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Your data matches 10 different statistics following compositions of up to 3 maps.
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Matching statistic: St001346
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
St001346: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 2
[2,1] => 1
[1,2,3] => 6
[1,3,2] => 2
[2,1,3] => 3
[2,3,1] => 2
[3,1,2] => 2
[3,2,1] => 1
[1,2,3,4] => 24
[1,2,4,3] => 6
[1,3,2,4] => 8
[1,3,4,2] => 4
[1,4,2,3] => 6
[1,4,3,2] => 2
[2,1,3,4] => 12
[2,1,4,3] => 3
[2,3,1,4] => 8
[2,3,4,1] => 6
[2,4,1,3] => 3
[2,4,3,1] => 2
[3,1,2,4] => 8
[3,1,4,2] => 4
[3,2,1,4] => 4
[3,2,4,1] => 3
[3,4,1,2] => 4
[3,4,2,1] => 2
[4,1,2,3] => 6
[4,1,3,2] => 2
[4,2,1,3] => 3
[4,2,3,1] => 2
[4,3,1,2] => 2
[4,3,2,1] => 1
[1,2,3,4,5] => 120
[1,2,3,5,4] => 24
[1,2,4,3,5] => 30
[1,2,4,5,3] => 12
[1,2,5,3,4] => 24
[1,2,5,4,3] => 6
[1,3,2,4,5] => 40
[1,3,2,5,4] => 8
[1,3,4,2,5] => 20
[1,3,4,5,2] => 12
[1,3,5,2,4] => 8
[1,3,5,4,2] => 4
[1,4,2,3,5] => 30
[1,4,2,5,3] => 12
[1,4,3,2,5] => 10
[1,4,3,5,2] => 6
[1,4,5,2,3] => 12
[1,4,5,3,2] => 4
Description
The number of parking functions that give the same permutation.
A '''parking function''' $(a_1,\dots,a_n)$ is a list of preferred parking spots of $n$ cars entering a one-way street. Once the cars have parked, the order of the cars gives a permutation of $\{1,\dots,n\}$. This statistic records the number of parking functions that yield the same permutation of cars.
Matching statistic: St000110
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> [2,1] => 2
[2,1] => [[.,.],.]
=> [1,2] => 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 6
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 3
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 2
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 2
[3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 24
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 12
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 8
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 8
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 8
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 4
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 6
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 3
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 4
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 6
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 4
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 3
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 6
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 3
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 4
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 2
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 6
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 3
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 2
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 120
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 60
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 40
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 40
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 40
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 20
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 30
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 15
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 20
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 30
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 20
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 15
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 30
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 15
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 10
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 10
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 20
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 10
Description
The number of permutations less than or equal to a permutation in left weak order.
This is the same as the number of permutations less than or equal to the given permutation in right weak order.
Matching statistic: St000100
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> [2,1] => ([],2)
=> 2
[2,1] => [[.,.],.]
=> [1,2] => ([(0,1)],2)
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => ([],3)
=> 6
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => ([(1,2)],3)
=> 3
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[2,3,1] => [[.,.],[.,.]]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([],4)
=> 24
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(2,3)],4)
=> 12
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(1,2),(1,3)],4)
=> 8
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(1,2),(1,3)],4)
=> 8
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> 8
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 4
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 6
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 3
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 6
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 6
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 3
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 3
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 6
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 3
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([],5)
=> 120
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(3,4)],5)
=> 60
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> 40
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> 40
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> 40
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> 20
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> 30
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> 15
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> 30
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> 30
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> 15
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> 15
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 20
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 20
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> 10
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> 10
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 20
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> 10
Description
The number of linear extensions of a poset.
