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Your data matches 14 different statistics following compositions of up to 3 maps.
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Matching statistic: St001714
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1,3] => [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 0
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 0
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 0
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 0
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 0
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 0
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 0
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 0
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 0
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 0
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 0
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 0
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 0
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 0
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 0
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 0
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 0
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 0
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 0
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 0
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 1
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 1
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 0
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 0
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 0
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 0
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 0
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 0
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 0
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 0
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 0
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 0
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 0
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 0
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 0
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 0
Description
The number of subpartitions of an integer partition that do not dominate the conjugate subpartition.
In particular, partitions with statistic $0$ are wide partitions.
Matching statistic: St000454
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 12%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 12%
Values
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 0 + 2
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 2
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 2
[2,3,1,4] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 2
[2,4,1,3] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 2
[2,4,3,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 2
[3,1,2,4] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 2
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 2
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[2,1,3,4,5] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 2
[2,1,3,5,4] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[2,1,4,3,5] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[2,1,4,5,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[2,1,5,3,4] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[2,1,5,4,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[2,3,1,4,5] => [2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[2,3,1,5,4] => [2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[2,3,4,1,5] => [2,5,4,1,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[2,3,5,1,4] => [2,5,4,1,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[2,3,5,4,1] => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[2,4,1,3,5] => [2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[2,4,1,5,3] => [2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[2,4,3,1,5] => [2,5,4,1,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[2,4,3,5,1] => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[2,4,5,1,3] => [2,5,4,1,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[2,4,5,3,1] => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[2,5,1,3,4] => [2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[2,5,1,4,3] => [2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[2,5,3,1,4] => [2,5,4,1,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[2,5,3,4,1] => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[2,5,4,1,3] => [2,5,4,1,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[2,5,4,3,1] => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[3,1,2,4,5] => [3,1,5,4,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[3,1,2,5,4] => [3,1,5,4,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[3,1,4,2,5] => [3,1,5,4,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[3,1,5,2,4] => [3,1,5,4,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[3,1,5,4,2] => [3,1,5,4,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[3,2,1,4,5] => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[3,2,1,5,4] => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[3,2,4,1,5] => [3,2,5,1,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[3,2,5,1,4] => [3,2,5,1,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[3,2,5,4,1] => [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[6,3,1,2,4,5,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,1,2,5,4,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,1,2,7,4,5] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,1,2,7,5,4] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,1,4,2,5,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,1,4,5,2,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,1,4,7,2,5] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,1,4,7,5,2] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,1,5,2,4,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,1,5,4,2,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,1,5,7,2,4] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,1,5,7,4,2] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,2,1,4,5,7] => [6,3,2,1,7,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,2,1,5,4,7] => [6,3,2,1,7,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,2,1,7,4,5] => [6,3,2,1,7,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,2,1,7,5,4] => [6,3,2,1,7,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,2,4,1,5,7] => [6,3,2,7,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,2,4,5,1,7] => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,2,4,7,1,5] => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,2,4,7,5,1] => [6,3,2,7,5,4,1] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,2,5,1,4,7] => [6,3,2,7,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,2,5,4,1,7] => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,2,5,7,1,4] => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,2,5,7,4,1] => [6,3,2,7,5,4,1] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,4,1,2,5,7] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,4,1,5,2,7] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,4,1,7,2,5] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,4,1,7,5,2] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,4,2,1,5,7] => [6,3,7,2,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,4,2,5,1,7] => [6,3,7,2,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,4,2,7,1,5] => [6,3,7,2,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,4,2,7,5,1] => [6,3,7,2,5,4,1] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,4,5,1,2,7] => [6,3,7,5,1,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,4,5,2,1,7] => [6,3,7,5,2,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,4,5,7,1,2] => [6,3,7,5,4,1,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,4,5,7,2,1] => [6,3,7,5,4,2,1] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,5,1,2,4,7] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,5,1,4,2,7] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,5,1,7,2,4] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,5,1,7,4,2] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,5,2,1,4,7] => [6,3,7,2,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,5,2,4,1,7] => [6,3,7,2,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,5,2,7,1,4] => [6,3,7,2,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,5,2,7,4,1] => [6,3,7,2,5,4,1] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,5,4,1,2,7] => [6,3,7,5,1,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,5,4,2,1,7] => [6,3,7,5,2,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,5,4,7,1,2] => [6,3,7,5,4,1,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,5,4,7,2,1] => [6,3,7,5,4,2,1] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,7,1,2,4,5] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[6,3,7,1,2,5,4] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000422
