Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St000957
(load all 4 compositions to match this statistic)
Mapping
St000957: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 2
[3,1,2] => 2
[3,2,1] => 2
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 2
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 3
[2,4,1,3] => 3
[2,4,3,1] => 3
[3,1,2,4] => 2
[3,1,4,2] => 3
[3,2,1,4] => 2
[3,2,4,1] => 3
[3,4,1,2] => 4
[3,4,2,1] => 3
[4,1,2,3] => 3
[4,1,3,2] => 3
[4,2,1,3] => 3
[4,2,3,1] => 4
[4,3,1,2] => 3
[4,3,2,1] => 3
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 2
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 2
[1,3,4,5,2] => 3
[1,3,5,2,4] => 3
[1,3,5,4,2] => 3
[1,4,2,3,5] => 2
[1,4,2,5,3] => 3
[1,4,3,2,5] => 2
[1,4,3,5,2] => 3
[1,4,5,2,3] => 4
[1,4,5,3,2] => 3
Description
The number of Bruhat lower covers of a permutation. This is, for a permutation $\pi$, the number of permutations $\tau$ with $\operatorname{inv}(\tau) = \operatorname{inv}(\pi) - 1$ such that $\tau*t = \pi$ for a transposition $t$. This is also the number of occurrences of the boxed pattern $21$: occurrences of the pattern $21$ such that any entry between the two matched entries is either larger or smaller than both of the matched entries.
Matching statistic: St000081
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => ([],2)
 => ([],2)
 => 0
[2,1] => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,2,3] => [3,2,1] => ([],3)
 => ([],3)
 => 0
[1,3,2] => [2,3,1] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
[2,1,3] => [3,1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
[2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
[3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 2
[1,2,3,4] => [4,3,2,1] => ([],4)
 => ([],4)
 => 0
[1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[1,4,3,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1
[2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 2
[2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
[2,4,3,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
[3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
[3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[3,2,4,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
[3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
[3,4,2,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,1,3,2] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
[4,2,1,3] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
[4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
[4,3,1,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
[1,2,3,4,5] => [5,4,3,2,1] => ([],5)
 => ([],5)
 => 0
[1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1
[1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1
[1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
 => ([(2,4),(3,4)],5)
 => 2
[1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 2
[1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 2
[1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1
[1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 2
[1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
 => ([(2,4),(3,4)],5)
 => 2
[1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 3
[1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 3
[1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 3
[1,4,2,3,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 2
[1,4,2,5,3] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 3
[1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 2
[1,4,3,5,2] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 3
[1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[1,4,5,3,2] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 3
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => ([(3,6),(4,5)],7)
 => ([(3,6),(4,5)],7)
 => ? = 2
[1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => ([(3,6),(4,5),(4,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ? = 3
[1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => ([(3,4),(3,5),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ? = 3
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => ([(3,6),(4,5),(4,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ? = 3
[1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => ([(3,4),(3,5),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ? = 3
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => ([(3,5),(3,6),(4,5),(4,6)],7)
 => ([(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 4
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => ([(3,6),(4,5),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ? = 3
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => ([(3,6),(4,5),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ? = 3
[1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => ([(3,4),(3,5),(4,6),(5,6)],7)
 => ([(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 4
[1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => ([(3,4),(4,6),(6,5)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ? = 3