Matching statistic: St000454
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 21%
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 21%
Values
[1,2] => {{1},{2}}
=> [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[2,1] => {{1,2}}
=> [2] => ([],2)
=> 0 = 1 - 1
[1,2,3] => {{1},{2},{3}}
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,3,2] => {{1},{2,3}}
=> [1,2] => ([(1,2)],3)
=> 1 = 2 - 1
[2,1,3] => {{1,2},{3}}
=> [2,1] => ([(0,2),(1,2)],3)
=> ? ∊ {2,2,6} - 1
[2,3,1] => {{1,2,3}}
=> [3] => ([],3)
=> 0 = 1 - 1
[3,1,2] => {{1,3},{2}}
=> [2,1] => ([(0,2),(1,2)],3)
=> ? ∊ {2,2,6} - 1
[3,2,1] => {{1,3},{2}}
=> [2,1] => ([(0,2),(1,2)],3)
=> ? ∊ {2,2,6} - 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3] => ([(2,3)],4)
=> 1 = 2 - 1
[1,4,2,3] => {{1},{2,4},{3}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[2,3,4,1] => {{1,2,3,4}}
=> [4] => ([],4)
=> 0 = 1 - 1
[2,4,1,3] => {{1,2,4},{3}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[3,1,2,4] => {{1,3},{2},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[3,1,4,2] => {{1,3,4},{2}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[3,4,2,1] => {{1,3},{2,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[4,1,2,3] => {{1,4},{2},{3}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[4,1,3,2] => {{1,4},{2},{3}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[4,2,1,3] => {{1,4},{2},{3}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[4,3,1,2] => {{1,4},{2,3}}
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[4,3,2,1] => {{1,4},{2,3}}
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,4] => ([(3,4)],5)
=> 1 = 2 - 1
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,4,5,3,2] => {{1},{2,4},{3,5}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,5,2,3,4] => {{1},{2,5},{3},{4}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,5,2,4,3] => {{1},{2,5},{3},{4}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,5,3,2,4] => {{1},{2,5},{3},{4}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,5,4,2,3] => {{1},{2,5},{3,4}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,5,4,3,2] => {{1},{2,5},{3,4}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[2,1,3,5,4] => {{1,2},{3},{4,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[2,1,4,3,5] => {{1,2},{3,4},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[2,1,4,5,3] => {{1,2},{3,4,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[2,1,5,3,4] => {{1,2},{3,5},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[2,1,5,4,3] => {{1,2},{3,5},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[2,3,1,4,5] => {{1,2,3},{4},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[2,3,1,5,4] => {{1,2,3},{4,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[2,3,4,1,5] => {{1,2,3,4},{5}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,3,4,5,1] => {{1,2,3,4,5}}
=> [5] => ([],5)
=> 0 = 1 - 1
[2,3,5,1,4] => {{1,2,3,5},{4}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,3,5,4,1] => {{1,2,3,5},{4}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,4,1,3,5] => {{1,2,4},{3},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[2,4,1,5,3] => {{1,2,4,5},{3}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,4,3,1,5] => {{1,2,4},{3},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[2,4,3,5,1] => {{1,2,4,5},{3}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,5,1,3,4] => {{1,2,5},{3},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[2,5,1,4,3] => {{1,2,5},{3},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[2,5,3,1,4] => {{1,2,5},{3},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[2,5,3,4,1] => {{1,2,5},{3},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[3,1,4,2,5] => {{1,3,4},{2},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[3,1,4,5,2] => {{1,3,4,5},{2}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,1,5,2,4] => {{1,3,5},{2},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[3,1,5,4,2] => {{1,3,5},{2},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[3,2,4,1,5] => {{1,3,4},{2},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[3,2,4,5,1] => {{1,3,4,5},{2}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,2,5,1,4] => {{1,3,5},{2},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[3,2,5,4,1] => {{1,3,5},{2},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[4,1,2,5,3] => {{1,4,5},{2},{3}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[4,1,3,5,2] => {{1,4,5},{2},{3}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[4,2,1,5,3] => {{1,4,5},{2},{3}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[4,2,3,5,1] => {{1,4,5},{2},{3}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,2,3,4,5,6] => {{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[1,2,3,4,6,5] => {{1},{2},{3},{4},{5,6}}
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,2,3,5,6,4] => {{1},{2},{3},{4,5,6}}
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,2,4,5,6,3] => {{1},{2},{3,4,5,6}}
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,3,4,5,6,2] => {{1},{2,3,4,5,6}}
=> [1,5] => ([(4,5)],6)
=> 1 = 2 - 1
[2,3,1,4,6,5] => {{1,2,3},{4},{5,6}}
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[2,3,4,1,6,5] => {{1,2,3,4},{5,6}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,3,4,5,6,1] => {{1,2,3,4,5,6}}
=> [6] => ([],6)
=> 0 = 1 - 1
[2,3,5,6,1,4] => {{1,2,3,5},{4,6}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,3,5,6,4,1] => {{1,2,3,5},{4,6}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,3,6,5,1,4] => {{1,2,3,6},{4,5}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,3,6,5,4,1] => {{1,2,3,6},{4,5}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
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: St001855
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001855: Signed permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 29%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001855: Signed permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 29%
Values
[1,2] => [.,[.,.]]