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 12%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 12%
Values
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 0 + 8
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 8
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 8
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 8
[2,3,1,4] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 8
[2,4,1,3] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 8
[2,4,3,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 8
[3,1,2,4] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 8
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 8
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 8
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[2,1,3,4,5] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 8
[2,1,3,5,4] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 8
[2,1,4,3,5] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[2,1,4,5,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[2,1,5,3,4] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 8
[2,1,5,4,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 8
[2,3,1,4,5] => [2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[2,3,1,5,4] => [2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[2,3,4,1,5] => [2,5,4,1,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[2,3,5,1,4] => [2,5,4,1,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[2,3,5,4,1] => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[2,4,1,3,5] => [2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[2,4,1,5,3] => [2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[2,4,3,1,5] => [2,5,4,1,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[2,4,3,5,1] => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[2,4,5,1,3] => [2,5,4,1,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 8
[2,4,5,3,1] => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 8
[2,5,1,3,4] => [2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[2,5,1,4,3] => [2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[2,5,3,1,4] => [2,5,4,1,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[2,5,3,4,1] => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[2,5,4,1,3] => [2,5,4,1,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[2,5,4,3,1] => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[3,1,2,4,5] => [3,1,5,4,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 8
[3,1,2,5,4] => [3,1,5,4,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[3,1,4,2,5] => [3,1,5,4,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[3,1,5,2,4] => [3,1,5,4,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[3,1,5,4,2] => [3,1,5,4,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[3,2,1,4,5] => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 8
[3,2,1,5,4] => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[3,2,4,1,5] => [3,2,5,1,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[3,2,5,1,4] => [3,2,5,1,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[3,2,5,4,1] => [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 8
[6,3,1,2,4,5,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,1,2,5,4,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,1,2,7,4,5] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,1,2,7,5,4] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,1,4,2,5,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,1,4,5,2,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,1,4,7,2,5] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,1,4,7,5,2] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,1,5,2,4,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,1,5,4,2,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,1,5,7,2,4] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,1,5,7,4,2] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,2,1,4,5,7] => [6,3,2,1,7,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,2,1,5,4,7] => [6,3,2,1,7,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,2,1,7,4,5] => [6,3,2,1,7,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,2,1,7,5,4] => [6,3,2,1,7,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,2,4,1,5,7] => [6,3,2,7,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,2,4,5,1,7] => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,2,4,7,1,5] => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,2,4,7,5,1] => [6,3,2,7,5,4,1] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,2,5,1,4,7] => [6,3,2,7,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,2,5,4,1,7] => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,2,5,7,1,4] => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,2,5,7,4,1] => [6,3,2,7,5,4,1] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,4,1,2,5,7] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,4,1,5,2,7] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,4,1,7,2,5] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,4,1,7,5,2] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,4,2,1,5,7] => [6,3,7,2,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,4,2,5,1,7] => [6,3,7,2,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,4,2,7,1,5] => [6,3,7,2,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,4,2,7,5,1] => [6,3,7,2,5,4,1] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,4,5,1,2,7] => [6,3,7,5,1,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,4,5,2,1,7] => [6,3,7,5,2,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,4,5,7,1,2] => [6,3,7,5,4,1,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,4,5,7,2,1] => [6,3,7,5,4,2,1] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,5,1,2,4,7] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,5,1,4,2,7] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,5,1,7,2,4] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,5,1,7,4,2] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,5,2,1,4,7] => [6,3,7,2,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,5,2,4,1,7] => [6,3,7,2,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,5,2,7,1,4] => [6,3,7,2,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,5,2,7,4,1] => [6,3,7,2,5,4,1] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,5,4,1,2,7] => [6,3,7,5,1,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,5,4,2,1,7] => [6,3,7,5,2,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,5,4,7,1,2] => [6,3,7,5,4,1,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,5,4,7,2,1] => [6,3,7,5,4,2,1] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,7,1,2,4,5] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
[6,3,7,1,2,5,4] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8 = 0 + 8
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St001876
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 12%
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 12%
Values
[2,1,3] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,3,2,4] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,3,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,1,4,3] => [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,3,1,4] => [3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,4,1,3] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 1
[2,4,3,1] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1)],2)
=> ? = 0 + 1
[3,1,2,4] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,2,1,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,2,4,3,5] => [2,3,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,3,2,4,5] => [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,3,2,5,4] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,3,4,2,5] => [2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,3,5,2,4] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,3,5,4,2] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,4,2,3,5] => [2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,4,3,2,5] => [2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,3,4,5] => [3,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ([(0,1)],2)