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => ([(3,6),(4,5)],7)
 => ([(3,6),(4,5)],7)
 => ? = 2
[1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => ([(3,6),(4,5)],7)
 => ([(3,6),(4,5)],7)
 => ? = 2
[1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => ([(2,6),(3,4),(3,5)],7)
 => ([(2,3),(4,6),(5,6)],7)
 => ? = 3
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => ([(2,6),(3,6),(4,5)],7)
 => ([(2,3),(4,6),(5,6)],7)
 => ? = 3
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => ([(2,4),(3,5),(5,6)],7)
 => ([(2,3),(4,6),(5,6)],7)
 => ? = 3
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => ([(2,6),(3,4),(3,5)],7)
 => ([(2,3),(4,6),(5,6)],7)
 => ? = 3
[1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => ([(2,6),(3,4),(3,5),(3,6)],7)
 => ([(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 4
[1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => ([(2,4),(2,5),(2,6),(6,3)],7)
 => ([(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 4
[1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => ([(3,6),(4,5),(4,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ? = 3
[1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => ([(2,5),(2,6),(3,4),(3,6)],7)
 => ([(2,6),(3,5),(4,5),(4,6)],7)
 => ? = 4
[1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => ([(3,4),(3,5),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ? = 3
[1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => ([(2,4),(2,5),(2,6),(6,3)],7)
 => ([(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 4
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => ([(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 5
[1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => ([(2,5),(2,6),(6,3),(6,4)],7)
 => ([(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 4
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => ([(2,6),(3,6),(4,5),(4,6)],7)
 => ([(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 4
[1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => ([(2,5),(3,4),(3,6),(5,6)],7)
 => ([(2,6),(3,5),(4,5),(4,6)],7)
 => ? = 4
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => ([(2,6),(3,4),(3,5),(5,6)],7)
 => ([(2,6),(3,5),(4,5),(4,6)],7)
 => ? = 4
[1,2,4,7,5,6,3] => [3,6,5,7,4,2,1] => ([(2,3),(2,4),(2,5),(4,6),(5,6)],7)
 => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 5
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => ([(2,6),(3,4),(3,6),(6,5)],7)
 => ([(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 4
[1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => ([(2,4),(2,6),(5,3),(6,5)],7)
 => ([(2,6),(3,5),(4,5),(4,6)],7)
 => ? = 4
[1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => ([(2,6),(3,6),(4,5)],7)
 => ([(2,3),(4,6),(5,6)],7)
 => ? = 3
[1,2,5,3,6,4,7] => [7,4,6,3,5,2,1] => ([(3,6),(4,5),(4,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ? = 3
[1,2,5,3,6,7,4] => [4,7,6,3,5,2,1] => ([(2,6),(3,4),(3,5),(3,6)],7)
 => ([(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 4
[1,2,5,3,7,4,6] => [6,4,7,3,5,2,1] => ([(2,6),(3,5),(4,5),(4,6)],7)
 => ([(2,6),(3,5),(4,5),(4,6)],7)
 => ? = 4
[1,2,5,3,7,6,4] => [4,6,7,3,5,2,1] => ([(2,6),(3,4),(3,6),(4,5)],7)
 => ([(2,6),(3,5),(4,5),(4,6)],7)
 => ? = 4
[1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => ([(2,4),(3,5),(5,6)],7)
 => ([(2,3),(4,6),(5,6)],7)
 => ? = 3
[1,2,5,4,6,3,7] => [7,3,6,4,5,2,1] => ([(3,4),(3,5),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => ? = 3
[1,2,5,4,6,7,3] => [3,7,6,4,5,2,1] => ([(2,4),(2,5),(2,6),(6,3)],7)
 => ([(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 4
[1,2,5,4,7,3,6] => [6,3,7,4,5,2,1] => ([(2,6),(3,4),(3,6),(4,5)],7)
 => ([(2,6),(3,5),(4,5),(4,6)],7)
 => ? = 4
[1,2,5,4,7,6,3] => [3,6,7,4,5,2,1] => ([(2,5),(2,6),(5,4),(6,3)],7)
 => ([(2,6),(3,5),(4,5),(4,6)],7)
 => ? = 4
[1,2,5,6,3,4,7] => [7,4,3,6,5,2,1] => ([(3,5),(3,6),(4,5),(4,6)],7)
 => ([(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 4
[1,2,5,6,3,7,4] => [4,7,3,6,5,2,1] => ([(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 5
[1,2,5,6,4,7,3] => [3,7,4,6,5,2,1] => ([(2,5),(2,6),(6,3),(6,4)],7)
 => ([(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 4
[1,2,5,6,7,3,4] => [4,3,7,6,5,2,1] => ([(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)
 => ? = 6
[1,2,5,7,3,4,6] => [6,4,3,7,5,2,1] => ([(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 5
[1,2,5,7,3,6,4] => [4,6,3,7,5,2,1] => ([(2,5),(2,6),(3,4),(3,5),(4,6)],7)
 => ([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ? = 5
[1,2,5,7,4,3,6] => [6,3,4,7,5,2,1] => ([(2,6),(3,4),(4,5),(4,6)],7)
 => ([(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 4
[1,2,5,7,4,6,3] => [3,6,4,7,5,2,1] => ([(2,3),(2,4),(3,6),(4,5),(4,6)],7)
 => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 5
[1,2,5,7,6,3,4] => [4,3,6,7,5,2,1] => ([(2,5),(2,6),(3,5),(3,6),(6,4)],7)
 => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 5
[1,2,5,7,6,4,3] => [3,4,6,7,5,2,1] => ([(2,6),(5,4),(6,3),(6,5)],7)
 => ([(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 4
Description
The number of edges of a graph.
