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Your data matches 18 different statistics following compositions of up to 3 maps.
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Matching statistic: St001948
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001948: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001948: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1,2] => [1,2] => 1
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,3,2] => [3,1,2] => 0
[.,[[.,.],.]]
=> [2,3,1] => [1,2,3] => [1,2,3] => 2
[[.,.],[.,.]]
=> [1,3,2] => [1,2,3] => [1,2,3] => 2
[[.,[.,.]],.]
=> [2,1,3] => [1,2,3] => [1,2,3] => 2
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 2
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,4,2,3] => [1,4,2,3] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,2,4,3] => [4,1,2,3] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,3,4,2] => [3,1,2,4] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 3
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,2,4,3] => [4,1,2,3] => 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 3
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 3
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 3
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 3
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 3
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 3
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 3
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,5,2,4,3] => [5,1,4,2,3] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,4,2,5,3] => [4,1,5,2,3] => 0
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,3,4,2,5] => [1,3,2,4,5] => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,4,2,3,5] => [1,2,4,3,5] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,3,5,2,4] => [3,5,1,2,4] => 2
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,5,2,3,4] => 2
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,2,4,3,5] => [1,4,2,3,5] => 2
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,3,5,2,4] => [3,5,1,2,4] => 2
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,4,5,2,3] => [4,5,1,2,3] => 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,3,2,4,5] => [1,3,4,2,5] => 2
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,2,4,5,3] => [4,1,2,3,5] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,3,4,5,2] => [3,1,2,4,5] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,2,5,3,4] => [1,5,2,3,4] => 2
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,2,4,3,5] => [1,4,2,3,5] => 2
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => 2
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,2,4,5,3] => [4,1,2,3,5] => 2
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => 2
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [1,3,2,4,5] => [1,3,4,2,5] => 2
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [1,4,2,3,5] => [1,2,4,3,5] => 2
Description
The number of augmented double ascents of a permutation.
An augmented double ascent of a permutation $\pi$ is a double ascent of the augmented permutation $\tilde\pi$ obtained from $\pi$ by adding an initial $0$.
A double ascent of $\tilde\pi$ then is a position $i$ such that $\tilde\pi(i) < \tilde\pi(i+1) < \tilde\pi(i+2)$.
Matching statistic: St001879
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 60%
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 60%
Values
[.,[.,.]]
=> [1,1,0,0]
=> [[.,.],.]
=> ([(0,1)],2)
=> ? = 1
[[.,.],.]
=> [1,0,1,0]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ? = 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 2
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 2
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 2
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 0
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 2
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 2
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 2
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 2
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2
[[.,[.,[[.,.],.]]],.]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2
[[.,[[.,[.,.]],.]],.]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2
[[.,[[[.,.],.],.]],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2
[[[.,.],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2
[[[.,[[.,.],.]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[[.,[.,.]],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[[[.,.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001880
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 60%
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 60%
Values
[.,[.,.]]
=> [1,1,0,0]
=> [[.,.],.]
=> ([(0,1)],2)
=> ? = 1 + 1
[[.,.],.]
=> [1,0,1,0]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ? = 1 + 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 0 + 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 1
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 1 + 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 1 + 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2 + 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2 + 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 2 + 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 2 + 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2 + 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2 + 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2 + 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 2 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 2 + 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 2 + 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2 + 1
[[.,[.,[[.,.],.]]],.]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2 + 1
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2 + 1
[[.,[[.,[.,.]],.]],.]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 1
[[.,[[[.,.],.],.]],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2 + 1
[[[.,.],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 1
[[[.,[[.,.],.]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[[[.,[.,.]],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[[[[.,.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000031
Mp00008: Binary trees —to complete tree⟶ Ordered trees
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 100%
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2 = 1 + 1
[[.,.],.]
=> [[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2 = 1 + 1
[.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> [.,[[.,[[.,[.,.]],.]],.]]
=> [4,3,5,2,6,1] => 1 = 0 + 1
[.,[[.,.],.]]
=> [[],[[[],[]],[]]]
=> [.,[[[.,[.,.]],[.,.]],.]]
=> [3,2,5,4,6,1] => 3 = 2 + 1
[[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> [[.,[.,.]],[[.,[.,.]],.]]