=> ? = 2 + 1
[2,1,3,5,4] => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,1,4,3,5] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,4,5,3] => [3,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([],1)
=> ? = 0 + 1
[2,1,5,3,4] => [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ([],1)
=> ? = 1 + 1
[2,1,5,4,3] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,3,1,4,5] => [3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,3,1,5,4] => [3,4,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,3,4,1,5] => [3,4,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,3,5,1,4] => [3,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([],1)
=> ? = 0 + 1
[2,3,5,4,1] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,4,1,3,5] => [3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,4,1,5,3] => [3,5,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([],1)
=> ? = 0 + 1
[2,4,3,1,5] => [3,5,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,4,3,5,1] => [3,5,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ([],1)
=> ? = 0 + 1
[2,4,5,1,3] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[2,4,5,3,1] => [3,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([],1)
=> ? = 1 + 1
[2,5,1,3,4] => [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([],1)
=> ? = 0 + 1
[2,5,1,4,3] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,5,3,1,4] => [3,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([],1)
=> ? = 0 + 1
[2,5,3,4,1] => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 0 + 1
[2,5,4,1,3] => [3,1,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ([],1)
=> ? = 0 + 1
[2,5,4,3,1] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ([(0,1)],2)
=> ? = 0 + 1
[3,1,2,4,5] => [4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ([(0,1)],2)
=> ? = 1 + 1
[3,1,2,5,4] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ([(0,1)],2)
=> ? = 0 + 1
[3,1,4,2,5] => [4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,1)],2)
=> ? = 0 + 1
[3,1,5,2,4] => [4,2,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([],1)
=> ? = 0 + 1
[3,1,5,4,2] => [4,2,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ([(0,1)],2)
=> ? = 0 + 1
[3,2,1,4,5] => [4,3,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ([(0,1)],2)
=> ? = 1 + 1
[3,2,1,5,4] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ([(0,1)],2)
=> ? = 0 + 1
[3,2,4,1,5] => [4,3,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ([(0,1)],2)
=> ? = 0 + 1
[3,2,5,1,4] => [4,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([],1)
=> ? = 0 + 1
[3,2,5,4,1] => [4,3,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ([(0,1)],2)
=> ? = 0 + 1
[3,4,1,2,5] => [4,5,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ([(0,1)],2)
=> ? = 0 + 1
[3,4,2,1,5] => [4,5,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ([(0,1)],2)
=> ? = 0 + 1
[4,1,2,3,5] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,1,3,2,5] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,1,5,2,3] => [5,2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ([(0,1)],2)
=> ? = 0 + 1
[4,2,1,3,5] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,2,3,1,5] => [5,3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,3,1,2,5] => [5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,3,2,1,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,1,2,3,4,6] => [6,2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,15),(6,13),(6,15),(8,14),(9,14),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,9)],16)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,1,2,4,3,6] => [6,2,3,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,11),(4,12),(5,12),(6,8),(6,11),(8,13),(9,7),(10,7),(11,13),(12,8),(13,9),(13,10)],14)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,1,3,2,4,6] => [6,2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,8),(3,11),(4,10),(5,13),(6,13),(8,12),(9,12),(10,7),(11,7),(12,10),(12,11),(13,8),(13,9)],14)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,1,3,4,2,6] => [6,2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,1,4,2,3,6] => [6,2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,1,4,3,2,6] => [6,2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,2,1,3,4,6] => [6,3,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,11),(4,12),(5,12),(6,8),(6,11),(8,13),(9,7),(10,7),(11,13),(12,8),(13,9),(13,10)],14)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,2,1,4,3,6] => [6,3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,11),(9,12),(10,12),(11,9),(11,10)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,2,3,1,4,6] => [6,3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,2,3,4,1,6] => [6,3,4,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,12),(3,11),(4,10),(5,12),(5,13),(6,11),(6,15),(8,7),(9,7),(10,9),(11,8),(12,14),(13,14),(14,10),(14,15),(15,8),(15,9)],16)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,2,4,1,3,6] => [6,3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,8),(6,7),(7,9),(8,9),(10,7),(10,8)],11)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,2,4,3,1,6] => [6,3,5,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,3,1,2,4,6] => [6,4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,3,1,4,2,6] => [6,4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,8),(6,7),(7,9),(8,9),(10,7),(10,8)],11)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,3,2,1,4,6] => [6,4,3,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,3,2,4,1,6] => [6,4,3,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,3,4,1,2,6] => [6,4,5,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,13),(5,14),(6,14),(8,7),(9,7),(10,8),(11,9),(12,8),(12,9),(13,10),(13,12),(14,11),(14,12)],15)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,3,4,2,1,6] => [6,4,5,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,17),(3,17),(4,12),(5,15),(5,16),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,9),(13,8),(14,10),(14,11),(15,9),(15,14),(16,8),(16,14),(17,12),(17,15)],18)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,4,1,2,3,6] => [6,5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,12),(3,11),(4,10),(5,12),(5,13),(6,11),(6,15),(8,7),(9,7),(10,9),(11,8),(12,14),(13,14),(14,10),(14,15),(15,8),(15,9)],16)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,4,1,3,2,6] => [6,5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,4,2,1,3,6] => [6,5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,4,2,3,1,6] => [6,5,3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,17),(2,17),(3,13),(4,12),(5,12),(5,15),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,14),(16,9),(16,14),(17,15),(17,16)],18)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,4,3,1,2,6] => [6,5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,17),(3,17),(4,12),(5,15),(5,16),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,9),(13,8),(14,10),(14,11),(15,9),(15,14),(16,8),(16,14),(17,12),(17,15)],18)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[7,5,4,6,2,1,3,8] => [8,6,5,7,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,14),(2,14),(3,15),(4,15),(5,13),(6,12),(7,17),(8,18),(10,9),(11,9),(12,10),(13,11),(14,17),(15,18),(16,10),(16,11),(17,12),(17,16),(18,13),(18,16)],19)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[7,3,6,2,5,1,4,8] => [8,4,7,3,6,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,10),(8,9),(9,11),(10,11),(12,9),(12,10)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[7,5,3,2,6,1,4,8] => [8,6,4,3,7,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,13),(3,13),(4,13),(5,11),(6,10),(7,9),(8,9),(9,13),(10,12),(11,12),(13,10),(13,11)],14)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[7,2,1,4,3,6,5,8] => [8,3,2,5,4,7,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,17),(2,17),(3,14),(4,14),(5,15),(6,15),(7,13),(8,12),(10,16),(11,16),(12,9),(13,9),(14,11),(15,10),(16,12),(16,13),(17,10),(17,11)],18)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[7,3,4,1,2,6,5,8] => [8,4,5,2,3,7,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,12),(3,11),(4,11),(5,10),(6,10),(7,9),(8,9),(9,15),