Matching statistic: St000327
(load all 2 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000327: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 70%
Values
[1,2] => [2,1] => ([],2)
 => 0
[2,1] => [1,2] => ([(0,1)],2)
 => 1
[1,2,3] => [3,2,1] => ([],3)
 => 0
[1,3,2] => [2,3,1] => ([(1,2)],3)
 => 1
[2,1,3] => [3,1,2] => ([(1,2)],3)
 => 1
[2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
 => 2
[3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
 => 2
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2
[1,2,3,4] => [4,3,2,1] => ([],4)
 => 0
[1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
 => 1
[1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
 => 1
[1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => 2
[1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 2
[1,4,3,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 2
[2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
 => 1
[2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 2
[2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => 2
[2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 3
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 3
[2,4,3,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => 3
[3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => 2
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => 3
[3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 2
[3,2,4,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 3
[3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
[3,4,2,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => 3
[4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,1,3,2] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 3
[4,2,1,3] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 3
[4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
[4,3,1,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 3
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3
[1,2,3,4,5] => [5,4,3,2,1] => ([],5)
 => 0
[1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
 => 1
[1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
 => 1
[1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
 => 2
[1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => 2
[1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => 2
[1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
 => 1
[1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => 2
[1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
 => 2
[1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => 3
[1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => 3
[1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
 => 3
[1,4,2,3,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => 2
[1,4,2,5,3] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => 3
[1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => 2
[1,4,3,5,2] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
 => 3
[1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[1,4,5,3,2] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
 => 3
[1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
 => ? = 0
[1,2,3,4,6,5] => [5,6,4,3,2,1] => ([(4,5)],6)
 => ? = 1
[1,2,3,5,4,6] => [6,4,5,3,2,1] => ([(4,5)],6)
 => ? = 1
[1,2,3,5,6,4] => [4,6,5,3,2,1] => ([(3,4),(3,5)],6)
 => ? = 2
[1,2,3,6,4,5] => [5,4,6,3,2,1] => ([(3,5),(4,5)],6)
 => ? = 2