=> [2,1,5,4,6,3] => 3 = 2 + 1
[[.,[.,.]],.]
=> [[[],[[],[]]],[]]
=> [[.,[[.,[.,.]],.]],[.,.]]
=> [3,2,4,1,6,5] => 3 = 2 + 1
[[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> [[[.,[.,.]],[.,.]],[.,.]]
=> [2,1,4,3,6,5] => 3 = 2 + 1
[.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [5,4,6,3,7,2,8,1] => 2 = 1 + 1
[.,[.,[[.,.],.]]]
=> [[],[[],[[[],[]],[]]]]
=> [.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [4,3,6,5,7,2,8,1] => 2 = 1 + 1
[.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> [.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [3,2,6,5,7,4,8,1] => 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [[],[[[],[[],[]]],[]]]
=> [.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [4,3,5,2,7,6,8,1] => 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> [.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [3,2,5,4,7,6,8,1] => 4 = 3 + 1
[[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> [[.,[.,.]],[[.,[[.,[.,.]],.]],.]]
=> [2,1,6,5,7,4,8,3] => 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> [[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [2,1,5,4,7,6,8,3] => 4 = 3 + 1
[[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> [[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [3,2,4,1,7,6,8,5] => 4 = 3 + 1
[[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> [[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> [2,1,4,3,7,6,8,5] => 4 = 3 + 1
[[.,[.,[.,.]]],.]
=> [[[],[[],[[],[]]]],[]]
=> [[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
=> [4,3,5,2,6,1,8,7] => 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [[[],[[[],[]],[]]],[]]
=> [[.,[[[.,[.,.]],[.,.]],.]],[.,.]]
=> [3,2,5,4,6,1,8,7] => 4 = 3 + 1
[[[.,.],[.,.]],.]
=> [[[[],[]],[[],[]]],[]]
=> [[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
=> [2,1,5,4,6,3,8,7] => 4 = 3 + 1
[[[.,[.,.]],.],.]
=> [[[[],[[],[]]],[]],[]]
=> [[[.,[[.,[.,.]],.]],[.,.]],[.,.]]
=> [3,2,4,1,6,5,8,7] => 4 = 3 + 1
[[[[.,.],.],.],.]
=> [[[[[],[]],[]],[]],[]]
=> [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [2,1,4,3,6,5,8,7] => 4 = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [[],[[],[[],[[],[[],[]]]]]]
=> [.,[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],.]]
=> [6,5,7,4,8,3,9,2,10,1] => ? = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [[],[[],[[],[[[],[]],[]]]]]
=> [.,[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],.]]
=> [5,4,7,6,8,3,9,2,10,1] => ? = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [[],[[],[[[],[]],[[],[]]]]]
=> [.,[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],.]]
=> [4,3,7,6,8,5,9,2,10,1] => ? = 2 + 1
[.,[.,[[.,[.,.]],.]]]
=> [[],[[],[[[],[[],[]]],[]]]]
=> [.,[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],.]]
=> [5,4,6,3,8,7,9,2,10,1] => ? = 2 + 1
[.,[.,[[[.,.],.],.]]]
=> [[],[[],[[[[],[]],[]],[]]]]
=> [.,[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],.]]
=> [4,3,6,5,8,7,9,2,10,1] => ? = 2 + 1
[.,[[.,.],[.,[.,.]]]]
=> [[],[[[],[]],[[],[[],[]]]]]
=> [.,[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],.]]
=> [3,2,7,6,8,5,9,4,10,1] => ? = 2 + 1
[.,[[.,.],[[.,.],.]]]
=> [[],[[[],[]],[[[],[]],[]]]]
=> [.,[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],.]]
=> [3,2,6,5,8,7,9,4,10,1] => ? = 2 + 1
[.,[[.,[.,.]],[.,.]]]
=> [[],[[[],[[],[]]],[[],[]]]]
=> [.,[[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],.]]
=> [4,3,5,2,8,7,9,6,10,1] => ? = 2 + 1
[.,[[[.,.],.],[.,.]]]
=> [[],[[[[],[]],[]],[[],[]]]]
=> [.,[[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],.]]
=> [3,2,5,4,8,7,9,6,10,1] => ? = 2 + 1
[.,[[.,[.,[.,.]]],.]]
=> [[],[[[],[[],[[],[]]]],[]]]
=> [.,[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],.]]