(10,14),(11,14),(12,16),(13,16),(14,15),(15,12),(15,13)],17)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[7,4,3,2,1,6,5,8] => [8,5,4,3,2,7,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,14),(2,13),(3,16),(4,15),(5,17),(6,17),(7,15),(7,19),(8,16),(8,19),(10,12),(11,12),(12,18),(13,9),(14,9),(15,10),(16,11),(17,18),(18,13),(18,14),(19,10),(19,11)],20)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[7,3,5,1,6,2,4,8] => [8,4,6,2,7,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,10),(8,9),(9,11),(10,11),(12,9),(12,10)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[7,2,1,5,6,3,4,8] => [8,3,2,6,7,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,12),(3,11),(4,11),(5,10),(6,10),(7,9),(8,9),(9,15),(10,14),(11,14),(12,16),(13,16),(14,15),(15,12),(15,13)],17)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[7,2,1,6,5,4,3,8] => [8,3,2,7,6,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,14),(2,13),(3,16),(4,15),(5,17),(6,17),(7,15),(7,19),(8,16),(8,19),(10,12),(11,12),(12,18),(13,9),(14,9),(15,10),(16,11),(17,18),(18,13),(18,14),(19,10),(19,11)],20)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[7,3,6,1,5,4,2,8] => [8,4,7,2,6,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,13),(3,13),(4,13),(5,11),(6,10),(7,9),(8,9),(9,13),(10,12),(11,12),(13,10),(13,11)],14)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[7,4,6,5,1,3,2,8] => [8,5,7,6,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,14),(2,14),(3,15),(4,15),(5,13),(6,12),(7,17),(8,18),(10,9),(11,9),(12,10),(13,11),(14,17),(15,18),(16,10),(16,11),(17,12),(17,16),(18,13),(18,16)],19)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[7,4,6,3,1,5,2,8] => [8,5,7,4,2,6,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,10),(8,9),(9,11),(10,11),(12,9),(12,10)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[7,4,6,1,5,2,3,8] => [8,5,7,2,6,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,13),(3,13),(4,13),(5,11),(6,10),(7,9),(8,9),(9,13),(10,12),(11,12),(13,10),(13,11)],14)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[7,3,6,1,5,2,4,8] => [8,4,7,2,6,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,10),(8,9),(9,11),(10,11),(12,9),(12,10)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[7,5,2,6,4,1,3,8] => [8,6,3,7,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,10),(8,9),(9,11),(10,11),(12,9),(12,10)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[7,4,2,6,1,5,3,8] => [8,5,3,7,2,6,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,10),(8,9),(9,11),(10,11),(12,9),(12,10)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[7,4,2,5,1,6,3,8] => [8,5,3,6,2,7,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,10),(8,9),(9,11),(10,11),(12,9),(12,10)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[7,3,1,4,6,2,5,8] => [8,4,2,5,7,3,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,10),(8,9),(9,11),(10,11),(12,9),(12,10)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St001613
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001613: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 12%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001613: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 12%
Values
[2,1,3] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 0 + 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 0 + 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 0 + 1
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 0 + 1
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 0 + 1
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 1 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 2 + 1
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 1 + 1
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[2,1,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 1 + 1
[2,1,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 1 + 1
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[2,3,5,1,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[2,4,1,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[2,4,1,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 0 + 1
[2,4,3,1,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[2,4,3,5,1] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 0 + 1
[2,4,5,1,3] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 1 + 1
[2,4,5,3,1] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 1 + 1
[2,5,1,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 0 + 1
[2,5,1,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[2,5,3,1,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 0 + 1
[2,5,3,4,1] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[2,5,4,1,3] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[2,5,4,3,1] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[3,1,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 1 + 1
[3,1,2,5,4] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[3,1,4,2,5] => [1,3,4,2,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 0 + 1
[3,1,5,2,4] => [1,3,5,4,2] => [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 0 + 1
[3,1,5,4,2] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 1
[3,2,1,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 1 + 1
[3,2,1,5,4] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[3,2,4,1,5] => [1,3,4,2,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 0 + 1
[3,2,5,1,4] => [1,3,5,4,2] => [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 0 + 1
[3,2,5,4,1] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 1
[3,4,1,2,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[3,4,2,1,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[4,1,2,3,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 0 + 1
[4,1,3,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 0 + 1
[4,1,5,2,3] => [1,4,2,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 0 + 1
[3,1,6,2,5,4] => [1,3,6,4,2,5] => [3,6,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[3,2,6,1,5,4] => [1,3,6,4,2,5] => [3,6,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,5,1,6,3,2] => [1,4,6,2,5,3] => [4,6,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,5,2,6,3,1] => [1,4,6,2,5,3] => [4,6,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,5,3,6,1,2] => [1,4,6,2,5,3] => [4,6,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,5,3,6,2,1] => [1,4,6,2,5,3] => [4,6,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,6,1,2,5,3] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,6,2,1,5,3] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,6,3,1,5,2] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,6,3,2,5,1] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,6,5,1,2,3] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,6,5,1,3,2] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,6,5,2,1,3] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,6,5,2,3,1] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[5,4,1,6,2,3] => [1,5,2,4,6,3] => [5,1,4,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[5,4,2,6,1,3] => [1,5,2,4,6,3] => [5,1,4,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[5,4,3,6,1,2] => [1,5,2,4,6,3] => [5,1,4,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[5,4,3,6,2,1] => [1,5,2,4,6,3] => [5,1,4,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[5,1,2,7,3,4,6] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,1,2,7,3,6,4] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,2,1,7,3,4,6] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,2,1,7,3,6,4] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,4,1,7,3,2,6] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,4,1,7,3,6,2] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,4,2,7,3,1,6] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,4,2,7,3,6,1] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,6,1,2,7,4,3] => [1,5,7,3,2,6,4] => [5,1,7,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,6,1,4,7,2,3] => [1,5,7,3,2,6,4] => [5,1,7,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,6,2,1,7,4,3] => [1,5,7,3,2,6,4] => [5,1,7,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,6,2,4,7,1,3] => [1,5,7,3,2,6,4] => [5,1,7,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
Description
The binary logarithm of the size of the center of a lattice.
An element of a lattice is central if it is neutral and has a complement. The subposet induced by central elements is a Boolean lattice.