[1,2,3,6,5,4] => [4,5,6,3,2,1] => ([(3,4),(4,5)],6)
 => ? = 2
[1,2,4,3,5,6] => [6,5,3,4,2,1] => ([(4,5)],6)
 => ? = 1
[1,2,4,3,6,5] => [5,6,3,4,2,1] => ([(2,5),(3,4)],6)
 => ? = 2
[1,2,4,5,3,6] => [6,3,5,4,2,1] => ([(3,4),(3,5)],6)
 => ? = 2
[1,2,4,5,6,3] => [3,6,5,4,2,1] => ([(2,3),(2,4),(2,5)],6)
 => ? = 3
[1,2,4,6,3,5] => [5,3,6,4,2,1] => ([(2,5),(3,4),(3,5)],6)
 => ? = 3
[1,2,4,6,5,3] => [3,5,6,4,2,1] => ([(2,3),(2,4),(4,5)],6)
 => ? = 3
[1,2,5,3,4,6] => [6,4,3,5,2,1] => ([(3,5),(4,5)],6)
 => ? = 2
[1,2,5,3,6,4] => [4,6,3,5,2,1] => ([(2,5),(3,4),(3,5)],6)
 => ? = 3
[1,2,5,4,3,6] => [6,3,4,5,2,1] => ([(3,4),(4,5)],6)
 => ? = 2
[1,2,5,4,6,3] => [3,6,4,5,2,1] => ([(2,3),(2,4),(4,5)],6)
 => ? = 3
[1,2,5,6,3,4] => [4,3,6,5,2,1] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 4
[1,2,5,6,4,3] => [3,4,6,5,2,1] => ([(2,3),(3,4),(3,5)],6)
 => ? = 3
[1,2,6,3,4,5] => [5,4,3,6,2,1] => ([(2,5),(3,5),(4,5)],6)
 => ? = 3
[1,2,6,3,5,4] => [4,5,3,6,2,1] => ([(2,5),(3,4),(4,5)],6)
 => ? = 3
[1,2,6,4,3,5] => [5,3,4,6,2,1] => ([(2,5),(3,4),(4,5)],6)
 => ? = 3
[1,2,6,4,5,3] => [3,5,4,6,2,1] => ([(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 4
[1,2,6,5,3,4] => [4,3,5,6,2,1] => ([(2,5),(3,5),(5,4)],6)
 => ? = 3
[1,2,6,5,4,3] => [3,4,5,6,2,1] => ([(2,3),(3,5),(5,4)],6)
 => ? = 3
[1,3,2,4,5,6] => [6,5,4,2,3,1] => ([(4,5)],6)
 => ? = 1
[1,3,2,4,6,5] => [5,6,4,2,3,1] => ([(2,5),(3,4)],6)
 => ? = 2
[1,3,2,5,4,6] => [6,4,5,2,3,1] => ([(2,5),(3,4)],6)
 => ? = 2
[1,3,2,5,6,4] => [4,6,5,2,3,1] => ([(1,5),(2,3),(2,4)],6)
 => ? = 3
[1,3,2,6,4,5] => [5,4,6,2,3,1] => ([(1,5),(2,5),(3,4)],6)
 => ? = 3
[1,3,2,6,5,4] => [4,5,6,2,3,1] => ([(1,3),(2,4),(4,5)],6)
 => ? = 3
[1,3,4,2,5,6] => [6,5,2,4,3,1] => ([(3,4),(3,5)],6)
 => ? = 2
[1,3,4,2,6,5] => [5,6,2,4,3,1] => ([(1,5),(2,3),(2,4)],6)
 => ? = 3
[1,3,4,5,2,6] => [6,2,5,4,3,1] => ([(2,3),(2,4),(2,5)],6)
 => ? = 3
[1,3,4,5,6,2] => [2,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5)],6)
 => ? = 4
[1,3,4,6,2,5] => [5,2,6,4,3,1] => ([(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 4
[1,3,4,6,5,2] => [2,5,6,4,3,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
 => ? = 4
[1,3,5,2,4,6] => [6,4,2,5,3,1] => ([(2,5),(3,4),(3,5)],6)
 => ? = 3
[1,3,5,2,6,4] => [4,6,2,5,3,1] => ([(1,4),(1,5),(2,3),(2,5)],6)
 => ? = 4
[1,3,5,4,2,6] => [6,2,4,5,3,1] => ([(2,3),(2,4),(4,5)],6)
 => ? = 3
[1,3,5,4,6,2] => [2,6,4,5,3,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
 => ? = 4
[1,3,5,6,2,4] => [4,2,6,5,3,1] => ([(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 5
[1,3,5,6,4,2] => [2,4,6,5,3,1] => ([(1,4),(1,5),(5,2),(5,3)],6)
 => ? = 4
[1,3,6,2,4,5] => [5,4,2,6,3,1] => ([(1,5),(2,5),(3,4),(3,5)],6)
 => ? = 4
[1,3,6,2,5,4] => [4,5,2,6,3,1] => ([(1,4),(2,3),(2,5),(4,5)],6)
 => ? = 4
[1,3,6,4,2,5] => [5,2,4,6,3,1] => ([(1,5),(2,3),(2,4),(4,5)],6)
 => ? = 4
[1,3,6,4,5,2] => [2,5,4,6,3,1] => ([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => ? = 5
[1,3,6,5,2,4] => [4,2,5,6,3,1] => ([(1,5),(2,3),(2,5),(5,4)],6)
 => ? = 4
[1,3,6,5,4,2] => [2,4,5,6,3,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
 => ? = 4
[1,4,2,3,5,6] => [6,5,3,2,4,1] => ([(3,5),(4,5)],6)
 => ? = 2
[1,4,2,3,6,5] => [5,6,3,2,4,1] => ([(1,5),(2,5),(3,4)],6)
 => ? = 3
Description
The number of cover relations in a poset. Equivalently, this is also the number of edges in the Hasse diagram [1].
