2
[2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3 = 1 + 2
[2,3,1] => [2,3,1] => ([(1,2)],3)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3 = 1 + 2
[3,1,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3 = 1 + 2
[3,2,1] => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 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
[1,2,4,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2 + 2
[1,3,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 4 = 2 + 2
[1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 2 + 2
[1,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 4 = 2 + 2
[1,4,3,2] => [4,3,1,2] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 1 + 2
[2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 4 = 2 + 2
[2,1,4,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
 => ? = 1 + 2
[2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 4 = 2 + 2
[2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2 + 2
[2,4,1,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 4 = 2 + 2
[2,4,3,1] => [4,2,3,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 1 + 2
[3,1,2,4] => [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 = 2 + 2
[3,1,4,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(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)
 => ? = 2 + 2
[3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ? = 1 + 2
[3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
 => ? = 1 + 2
[3,4,1,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 4 = 2 + 2
[3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 1 + 2
[4,1,2,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 4 = 2 + 2
[4,1,3,2] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 2
[4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 1 + 2
[4,2,3,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 2
[4,3,1,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => ? = 1 + 2
[4,3,2,1] => [4,3,2,1] => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 0 + 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
[1,2,3,5,4] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ([(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)
 => ? = 3 + 2
[1,2,4,3,5] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 3 + 2
[1,2,4,5,3] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
 => ? = 3 + 2
[1,2,5,3,4] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? = 3 + 2
[1,2,5,4,3] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
 => ? = 2 + 2
[1,3,2,4,5] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 3 + 2
[1,3,2,5,4] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
 => ? = 2 + 2
[1,3,4,2,5] => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ? = 3 + 2
[1,3,4,5,2] => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ? = 3 + 2
[1,3,5,2,4] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
 => ? = 3 + 2
[1,3,5,4,2] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,8),(2,10),(3,9),(3,11),(4,9),(4,12),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,15),(10,6),(10,13),(11,8),(11,15),(12,1),(12,10),(12,15),(13,14),(15,7),(15,13)],16)
 => ? = 2 + 2
[1,4,2,3,5] => [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)
 => ? = 3 + 2
[1,4,2,5,3] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => ([(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)
 => ? = 3 + 2
[1,4,3,2,5] => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
 => ? = 2 + 2
[1,4,3,5,2] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,1),(12,13),(13,8)],14)
 => ? = 2 + 2
[1,4,5,2,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? = 3 + 2
[1,4,5,3,2] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
 => ([(0,2),(0,3),(1,11),(2,1),(2,12),(3,4),(3,5),(3,12),(4,8),(4,10),(5,8),(5,9),(6,14),(7,14),(8,13),(9,6),(9,13),(10,7),(10,13),(11,6),(11,7),(12,9),(12,10),(12,11),(13,14)],15)
 => ? = 2 + 2
[1,5,2,3,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 3 + 2
[1,5,2,4,3] => [5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
 => ([(0,1),(0,2),(1,11),(2,3),(2,4),(2,11),(3,8),(3,10),(4,5),(4,9),(4,10),(5,6),(5,7),(6,13),(7,13),(8,12),(9,7),(9,12),(10,6),(10,12),(11,8),(11,9),(12,13)],14)
 => ? = 2 + 2
[1,5,3,2,4] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 2 + 2
[1,5,3,4,2] => [5,1,3,4,2] => ([(1,3),(1,4),(4,2)],5)
 => ([(0,1),(0,2),(1,11),(2,3),(2,4),(2,11),(3,8),(3,10),(4,5),(4,9),(4,10),(5,6),(5,7),(6,13),(7,13),(8,12),(9,7),(9,12),(10,6),(10,12),(11,8),(11,9),(12,13)],14)
 => ? = 2 + 2
[1,5,4,2,3] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
 => ? = 2 + 2
[1,5,4,3,2] => [5,4,3,1,2] => ([(3,4)],5)
 => ?
 => ? = 1 + 2
[2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 5 = 3 + 2
[2,1,3,5,4] => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
 => ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
 => ? = 2 + 2
[2,1,4,3,5] => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 2 + 2
[2,1,4,5,3] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
 => ? = 2 + 2
[2,1,5,3,4] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ? = 2 + 2
[2,1,5,4,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
 => ? = 1 + 2
[2,3,1,4,5] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 3 + 2
[2,3,1,5,4] => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
 => ? = 2 + 2
[2,3,4,1,5] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 3 + 2
[2,3,4,5,1] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(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)
 => ? = 3 + 2
[2,3,5,1,4] => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
 => ? = 3 + 2
[2,3,5,4,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
 => ? = 2 + 2
[3,1,2,4,5] => [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 = 3 + 2
[4,1,2,3,5] => [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 = 3 + 2
[5,1,2,3,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 5 = 3 + 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
Description
The number of maximal proper sublattices of a lattice.