Matching statistic: St001719
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 12%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 12%
Values
[2,1,3] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 0 + 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 0 + 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 0 + 1
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 0 + 1
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 0 + 1
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 1 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 2 + 1
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 1 + 1
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[2,1,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 1 + 1
[2,1,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 1 + 1
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[2,3,5,1,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[2,4,1,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[2,4,1,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 0 + 1
[2,4,3,1,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[2,4,3,5,1] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 0 + 1
[2,4,5,1,3] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 1 + 1
[2,4,5,3,1] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 1 + 1
[2,5,1,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 0 + 1
[2,5,1,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[2,5,3,1,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 0 + 1
[2,5,3,4,1] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[2,5,4,1,3] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[2,5,4,3,1] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[3,1,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 1 + 1
[3,1,2,5,4] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[3,1,4,2,5] => [1,3,4,2,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 0 + 1
[3,1,5,2,4] => [1,3,5,4,2] => [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 0 + 1
[3,1,5,4,2] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 1
[3,2,1,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 1 + 1
[3,2,1,5,4] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[3,2,4,1,5] => [1,3,4,2,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 0 + 1
[3,2,5,1,4] => [1,3,5,4,2] => [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 0 + 1
[3,2,5,4,1] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 1
[3,4,1,2,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[3,4,2,1,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[4,1,2,3,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 0 + 1
[4,1,3,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 0 + 1
[4,1,5,2,3] => [1,4,2,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 0 + 1
[3,1,6,2,5,4] => [1,3,6,4,2,5] => [3,6,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[3,2,6,1,5,4] => [1,3,6,4,2,5] => [3,6,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,5,1,6,3,2] => [1,4,6,2,5,3] => [4,6,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,5,2,6,3,1] => [1,4,6,2,5,3] => [4,6,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,5,3,6,1,2] => [1,4,6,2,5,3] => [4,6,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,5,3,6,2,1] => [1,4,6,2,5,3] => [4,6,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,6,1,2,5,3] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,6,2,1,5,3] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,6,3,1,5,2] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,6,3,2,5,1] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,6,5,1,2,3] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,6,5,1,3,2] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,6,5,2,1,3] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,6,5,2,3,1] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[5,4,1,6,2,3] => [1,5,2,4,6,3] => [5,1,4,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[5,4,2,6,1,3] => [1,5,2,4,6,3] => [5,1,4,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[5,4,3,6,1,2] => [1,5,2,4,6,3] => [5,1,4,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[5,4,3,6,2,1] => [1,5,2,4,6,3] => [5,1,4,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[5,1,2,7,3,4,6] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,1,2,7,3,6,4] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,2,1,7,3,4,6] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,2,1,7,3,6,4] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,4,1,7,3,2,6] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,4,1,7,3,6,2] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,4,2,7,3,1,6] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,4,2,7,3,6,1] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,6,1,2,7,4,3] => [1,5,7,3,2,6,4] => [5,1,7,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,6,1,4,7,2,3] => [1,5,7,3,2,6,4] => [5,1,7,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,6,2,1,7,4,3] => [1,5,7,3,2,6,4] => [5,1,7,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,6,2,4,7,1,3] => [1,5,7,3,2,6,4] => [5,1,7,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
Description
The number of shortest chains of small intervals from the bottom to the top in a lattice.
An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.
Matching statistic: St001881
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001881: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 12%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001881: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 12%
Values
[2,1,3] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 0 + 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 0 + 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 0 + 1
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 0 + 1
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 0 + 1
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 1 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 2 + 1
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 1 + 1
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[2,1,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 1 + 1
[2,1,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 1 + 1
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[2,3,5,1,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[2,4,1,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[2,4,1,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 0 + 1
[2,4,3,1,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[2,4,3,5,1] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 0 + 1
[2,4,5,1,3] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 1 + 1
[2,4,5,3,1] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 1 + 1
[2,5,1,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 0 + 1
[2,5,1,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[2,5,3,1,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 0 + 1
[2,5,3,4,1] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[2,5,4,1,3] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[2,5,4,3,1] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[3,1,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 1 + 1
[3,1,2,5,4] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[3,1,4,2,5] => [1,3,4,2,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 0 + 1
[3,1,5,2,4] => [1,3,5,4,2] => [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 0 + 1
[3,1,5,4,2] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 1
[3,2,1,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 1 + 1
[3,2,1,5,4] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[3,2,4,1,5] => [1,3,4,2,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 0 + 1
[3,2,5,1,4] => [1,3,5,4,2] => [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 0 + 1
[3,2,5,4,1] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 1
[3,4,1,2,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[3,4,2,1,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[4,1,2,3,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 0 + 1
[4,1,3,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 0 + 1
[4,1,5,2,3] => [1,4,2,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 0 + 1
[3,1,6,2,5,4] => [1,3,6,4,2,5] => [3,6,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[3,2,6,1,5,4] => [1,3,6,4,2,5] => [3,6,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,5,1,6,3,2] => [1,4,6,2,5,3] => [4,6,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,5,2,6,3,1] => [1,4,6,2,5,3] => [4,6,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,5,3,6,1,2] => [1,4,6,2,5,3] => [4,6,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,5,3,6,2,1] => [1,4,6,2,5,3] => [4,6,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,6,1,2,5,3] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,6,2,1,5,3] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,6,3,1,5,2] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,6,3,2,5,1] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,6,5,1,2,3] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,6,5,1,3,2] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,6,5,2,1,3] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[4,6,5,2,3,1] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[5,4,1,6,2,3] => [1,5,2,4,6,3] => [5,1,4,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[5,4,2,6,1,3] => [1,5,2,4,6,3] => [5,1,4,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[5,4,3,6,1,2] => [1,5,2,4,6,3] => [5,1,4,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[5,4,3,6,2,1] => [1,5,2,4,6,3] => [5,1,4,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[5,1,2,7,3,4,6] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,1,2,7,3,6,4] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,2,1,7,3,4,6] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,2,1,7,3,6,4] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,4,1,7,3,2,6] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,4,1,7,3,6,2] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,4,2,7,3,1,6] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,4,2,7,3,6,1] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,6,1,2,7,4,3] => [1,5,7,3,2,6,4] => [5,1,7,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,6,1,4,7,2,3] => [1,5,7,3,2,6,4] => [5,1,7,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,6,2,1,7,4,3] => [1,5,7,3,2,6,4] => [5,1,7,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[5,6,2,4,7,1,3] => [1,5,7,3,2,6,4] => [5,1,7,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
Description
The number of factors of a lattice as a Cartesian product of lattices.
Since the cardinality of a lattice is the product of the cardinalities of its factors, this statistic is one whenever the cardinality of the lattice is prime.