Matching statistic: St001630
(load all 6 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 10%
Values
[1,2] => ([],2)
 => ([],1)
 => ? = 0
[2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 1
[1,2,3] => ([],3)
 => ([],1)
 => ? = 0
[1,3,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? = 1
[2,1,3] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? = 1
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,3,4] => ([],4)
 => ([],1)
 => ? = 0
[1,2,4,3] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 1
[1,3,2,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[2,1,3,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(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)
 => ? = 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(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)
 => ? = 3
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(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)
 => ? = 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],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),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 4
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(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)
 => ? = 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 4
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 3
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ? = 3
[1,2,3,4,5] => ([],5)
 => ([],1)
 => ? = 0
[1,2,3,5,4] => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 1
[1,2,4,3,5] => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,3,2,4,5] => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(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)
 => ? = 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(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)
 => ? = 3
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(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)
 => ? = 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(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),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 4
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 3
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(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)
 => ? = 3
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 4
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 3
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ? = 3
[2,1,3,4,5] => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ([(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)
 => ? = 3
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(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)
 => ? = 3
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(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)
 => ? = 3
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(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)
 => ? = 3
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(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)
 => ? = 4
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(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)
 => ? = 4
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 4
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ([(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)
 => ? = 3
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(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)
 => ? = 4
[3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,4,2,5,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,2,3,5,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[2,1,3,4,6,5] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,3,5,4,6] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,4,3,5,6] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,3,1,4,5,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,1,2,4,5,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,5,6,4,7] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,6,4,5,7] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,4,5,3,6,7] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001878
(load all 6 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 10%
Values
[1,2] => ([],2)
 => ([],1)
 => ? = 0
[2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 1
[1,2,3] => ([],3)
 => ([],1)
 => ? = 0
[1,3,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? = 1
[2,1,3] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? = 1
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,3,4] => ([],4)
 => ([],1)
 => ? = 0
[1,2,4,3] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 1
[1,3,2,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[2,1,3,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(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)
 => ? = 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(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)
 => ? = 3
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(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)
 => ? = 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],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),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 4
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(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)
 => ? = 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 4
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 3
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ? = 3
[1,2,3,4,5] => ([],5)
 => ([],1)
 => ? = 0
[1,2,3,5,4] => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 1
[1,2,4,3,5] => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,3,2,4,5] => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(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)