=> [5,4,6,3,7,2,9,8,10,1] => ? = 2 + 1
[.,[[.,[[.,.],.]],.]]
=> [[],[[[],[[[],[]],[]]],[]]]
=> [.,[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],.]]
=> [4,3,6,5,7,2,9,8,10,1] => ? = 2 + 1
[.,[[[.,.],[.,.]],.]]
=> [[],[[[[],[]],[[],[]]],[]]]
=> [.,[[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],.]]
=> [3,2,6,5,7,4,9,8,10,1] => ? = 2 + 1
[.,[[[.,[.,.]],.],.]]
=> [[],[[[[],[[],[]]],[]],[]]]
=> [.,[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],.]]
=> [4,3,5,2,7,6,9,8,10,1] => ? = 2 + 1
[.,[[[[.,.],.],.],.]]
=> [[],[[[[[],[]],[]],[]],[]]]
=> [.,[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],.]]
=> [3,2,5,4,7,6,9,8,10,1] => ? = 4 + 1
[[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> [[.,[.,.]],[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [2,1,7,6,8,5,9,4,10,3] => ? = 2 + 1
[[.,.],[.,[[.,.],.]]]
=> [[[],[]],[[],[[[],[]],[]]]]
=> [[.,[.,.]],[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [2,1,6,5,8,7,9,4,10,3] => ? = 2 + 1
[[.,.],[[.,.],[.,.]]]
=> [[[],[]],[[[],[]],[[],[]]]]
=> [[.,[.,.]],[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [2,1,5,4,8,7,9,6,10,3] => ? = 2 + 1
[[.,.],[[.,[.,.]],.]]
=> [[[],[]],[[[],[[],[]]],[]]]
=> [[.,[.,.]],[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [2,1,6,5,7,4,9,8,10,3] => ? = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [[[],[]],[[[[],[]],[]],[]]]
=> [[.,[.,.]],[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [2,1,5,4,7,6,9,8,10,3] => ? = 4 + 1
[[.,[.,.]],[.,[.,.]]]
=> [[[],[[],[]]],[[],[[],[]]]]
=> [[.,[[.,[.,.]],.]],[[.,[[.,[.,.]],.]],.]]
=> [3,2,4,1,8,7,9,6,10,5] => ? = 2 + 1
[[.,[.,.]],[[.,.],.]]
=> [[[],[[],[]]],[[[],[]],[]]]
=> [[.,[[.,[.,.]],.]],[[[.,[.,.]],[.,.]],.]]
=> [3,2,4,1,7,6,9,8,10,5] => ? = 4 + 1
[[[.,.],.],[.,[.,.]]]
=> [[[[],[]],[]],[[],[[],[]]]]
=> [[[.,[.,.]],[.,.]],[[.,[[.,[.,.]],.]],.]]
=> [2,1,4,3,8,7,9,6,10,5] => ? = 2 + 1
[[[.,.],.],[[.,.],.]]
=> [[[[],[]],[]],[[[],[]],[]]]
=> [[[.,[.,.]],[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [2,1,4,3,7,6,9,8,10,5] => ? = 4 + 1
[[.,[.,[.,.]]],[.,.]]
=> [[[],[[],[[],[]]]],[[],[]]]
=> [[.,[[.,[[.,[.,.]],.]],.]],[[.,[.,.]],.]]
=> [4,3,5,2,6,1,9,8,10,7] => ? = 2 + 1
[[.,[[.,.],.]],[.,.]]
=> [[[],[[[],[]],[]]],[[],[]]]
=> [[.,[[[.,[.,.]],[.,.]],.]],[[.,[.,.]],.]]
=> [3,2,5,4,6,1,9,8,10,7] => ? = 4 + 1
[[[.,.],[.,.]],[.,.]]
=> [[[[],[]],[[],[]]],[[],[]]]
=> [[[.,[.,.]],[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [2,1,5,4,6,3,9,8,10,7] => ? = 4 + 1
[[[.,[.,.]],.],[.,.]]
=> [[[[],[[],[]]],[]],[[],[]]]
=> [[[.,[[.,[.,.]],.]],[.,.]],[[.,[.,.]],.]]
=> [3,2,4,1,6,5,9,8,10,7] => ? = 4 + 1
[[[[.,.],.],.],[.,.]]