Matching statistic: St001616
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 12%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 12%
Values
[2,1,3] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 0 + 2
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 2
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 0 + 2
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 0 + 2
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2 = 0 + 2
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2 = 0 + 2
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 2 = 0 + 2
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 2 = 0 + 2
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 2
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 1 + 2
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 2
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 2
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 2
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 2
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 2
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 2
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 2 + 2
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 1 + 2
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 2
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 2
[2,1,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 1 + 2
[2,1,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 1 + 2
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 2
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 2
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 2
[2,3,5,1,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 2
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 2
[2,4,1,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 2
[2,4,1,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 0 + 2
[2,4,3,1,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 2
[2,4,3,5,1] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 0 + 2
[2,4,5,1,3] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 1 + 2
[2,4,5,3,1] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 1 + 2
[2,5,1,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 0 + 2
[2,5,1,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 2
[2,5,3,1,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 0 + 2
[2,5,3,4,1] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 2
[2,5,4,1,3] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 2
[2,5,4,3,1] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 2
[3,1,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 1 + 2
[3,1,2,5,4] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 2
[3,1,4,2,5] => [1,3,4,2,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 0 + 2
[3,1,5,2,4] => [1,3,5,4,2] => [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 0 + 2
[3,1,5,4,2] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 2
[3,2,1,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 1 + 2
[3,2,1,5,4] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 2
[3,2,4,1,5] => [1,3,4,2,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 0 + 2
[3,2,5,1,4] => [1,3,5,4,2] => [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 0 + 2
[3,2,5,4,1] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 2
[3,4,1,2,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 2
[3,4,2,1,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 2
[4,1,2,3,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 0 + 2
[4,1,3,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 0 + 2
[4,1,5,2,3] => [1,4,2,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 0 + 2
[3,1,6,2,5,4] => [1,3,6,4,2,5] => [3,6,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[3,2,6,1,5,4] => [1,3,6,4,2,5] => [3,6,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[4,5,1,6,3,2] => [1,4,6,2,5,3] => [4,6,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[4,5,2,6,3,1] => [1,4,6,2,5,3] => [4,6,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[4,5,3,6,1,2] => [1,4,6,2,5,3] => [4,6,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[4,5,3,6,2,1] => [1,4,6,2,5,3] => [4,6,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[4,6,1,2,5,3] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[4,6,2,1,5,3] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[4,6,3,1,5,2] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[4,6,3,2,5,1] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[4,6,5,1,2,3] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[4,6,5,1,3,2] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[4,6,5,2,1,3] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[4,6,5,2,3,1] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[5,4,1,6,2,3] => [1,5,2,4,6,3] => [5,1,4,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[5,4,2,6,1,3] => [1,5,2,4,6,3] => [5,1,4,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[5,4,3,6,1,2] => [1,5,2,4,6,3] => [5,1,4,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[5,4,3,6,2,1] => [1,5,2,4,6,3] => [5,1,4,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[5,1,2,7,3,4,6] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[5,1,2,7,3,6,4] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[5,2,1,7,3,4,6] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[5,2,1,7,3,6,4] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[5,4,1,7,3,2,6] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[5,4,1,7,3,6,2] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[5,4,2,7,3,1,6] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[5,4,2,7,3,6,1] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[5,6,1,2,7,4,3] => [1,5,7,3,2,6,4] => [5,1,7,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[5,6,1,4,7,2,3] => [1,5,7,3,2,6,4] => [5,1,7,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[5,6,2,1,7,4,3] => [1,5,7,3,2,6,4] => [5,1,7,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[5,6,2,4,7,1,3] => [1,5,7,3,2,6,4] => [5,1,7,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
Description
The number of neutral elements in a lattice.
An element $e$ of the lattice $L$ is neutral if the sublattice generated by $e$, $x$ and $y$ is distributive for all $x, y \in L$.
Matching statistic: St001720
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 12%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 12%
Values
[2,1,3] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 0 + 2
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 2
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 0 + 2
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 0 + 2
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2 = 0 + 2
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2 = 0 + 2
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 2 = 0 + 2
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 2 = 0 + 2
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 2
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 1 + 2
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 2
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 2
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 2
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 2
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 2
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 2
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 2 + 2
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 1 + 2
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 2
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 2
[2,1,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 1 + 2
[2,1,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 1 + 2
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 2
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 2
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 2
[2,3,5,1,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 2
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 2
[2,4,1,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 2
[2,4,1,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 0 + 2
[2,4,3,1,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 2
[2,4,3,5,1] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 0 + 2
[2,4,5,1,3] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 1 + 2
[2,4,5,3,1] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 1 + 2
[2,5,1,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 0 + 2
[2,5,1,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 2
[2,5,3,1,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 0 + 2
[2,5,3,4,1] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 2
[2,5,4,1,3] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 2
[2,5,4,3,1] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 2
[3,1,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 1 + 2
[3,1,2,5,4] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 2
[3,1,4,2,5] => [1,3,4,2,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 0 + 2
[3,1,5,2,4] => [1,3,5,4,2] => [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 0 + 2
[3,1,5,4,2] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 2
[3,2,1,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 1 + 2
[3,2,1,5,4] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 2
[3,2,4,1,5] => [1,3,4,2,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 0 + 2
[3,2,5,1,4] => [1,3,5,4,2] => [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 0 + 2
[3,2,5,4,1] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 2
[3,4,1,2,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 2
[3,4,2,1,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 2
[4,1,2,3,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 0 + 2
[4,1,3,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 0 + 2
[4,1,5,2,3] => [1,4,2,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 0 + 2
[3,1,6,2,5,4] => [1,3,6,4,2,5] => [3,6,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[3,2,6,1,5,4] => [1,3,6,4,2,5] => [3,6,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[4,5,1,6,3,2] => [1,4,6,2,5,3] => [4,6,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[4,5,2,6,3,1] => [1,4,6,2,5,3] => [4,6,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[4,5,3,6,1,2] => [1,4,6,2,5,3] => [4,6,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[4,5,3,6,2,1] => [1,4,6,2,5,3] => [4,6,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[4,6,1,2,5,3] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[4,6,2,1,5,3] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[4,6,3,1,5,2] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[4,6,3,2,5,1] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[4,6,5,1,2,3] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[4,6,5,1,3,2] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[4,6,5,2,1,3] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[4,6,5,2,3,1] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[5,4,1,6,2,3] => [1,5,2,4,6,3] => [5,1,4,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[5,4,2,6,1,3] => [1,5,2,4,6,3] => [5,1,4,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[5,4,3,6,1,2] => [1,5,2,4,6,3] => [5,1,4,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[5,4,3,6,2,1] => [1,5,2,4,6,3] => [5,1,4,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[5,1,2,7,3,4,6] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[5,1,2,7,3,6,4] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[5,2,1,7,3,4,6] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[5,2,1,7,3,6,4] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[5,4,1,7,3,2,6] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[5,4,1,7,3,6,2] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[5,4,2,7,3,1,6] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[5,4,2,7,3,6,1] => [1,5,3,2,4,7,6] => [5,1,3,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[5,6,1,2,7,4,3] => [1,5,7,3,2,6,4] => [5,1,7,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[5,6,1,4,7,2,3] => [1,5,7,3,2,6,4] => [5,1,7,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[5,6,2,1,7,4,3] => [1,5,7,3,2,6,4] => [5,1,7,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[5,6,2,4,7,1,3] => [1,5,7,3,2,6,4] => [5,1,7,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
Description
The minimal length of a chain of small intervals in a lattice.