 => ? = 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(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)
 => ? = 3
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(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)
 => ? = 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(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),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 4
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 3
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(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)
 => ? = 3
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 4
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 3
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ? = 3
[2,1,3,4,5] => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ([(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)
 => ? = 3
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(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)
 => ? = 3
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(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)
 => ? = 3
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(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)
 => ? = 3
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(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)
 => ? = 4
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(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)
 => ? = 4
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 4
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ([(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)
 => ? = 3
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(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)
 => ? = 4
[3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,4,2,5,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,2,3,5,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[2,1,3,4,6,5] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,3,5,4,6] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,4,3,5,6] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,3,1,4,5,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,1,2,4,5,6] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,5,6,4,7] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,6,4,5,7] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,4,5,3,6,7] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001861
(load all 9 compositions to match this statistic)
Mapping
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001861: Signed permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Values
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [2,3,1] => 2
[3,1,2] => [3,1,2] => 2
[3,2,1] => [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,3,4,2] => 2
[1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [2,3,1,4] => 2
[2,3,4,1] => [2,3,4,1] => 3
[2,4,1,3] => [2,4,1,3] => 3
[2,4,3,1] => [2,4,3,1] => 3
[3,1,2,4] => [3,1,2,4] => 2
[3,1,4,2] => [3,1,4,2] => 3
[3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [3,2,4,1] => 3
[3,4,1,2] => [3,4,1,2] => 4
[3,4,2,1] => [3,4,2,1] => 3
[4,1,2,3] => [4,1,2,3] => 3
[4,1,3,2] => [4,1,3,2] => 3
[4,2,1,3] => [4,2,1,3] => 3
[4,2,3,1] => [4,2,3,1] => 4
[4,3,1,2] => [4,3,1,2] => 3
[4,3,2,1] => [4,3,2,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,4,5,3] => 2
[1,2,5,3,4] => [1,2,5,3,4] => 2
[1,2,5,4,3] => [1,2,5,4,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,3,4,2,5] => 2
[1,3,4,5,2] => [1,3,4,5,2] => 3
[1,3,5,2,4] => [1,3,5,2,4] => 3
[1,3,5,4,2] => [1,3,5,4,2] => 3
[1,4,2,3,5] => [1,4,2,3,5] => 2
[1,4,2,5,3] => [1,4,2,5,3] => 3
[1,4,3,2,5] => [1,4,3,2,5] => 2
[1,4,3,5,2] => [1,4,3,5,2] => 3
[1,4,5,2,3] => [1,4,5,2,3] => 4
[1,4,5,3,2] => [1,4,5,3,2] => 3
[2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[2,1,3,5,4] => [2,1,3,5,4] => ? = 2
[2,1,4,3,5] => [2,1,4,3,5] => ? = 2
[2,1,4,5,3] => [2,1,4,5,3] => ? = 3
[2,1,5,3,4] => [2,1,5,3,4] => ? = 3
[2,1,5,4,3] => [2,1,5,4,3] => ? = 3
[2,3,1,4,5] => [2,3,1,4,5] => ? = 2
[2,3,1,5,4] => [2,3,1,5,4] => ? = 3
[2,3,4,1,5] => [2,3,4,1,5] => ? = 3
[2,3,4,5,1] => [2,3,4,5,1] => ? = 4
[2,3,5,1,4] => [2,3,5,1,4] => ? = 4
[2,3,5,4,1] => [2,3,5,4,1] => ? = 4
[2,4,1,3,5] => [2,4,1,3,5] => ? = 3
[2,4,1,5,3] => [2,4,1,5,3] => ? = 4
[2,4,3,1,5] => [2,4,3,1,5] => ? = 3
[2,4,3,5,1] => [2,4,3,5,1] => ? = 4
[2,4,5,1,3] => [2,4,5,1,3] => ? = 5
[2,4,5,3,1] => [2,4,5,3,1] => ? = 4
[2,5,1,3,4] => [2,5,1,3,4] => ? = 4
[2,5,1,4,3] => [2,5,1,4,3] => ? = 4
[2,5,3,1,4] => [2,5,3,1,4] => ? = 4
[2,5,3,4,1] => [2,5,3,4,1] => ? = 5
[2,5,4,1,3] => [2,5,4,1,3] => ? = 4
[2,5,4,3,1] => [2,5,4,3,1] => ? = 4
[3,1,2,4,5] => [3,1,2,4,5] => ? = 2
[3,1,2,5,4] => [3,1,2,5,4] => ? = 3
[3,1,4,2,5] => [3,1,4,2,5] => ? = 3
[3,1,4,5,2] => [3,1,4,5,2] => ? = 4
[3,1,5,2,4] => [3,1,5,2,4] => ? = 4
[3,1,5,4,2] => [3,1,5,4,2] => ? = 4
[3,2,1,4,5] => [3,2,1,4,5] => ? = 2
[3,2,1,5,4] => [3,2,1,5,4] => ? = 3
[3,2,4,1,5] => [3,2,4,1,5] => ? = 3
[3,2,4,5,1] => [3,2,4,5,1] => ? = 4
[3,2,5,1,4] => [3,2,5,1,4] => ? = 4
[3,2,5,4,1] => [3,2,5,4,1] => ? = 4
[3,4,1,2,5] => [3,4,1,2,5] => ? = 4
[3,4,1,5,2] => [3,4,1,5,2] => ? = 5
[3,4,2,1,5] => [3,4,2,1,5] => ? = 3
[3,4,2,5,1] => [3,4,2,5,1] => ? = 4
[3,4,5,1,2] => [3,4,5,1,2] => ? = 6
[3,4,5,2,1] => [3,4,5,2,1] => ? = 4
[3,5,1,2,4] => [3,5,1,2,4] => ? = 5
[3,5,1,4,2] => [3,5,1,4,2] => ? = 5
[3,5,2,1,4] => [3,5,2,1,4] => ? = 4
[3,5,2,4,1] => [3,5,2,4,1] => ? = 5
[3,5,4,1,2] => [3,5,4,1,2] => ? = 5
[3,5,4,2,1] => [3,5,4,2,1] => ? = 4
[4,1,2,3,5] => [4,1,2,3,5] => ? = 3
[4,1,2,5,3] => [4,1,2,5,3] => ? = 4
Description
The number of Bruhat lower covers of a permutation. This is, for a signed permutation $\pi$, the number of signed permutations $\tau$ having a reduced word which is obtained by deleting a letter from a reduced word from $\pi$.