=> [2,1] => [2,1] => 2
[2,1] => [[.,.],.]
=> [1,2] => [1,2] => 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 6
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [2,3,1] => 3
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => 2
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 2
[3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 24
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,4,2,1] => 12
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,4,3,1] => 8
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,4,1] => 8
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,4,3,1] => 8
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,4,1] => 4
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => 6
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,4,2] => 3
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 4
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 6
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 4
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1,4] => 3
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => 6
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,4,2] => 3
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 4
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 2
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => 6
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,4,2] => 3
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => 2
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,5,3,2,1] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [3,5,4,2,1] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,5,2,1] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [3,5,4,2,1] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,5,2,1] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,5,4,3,1] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,4,5,3,1] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,5,4,1] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,5,1] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,5,4,1] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,5,1] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,5,4,3,1] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,4,5,3,1] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [2,3,5,4,1] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,4,3,5,1] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,5,4,1] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,5,1] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,5,4,3,1] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,4,5,3,1] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [2,3,5,4,1] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,4,3,5,1] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [2,3,5,4,1] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => 24
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,4,5,3,2] => 12
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,5,4,2] => 8
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,4,3,5,2] => 8
[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,5,4,2] => 8
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,3,4,5,2] => 4
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,4,5,3] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,4,2,1,5] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,4,5,3] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,3,1,5,4] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [2,4,3,1,5] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [3,2,4,1,5] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,4,5,3] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,3,1,5,4] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[2,5,3,4,1] => [[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [2,4,3,1,5] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,3,1,5,4] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [2,3,4,1,5] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => 24
[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,4,5,3,2] => 12
[3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,5,4,2] => 8
[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,4,3,5,2] => 8
[3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,5,4,2] => 8
[3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,3,4,5,2] => 4
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,4,3] => 6
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,4,5,3] => 3
[3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => 4
[3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => [1,4,3,2,5] => 6
[3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => 4
[3,2,5,4,1] => [[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => [1,3,4,2,5] => 3
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,4,5,3] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => [2,1,4,3,5] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [3,2,1,4,5] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
[3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,4,5,3] => ? ∊ {2,3,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,30,30,30,30,40,40,40,60,120}
Description
The number of signed permutations less than or equal to a signed permutation in left weak order.