An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.
Matching statistic: St000181
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 12%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 12%
Values
[2,1,3] => [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2,4] => [2,4,3,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,3,4] => [3,2,4,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[2,1,4,3] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[2,3,1,4] => [4,2,3,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,4,1,3] => [4,2,1,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[3,1,2,4] => [3,4,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,2,1,4] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3,5] => [2,3,5,4,1] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0 + 1
[1,3,2,4,5] => [2,4,3,5,1] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1 + 1
[1,3,2,5,4] => [2,4,3,1,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[1,3,4,2,5] => [2,5,3,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,5,2,4] => [2,5,3,1,4] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 + 1
[1,3,5,4,2] => [2,1,3,5,4] => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 + 1
[1,4,2,3,5] => [2,4,5,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,4,3,2,5] => [2,5,4,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,3,4,5] => [3,2,4,5,1] => [5,2,3,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 2 + 1
[2,1,3,5,4] => [3,2,4,1,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 1
[2,1,4,3,5] => [3,2,5,4,1] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0 + 1
[2,1,4,5,3] => [3,2,1,4,5] => [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0 + 1
[2,1,5,3,4] => [3,2,5,1,4] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 1 + 1
[2,1,5,4,3] => [3,2,1,5,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? = 1 + 1
[2,3,1,4,5] => [4,2,3,5,1] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0 + 1
[2,3,1,5,4] => [4,2,3,1,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[2,3,4,1,5] => [5,2,3,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,5,1,4] => [5,2,3,1,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[2,4,1,3,5] => [4,2,5,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,4,1,5,3] => [4,2,1,3,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[2,4,3,1,5] => [5,2,4,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,4,3,5,1] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[2,4,5,1,3] => [5,2,1,3,4] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1 + 1
[2,4,5,3,1] => [1,2,5,3,4] => [1,2,5,4,3] => ([(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)
=> ? = 1 + 1
[2,5,1,3,4] => [4,2,5,1,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,5,1,4,3] => [4,2,1,5,3] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0 + 1
[2,5,3,1,4] => [5,2,4,1,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,5,3,4,1] => [1,2,4,5,3] => [1,2,5,4,3] => ([(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 + 1
[2,5,4,1,3] => [5,2,1,4,3] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0 + 1
[2,5,4,3,1] => [1,2,5,4,3] => [1,2,5,4,3] => ([(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 + 1
[3,1,2,4,5] => [3,4,2,5,1] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1 + 1
[3,1,2,5,4] => [3,4,2,1,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[3,1,4,2,5] => [3,5,2,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,1,5,2,4] => [3,5,2,1,4] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 + 1
[3,1,5,4,2] => [3,1,2,5,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? = 0 + 1
[3,2,1,4,5] => [4,3,2,5,1] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1 + 1
[3,2,1,5,4] => [4,3,2,1,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[3,2,4,1,5] => [5,3,2,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,2,5,1,4] => [5,3,2,1,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,2,5,4,1] => [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 + 1
[3,4,1,2,5] => [4,5,2,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,4,2,1,5] => [5,4,2,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,1,2,3,5] => [3,4,5,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,1,3,2,5] => [3,5,4,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,1,5,2,3] => [3,5,1,2,4] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 + 1
[4,1,5,3,2] => [3,1,5,2,4] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 0 + 1
[4,2,1,3,5] => [4,3,5,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,2,3,1,5] => [5,3,4,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,2,5,1,3] => [5,3,1,2,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,2,5,3,1] => [1,3,5,2,4] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0 + 1