Matching statistic: St001596
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 30%
Values
[1,2] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [[3],[]]
 => 0
[2,1] => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [[2,2],[]]
 => 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => 0
[1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => 1
[2,1,3] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => 1
[2,3,1] => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => 2
[3,1,2] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => 2
[3,2,1] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [[4,4],[2]]
 => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [[4,3],[1]]
 => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[2,2,2,2],[]]
 => ? = 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2],[]]
 => ? = 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2],[]]
 => ? = 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[4,3],[]]
 => 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [[3,3,3],[1]]
 => ? = 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[4,3],[]]
 => 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [[3,3,3],[1]]
 => ? = 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[4,4],[]]
 => ? = 4
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[4,4],[]]
 => ? = 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3],[]]
 => ? = 3
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3],[]]
 => ? = 3
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3],[]]
 => ? = 3
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3],[]]
 => ? = 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3],[]]
 => ? = 3
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3],[]]
 => ? = 3
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [[6],[]]
 => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [[5,5],[3]]
 => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [[5,4],[2]]
 => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [[4,4,4],[2,2]]
 => ? = 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [[5,5],[2]]
 => ? = 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [[5,5],[2]]
 => ? = 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [[5,3],[1]]
 => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [[4,4,3],[2,1]]
 => ? = 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [[4,3,3],[1,1]]
 => ? = 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [[3,3,3,3],[1,1,1]]
 => ? = 3
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [[4,4,3],[1,1]]
 => ? = 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [[4,4,3],[1,1]]
 => ? = 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [[5,4],[1]]
 => ? = 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [[4,4,4],[2,1]]
 => ? = 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [[5,4],[1]]
 => ? = 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [[4,4,4],[2,1]]
 => ? = 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [[5,5],[1]]
 => ? = 4
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [[5,5],[1]]
 => ? = 3
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [[4,4,4],[1,1]]
 => ? = 3
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [[4,4,4],[1,1]]
 => ? = 3
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [[4,4,4],[1,1]]
 => ? = 3
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [[4,4,4],[1,1]]
 => ? = 4
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [[4,4,4],[1,1]]
 => ? = 3
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [[4,4,4],[1,1]]
 => ? = 3
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [[5,2],[]]
 => 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [[4,4,2],[2]]
 => ? = 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [[4,3,2],[1]]
 => ? = 2
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [[3,3,3,2],[1,1]]
 => ? = 3
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [[4,4,2],[1]]
 => ? = 3
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [[4,4,2],[1]]
 => ? = 3
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [[4,2,2],[]]
 => ? = 2
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [[3,3,2,2],[1]]
 => ? = 3
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [[3,2,2,2],[]]
 => ? = 3
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [[2,2,2,2,2],[]]
 => ? = 4
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [[3,3,2,2],[]]
 => ? = 4
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [[3,3,2,2],[]]
 => ? = 4
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [[4,3,2],[]]
 => ? = 3
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [[3,3,3,2],[1]]
 => ? = 4
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [[4,3,2],[]]
 => ? = 3
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [[3,3,3,2],[1]]
 => ? = 4
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [[4,4,2],[]]
 => ? = 5
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [[4,4,2],[]]
 => ? = 4
[1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[7],[]]
 => 0
Description
The number of two-by-two squares inside a skew partition. This is, the number of cells $(i,j)$ in a skew partition for which the box $(i+1,j+1)$ is also a cell inside the skew partition.