Matching statistic: St000699
Values
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,2}
[2,1] => ([],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,2}
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,3,6}
[1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,3,6}
[2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,3,6}
[2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,3,6}
[3,1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,3,6}
[3,2,1] => ([],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,2,2,2,3,6}
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[3,4,2,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[4,1,3,2] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[4,3,1,2] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[4,3,2,1] => ([],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,3,4,2,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,3,4,6,2,5] => ([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,3,5,2,6,4] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,3,5,4,2,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,3,6,4,2,5] => ([(0,3),(0,4),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,4,2,3,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,4,2,6,3,5] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,4,5,2,6,3] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,4,5,3,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,4,5,3,6,2] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,4,6,2,3,5] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,5,2,3,6,4] => ([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,5,2,4,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,5,2,4,6,3] => ([(0,3),(0,4),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,5,3,2,4,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,5,3,4,6,2] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,5,4,2,3,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,6,3,4,2,5] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,6,4,2,3,5] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,1,4,5,3,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,1,5,3,4,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,3,5,1,6,4] => ([(0,4),(0,5),(1,2),(2,3),(2,5),(3,4)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,4,1,5,3,6] => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,4,1,5,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,4,1,6,3,5] => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,4,1,6,5,3] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,4,5,1,3,6] => ([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,4,5,3,1,6] => ([(0,5),(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,4,5,3,6,1] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,4,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,5,1,3,4,6] => ([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,5,1,6,3,4] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(4,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,5,1,6,4,3] => ([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,5,3,1,4,6] => ([(0,4),(1,2),(1,3),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,5,3,4,1,6] => ([(0,5),(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,5,3,4,6,1] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,5,4,1,6,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,5),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,6,4,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[3,1,4,5,2,6] => ([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[3,1,5,2,4,6] => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
Description
The toughness times the least common multiple of 1,...,n-1 of a non-complete graph.
A graph $G$ is $t$-tough if $G$ cannot be split into $k$ different connected components by the removal of fewer than $tk$ vertices for all integers $k>1$.
The toughness of $G$ is the maximal number $t$ such that $G$ is $t$-tough. It is a rational number except for the complete graph, where it is infinity. The toughness of a disconnected graph is zero.
This statistic is the toughness multiplied by the least common multiple of $1,\dots,n-1$, where $n$ is the number of vertices of $G$.
Matching statistic: St001645
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 21%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 21%
Values
[1,2] => [2] => ([],2)
=> ([],1)
=> 1
[2,1] => [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,2,3] => [3] => ([],3)
=> ([],1)
=> 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[2,1,3] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {2,6}
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[3,1,2] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {2,6}
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,3,4] => [4] => ([],4)
=> ([],1)
=> 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,8,8,12,24}
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,8,8,12,24}
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,8,8,12,24}
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,8,8,12,24}
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,8,8,12,24}
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,8,8,12,24}
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,8,8,12,24}
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,8,8,12,24}
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,8,8,12,24}
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,8,8,12,24}
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,8,8,12,24}
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,8,8,12,24}
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,8,8,12,24}
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,8,8,12,24}
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,8,8,12,24}
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,4,4,4,6,6,6,6,8,8,8,12,24}
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,2,3,4,5] => [5] => ([],5)
=> ([],1)
=> 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,1,3,4,5] => [1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[2,1,4,3,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[2,1,5,3,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[2,4,3,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[2,5,4,3,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[3,5,4,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,5,3,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[5,4,3,2,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,2,3,4,5,6] => [6] => ([],6)
=> ([],1)
=> 1
[1,2,3,4,6,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2
[1,2,3,5,6,4] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2
[1,2,3,6,5,4] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,4,5,6,3] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2
[1,2,4,6,5,3] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,5,6,4,3] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,6,5,4,3] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,3,4,5,6,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2
[1,3,4,6,5,2] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,3,5,6,4,2] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,3,6,5,4,2] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,4,5,6,3,2] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,4,6,5,3,2] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,5,6,4,3,2] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,6,5,4,3,2] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[2,1,6,5,4,3] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2
[2,3,4,6,5,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,3,5,6,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
Description
The pebbling number of a connected graph.
Matching statistic: St000422
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 11%
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 11%
Values
[1,2] => [.,[.,.]]
=> [[[]]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,2}
[2,1] => [[.,.],.]
=> [[],[]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,2}
[1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,2,2,2,3,6}
[1,3,2] => [.,[[.,.],.]]