[4,3,1,2,5] => [4,5,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,3,2,1,5] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,3,5,1,2] => [5,1,3,2,4] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0 + 1
[4,3,5,2,1] => [1,5,3,2,4] => [1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[1,2,3,5,4,6] => [2,3,4,6,5,1] => [6,2,3,5,4,1] => ([(0,2),(0,3),(0,4),(0,5),(1,17),(2,8),(2,9),(2,12),(3,7),(3,9),(3,11),(4,7),(4,8),(4,10),(5,1),(5,10),(5,11),(5,12),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(10,17),(11,15),(11,16),(11,17),(12,14),(12,15),(12,17),(13,19),(14,18),(14,19),(15,18),(15,19),(16,18),(16,19),(17,18),(18,6),(19,6)],20)
=> ? = 0 + 1
[1,2,4,3,5,6] => [2,3,5,4,6,1] => [6,2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(1,8),(1,18),(2,12),(2,13),(2,14),(2,18),(3,10),(3,11),(3,14),(3,18),(4,8),(4,9),(4,11),(4,13),(5,7),(5,9),(5,10),(5,12),(7,15),(7,19),(8,15),(8,20),(9,15),(9,16),(9,17),(10,16),(10,19),(10,23),(11,16),(11,20),(11,23),(12,17),(12,19),(12,23),(13,17),(13,20),(13,23),(14,23),(15,22),(16,21),(16,22),(17,21),(17,22),(18,19),(18,20),(18,23),(19,21),(19,22),(20,21),(20,22),(21,6),(22,6),(23,21)],24)
=> ? = 1 + 1
[1,2,4,3,6,5] => [2,3,5,4,1,6] => [5,2,4,3,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(1,19),(2,10),(2,12),(2,13),(2,19),(3,9),(3,11),(3,13),(3,19),(4,8),(4,11),(4,12),(4,19),(5,6),(5,8),(5,9),(5,10),(6,20),(8,15),(8,16),(8,20),(9,15),(9,17),(9,20),(10,16),(10,17),(10,20),(11,14),(11,15),(11,18),(12,14),(12,16),(12,18),(13,14),(13,17),(13,18),(14,22),(15,21),(15,22),(16,21),(16,22),(17,21),(17,22),(18,21),(18,22),(19,18),(19,20),(20,21),(21,7),(22,7)],23)
=> ? = 0 + 1
[1,2,4,5,3,6] => [2,3,6,4,5,1] => [6,2,5,4,3,1] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 0 + 1
[1,2,4,6,3,5] => [2,3,6,4,1,5] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,15),(1,16),(2,8),(2,11),(2,16),(2,18),(3,8),(3,10),(3,15),(3,17),(4,9),(4,13),(4,14),(4,17),(4,18),(5,10),(5,12),(5,13),(5,16),(5,18),(6,11),(6,12),(6,14),(6,15),(6,17),(8,21),(8,25),(9,23),(9,24),(10,21),(10,23),(10,25),(11,21),(11,24),(11,25),(12,23),(12,24),(12,25),(13,19),(13,23),(13,25),(14,19),(14,24),(14,25),(15,24),(15,25),(16,23),(16,25),(17,19),(17,21),(17,23),(17,24),(18,19),(18,21),(18,23),(18,24),(19,20),(19,22),(20,7),(21,20),(21,22),(22,7),(23,20),(23,22),(24,20),(24,22),(25,22)],26)
=> ? = 0 + 1
[1,2,4,6,5,3] => [2,3,1,4,6,5] => [3,2,1,4,6,5] => ([(0,3),(0,4),(0,5),(1,14),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,13),(7,14),(9,1),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(12,14),(13,8),(14,8)],15)
=> ? = 0 + 1
[1,2,5,3,4,6] => [2,3,5,6,4,1] => [6,2,5,4,3,1] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 0 + 1
[1,2,5,4,3,6] => [2,3,6,5,4,1] => [6,2,5,4,3,1] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 0 + 1
[1,3,2,4,5,6] => [2,4,3,5,6,1] => [6,3,2,4,5,1] => ([(0,2),(0,3),(0,4),(0,5),(1,17),(2,8),(2,9),(2,12),(3,7),(3,9),(3,11),(4,7),(4,8),(4,10),(5,1),(5,10),(5,11),(5,12),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(10,17),(11,15),(11,16),(11,17),(12,14),(12,15),(12,17),(13,19),(14,18),(14,19),(15,18),(15,19),(16,18),(16,19),(17,18),(18,6),(19,6)],20)
=> ? = 2 + 1
[1,3,2,4,6,5] => [2,4,3,5,1,6] => [5,3,2,4,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(1,17),(2,9),(2,11),(2,13),(2,17),(3,8),(3,11),(3,12),(3,17),(4,10),(4,12),(4,13),(4,17),(5,6),(5,8),(5,9),(5,10),(6,20),(8,15),(8,20),(8,21),(9,16),(9,20),(9,21),(10,15),(10,16),(10,20),(11,14),(11,21),(12,14),(12,15),(12,21),(13,14),(13,16),(13,21),(14,19),(15,18),(15,19),(16,18),(16,19),(17,20),(17,21),(18,7),(19,7),(20,18),(21,18),(21,19)],22)
=> ? = 1 + 1
[1,3,2,5,4,6] => [2,4,3,6,5,1] => [6,3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,5),(3,7),(3,9),(3,11),(4,5),(4,7),(4,8),(4,10),(5,19),(7,13),(7,14),(7,19),(8,13),(8,15),(8,19),(9,13),(9,16),(9,19),(10,14),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(13,17),(13,18),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,18)],20)
=> ? = 0 + 1
[1,3,2,5,6,4] => [2,4,3,1,5,6] => [4,3,2,1,5,6] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 0 + 1
[1,3,2,6,4,5] => [2,4,3,6,1,5] => [5,3,2,6,1,4] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,18),(1,19),(2,7),(2,13),(2,15),(3,7),(3,12),(3,15),(4,8),(4,9),(4,12),(4,15),(5,8),(5,10),(5,13),(5,15),(6,1),(6,9),(6,10),(6,12),(6,13),(7,17),(8,14),(8,21),(9,14),(9,18),(9,19),(9,21),(10,14),(10,18),(10,19),(10,21),(12,17),(12,18),(12,21),(13,17),(13,19),(13,21),(14,16),(14,20),(15,17),(15,21),(16,11),(17,20),(18,16),(18,20),(19,16),(19,20),(20,11),(21,16),(21,20)],22)
=> ? = 1 + 1
Description
The number of connected components of the Hasse diagram for the poset.
The following 4 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001568The smallest positive integer that does not appear twice in the partition.
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