Matching statistic: St001877
(load all 10 compositions to match this statistic)
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 30%
Values
[1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 0
[2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[2,3,1] => ([(1,2)],3)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[3,1,2] => ([(1,2)],3)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[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)
 => ? = 2
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 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)
 => 1
[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)
 => 2
[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)
 => 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)
 => ? = 2
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1
[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
[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)
 => 2
[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)
 => ? = 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)
 => ? = 3
[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)
 => ? = 3
[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)
 => 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)
 => ? = 3
[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)
 => ? = 2
[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)
 => ? = 3
[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)
 => ? = 4
[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)
 => ? = 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)
 => ? = 3
[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)
 => ? = 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)
 => ? = 3
[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)
 => ? = 4
[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)
 => ? = 3
[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)
 => ? = 3
[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)
 => 0
[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)
 => 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)
 => 1
[1,2,4,5,3] => ([(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)
 => ? = 2
[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)
 => ? = 2
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 2
[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)
 => 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ? = 2
[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)
 => ? = 2
[1,3,4,5,2] => ([(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
[1,3,5,2,4] => ([(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
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 3
[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)
 => ? = 2
[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
[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)
 => ? = 2
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
 => ? = 3
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 4
[1,4,5,3,2] => ([(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)
 => ? = 3
[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
[1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 3
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
 => ? = 3
[1,5,3,4,2] => ([(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)
 => ? = 4
[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)
 => ? = 3
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 3
[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)
 => 1
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
 => ? = 2
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
 => ? = 2
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => ? = 3
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => ? = 3
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
 => ? = 3
[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)
 => ? = 2
[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
[2,3,4,1,5] => ([(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
[2,3,4,5,1] => ([(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)
 => ? = 4
[2,3,5,1,4] => ([(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)
 => ? = 4
[2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
 => ([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12)
 => ? = 4
[2,4,1,3,5] => ([(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
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? = 4
[2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
 => ? = 3
[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)
 => 0
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St001633
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001633: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 30%
Values
[1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0
[2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[2,3,1] => ([(1,2)],3)
 => ([(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
[3,1,2] => ([(1,2)],3)
 => ([(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
[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)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(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
[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)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 1
[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)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 2
[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)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 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)
 => ([(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)
 => ? = 2
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1
[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)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 2
[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)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2
[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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 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)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 3
[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)
 => ([(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)
 => ? = 3
[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)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 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)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 3
[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)
 => ([(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)
 => ? = 2
[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)
 => ([(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)
 => ? = 3
[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)
 => ([(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
[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)
 => ([(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)
 => ? = 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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 3
[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)
 => ([(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)
 => ? = 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)
 => ([(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)
 => ? = 3
[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)
 => ([(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)
 => ? = 4
[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)
 => ([(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)
 => ? = 3
[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)
 => ([(0,1),(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)
 => ? = 3
[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)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[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)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 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)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 1
[1,2,4,5,3] => ([(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)
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[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)
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 2
[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)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ? = 2
[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)
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 2
[1,3,4,5,2] => ([(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)
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? = 3
[1,3,5,2,4] => ([(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)
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? = 3
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 3
[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)
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 2
[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)
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? = 3
[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)
 => ([(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)
 => ? = 2
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
 => ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
 => ? = 3
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 4
[1,4,5,3,2] => ([(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)
 => ([(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)
 => ? = 3
[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)
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? = 3
[1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 3
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
 => ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
 => ? = 3
[1,5,3,4,2] => ([(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)
 => ([(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)
 => ? = 4
[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)
 => ([(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)
 => ? = 3
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 3
[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)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 1
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
 => ? = 2
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
 => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
 => ? = 2
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => ? = 3
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => ? = 3
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
 => ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
 => ? = 3
[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)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[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)
 => ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
 => ? = 3
[2,3,4,1,5] => ([(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)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 3
[2,3,4,5,1] => ([(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)
 => ([(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)
 => ? = 4
[2,3,5,1,4] => ([(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)
 => ([(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)
 => ? = 4
[2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
 => ([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12)
 => ([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12)
 => ? = 4
[2,4,1,3,5] => ([(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)
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ? = 3
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? = 4
[2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
 => ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
 => ? = 3
[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)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 0
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.