=> [[[],[]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,3,6}
[2,1,3] => [[.,.],[.,.]]
=> [[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,2,2,2,3,6}
[2,3,1] => [[.,.],[.,.]]
=> [[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,2,2,2,3,6}
[3,1,2] => [[.,[.,.]],.]
=> [[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,2,2,2,3,6}
[3,2,1] => [[[.,.],.],.]
=> [[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,3,6}
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,6,6,6,6,8,8,8,12,24}
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,6,6,6,6,8,8,8,12,24}
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,6,6,6,6,8,8,8,12,24}
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,6,6,6,6,8,8,8,12,24}
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,6,6,6,6,8,8,8,12,24}
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,6,6,6,6,8,8,8,12,24}
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,6,6,6,6,8,8,8,12,24}
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,6,6,6,6,8,8,8,12,24}
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,6,6,6,6,8,8,8,12,24}
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,6,6,6,6,8,8,8,12,24}
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,6,6,6,6,8,8,8,12,24}
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,6,6,6,6,8,8,8,12,24}
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,6,6,6,6,8,8,8,12,24}
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,6,6,6,6,8,8,8,12,24}
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,6,6,6,6,8,8,8,12,24}
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,6,6,6,6,8,8,8,12,24}
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,6,6,6,6,8,8,8,12,24}
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [[[[]]],[]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,6,6,6,6,8,8,8,12,24}
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [[[],[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,6,6,6,6,8,8,8,12,24}
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,6,6,6,6,8,8,8,12,24}
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,6,6,6,6,8,8,8,12,24}
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,6,6,6,6,8,8,8,12,24}
[4,3,2,1] => [[[[.,.],.],.],.]
=> [[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [[[[[]],[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [[[],[[],[]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [[[],[[],[]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [[[],[[],[]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [[[],[],[[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [[[],[],[[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [[[],[],[[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> [[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
=> [[[[],[]],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[1,5,3,2,4] => [.,[[[.,.],[.,.]],.]]
=> [[[],[[]],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> [[[],[[]],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,5,4,2,3] => [.,[[[.,[.,.]],.],.]]
=> [[[[]],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[3,2,5,1,4] => [[[.,.],.],[[.,.],.]]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[3,5,2,1,4] => [[[.,.],.],[[.,.],.]]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[3,5,2,4,1] => [[[.,.],.],[[.,.],.]]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[5,2,1,4,3] => [[[.,.],[[.,.],.]],.]
=> [[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[5,2,4,1,3] => [[[.,.],[[.,.],.]],.]
=> [[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> [[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[5,4,1,3,2] => [[[.,[[.,.],.]],.],.]
=> [[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[1,2,5,3,4,6] => [.,[.,[[.,[.,.]],[.,.]]]]
=> [[[[[]],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[1,2,5,3,6,4] => [.,[.,[[.,[.,.]],[.,.]]]]
=> [[[[[]],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[1,2,5,6,3,4] => [.,[.,[[.,[.,.]],[.,.]]]]
=> [[[[[]],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[2,1,5,3,4,6] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[2,1,5,3,6,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[2,1,5,6,3,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[2,5,1,3,4,6] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[2,5,1,3,6,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[2,5,1,6,3,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[2,5,3,1,4,6] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[2,5,3,1,6,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[2,5,3,4,1,6] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[2,5,3,4,6,1] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[2,5,3,6,1,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[2,5,3,6,4,1] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[2,5,6,1,3,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[2,5,6,3,1,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[2,5,6,3,4,1] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[5,3,1,2,4,6] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[5,3,1,2,6,4] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[5,3,1,4,2,6] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[5,3,1,4,6,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[5,3,1,6,2,4] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[5,3,1,6,4,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[5,3,4,1,2,6] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[5,3,4,1,6,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[5,3,4,6,1,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[5,3,6,1,2,4] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[5,3,6,1,4,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[5,3,6,4,1,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[5,6,3,1,2,4] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[5,6,3,1,4,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[5,6,3,4,1,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[6,1,4,2,3,5] => [[.,[[.,[.,.]],[.,.]]],.]
=> [[[[]],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 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: St000264
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 4%
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 4%
Values
[1,2] => [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2}
[2,1] => [2,1] => [1,2] => ([],2)
=> ? ∊ {1,2}
[1,2,3] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,2,2,2,3,6}
[1,3,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,2,2,2,3,6}
[2,1,3] => [2,1,3] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,2,2,2,3,6}
[2,3,1] => [2,3,1] => [1,2,3] => ([],3)
=> ? ∊ {1,2,2,2,3,6}
[3,1,2] => [3,1,2] => [1,2,3] => ([],3)
=> ? ∊ {1,2,2,2,3,6}
[3,2,1] => [3,2,1] => [1,2,3] => ([],3)
=> ? ∊ {1,2,2,2,3,6}
[1,2,3,4] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[1,2,4,3] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[1,3,2,4] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[1,3,4,2] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[1,4,2,3] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[1,4,3,2] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[2,1,3,4] => [2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[2,1,4,3] => [2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[2,3,1,4] => [2,4,1,3] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[2,3,4,1] => [2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[2,4,1,3] => [2,4,1,3] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[2,4,3,1] => [2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[3,1,2,4] => [3,1,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[3,1,4,2] => [3,1,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[3,2,1,4] => [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[3,2,4,1] => [3,2,4,1] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[3,4,1,2] => [3,4,1,2] => [1,2,3,4] => ([],4)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[3,4,2,1] => [3,4,2,1] => [1,2,3,4] => ([],4)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[4,1,2,3] => [4,1,3,2] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[4,1,3,2] => [4,1,3,2] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[4,2,1,3] => [4,2,1,3] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[4,2,3,1] => [4,2,3,1] => [1,2,3,4] => ([],4)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[4,3,1,2] => [4,3,1,2] => [1,2,3,4] => ([],4)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => ([],4)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,12,24}
[1,2,3,4,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,2,3,5,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,2,4,3,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,2,4,5,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,2,5,3,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,2,5,4,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,3,2,4,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,3,2,5,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,3,4,2,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,3,4,5,2] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,3,5,2,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,4,2,3,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,4,2,5,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,4,3,2,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,4,3,5,2] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[1,4,5,3,2] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120}
[2,3,1,4,5,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,3,1,4,6,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,3,1,5,4,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,3,1,5,6,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,3,1,6,4,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,3,1,6,5,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,4,1,3,5,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,4,1,3,6,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,4,1,5,3,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,4,1,5,6,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,4,1,6,3,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,4,1,6,5,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,5,1,3,4,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,5,1,3,6,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,5,1,4,3,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,5,1,4,6,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,5,1,6,3,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,5,1,6,4,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,6,1,3,4,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,6,1,3,5,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,6,1,4,3,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,6,1,4,5,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,6,1,5,3,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[2,6,1,5,4,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[3,2,4,1,5,6] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[3,2,4,1,6,5] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[3,2,5,1,4,6] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[3,2,5,1,6,4] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[3,2,6,1,4,5] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[3,2,6,1,5,4] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[4,2,3,1,5,6] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[4,2,3,1,6,5] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[4,2,5,1,3,6] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[4,2,5,1,6,3] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[4,2,6,1,3,5] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[4,2,6,1,5,3] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[4,3,2,5,1,6] => [4,3,2,6,1,5] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[4,3,2,6,1,5] => [4,3,2,6,1,5] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001232
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 21%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 21%
Values
[1,2] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[2,1] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,6} - 1
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,6} - 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,6} - 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,3,3,3,4,4,4,6,6,6,6,8,8,8,12,24} - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,5,2,3,4] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,5,2,4,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,5,3,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,5,4,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,15,15,15,15,20,20,20,20,24,24,24,24,24,30,30,30,30,40,40,40,60,120} - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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