=> [[[[[],[]],[]],[]],[[],[]]]
=> [[[[.,[.,.]],[.,.]],[.,.]],[[.,[.,.]],.]]
=> [2,1,4,3,6,5,9,8,10,7] => ? = 4 + 1
[[.,[.,[.,[.,.]]]],.]
=> [[[],[[],[[],[[],[]]]]],[]]
=> [[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],[.,.]]
=> [5,4,6,3,7,2,8,1,10,9] => ? = 2 + 1
[[.,[.,[[.,.],.]]],.]
=> [[[],[[],[[[],[]],[]]]],[]]
=> [[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],[.,.]]
=> [4,3,6,5,7,2,8,1,10,9] => ? = 2 + 1
[[.,[[.,.],[.,.]]],.]
=> [[[],[[[],[]],[[],[]]]],[]]
=> [[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],[.,.]]
=> [3,2,6,5,7,4,8,1,10,9] => ? = 2 + 1
[[.,[[.,[.,.]],.]],.]
=> [[[],[[[],[[],[]]],[]]],[]]
=> [[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],[.,.]]
=> [4,3,5,2,7,6,8,1,10,9] => ? = 2 + 1
[[.,[[[.,.],.],.]],.]
=> [[[],[[[[],[]],[]],[]]],[]]
=> [[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],[.,.]]
=> [3,2,5,4,7,6,8,1,10,9] => ? = 4 + 1
[[[.,.],[.,[.,.]]],.]
=> [[[[],[]],[[],[[],[]]]],[]]
=> [[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],[.,.]]
=> [2,1,6,5,7,4,8,3,10,9] => ? = 2 + 1
[[[.,.],[[.,.],.]],.]
=> [[[[],[]],[[[],[]],[]]],[]]
=> [[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],[.,.]]
=> [2,1,5,4,7,6,8,3,10,9] => ? = 4 + 1
[[[.,[.,.]],[.,.]],.]
=> [[[[],[[],[]]],[[],[]]],[]]
=> [[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],[.,.]]
=> [3,2,4,1,7,6,8,5,10,9] => ? = 4 + 1
[[[[.,.],.],[.,.]],.]
=> [[[[[],[]],[]],[[],[]]],[]]
=> [[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],[.,.]]
=> [2,1,4,3,7,6,8,5,10,9] => ? = 4 + 1
[[[.,[.,[.,.]]],.],.]
=> [[[[],[[],[[],[]]]],[]],[]]
=> [[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],[.,.]]
=> [4,3,5,2,6,1,8,7,10,9] => ? = 2 + 1
[[[.,[[.,.],.]],.],.]
=> [[[[],[[[],[]],[]]],[]],[]]
=> [[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],[.,.]]
=> [3,2,5,4,6,1,8,7,10,9] => ? = 4 + 1
[[[[.,.],[.,.]],.],.]
=> [[[[[],[]],[[],[]]],[]],[]]
=> [[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],[.,.]]
=> [2,1,5,4,6,3,8,7,10,9] => ? = 4 + 1
[[[[.,[.,.]],.],.],.]
=> [[[[[],[[],[]]],[]],[]],[]]
=> [[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],[.,.]]
=> [3,2,4,1,6,5,8,7,10,9] => ? = 4 + 1
[[[[[.,.],.],.],.],.]
=> [[[[[[],[]],[]],[]],[]],[]]
=> [[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]]
=> [2,1,4,3,6,5,8,7,10,9] => 5 = 4 + 1
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St001875
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 80%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 80%
Values
[.,[.,.]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[[.,.],.]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[.,[.,[.,.]]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[.,[[.,.],.]]
=> [2,3,1] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 2 + 2
[[.,.],[.,.]]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4 = 2 + 2
[[.,[.,.]],.]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4 = 2 + 2
[[[.,.],.],.]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1 + 2
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 1 + 2
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 1 + 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 1 + 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 + 2
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ? = 1 + 2
[[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 5 = 3 + 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 5 = 3 + 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 5 = 3 + 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? = 1 + 2
[[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 5 = 3 + 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 5 = 3 + 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5 = 3 + 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([],5)
=> ?
=> ? = 0 + 2
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(3,4)],5)
=> ?
=> ? = 0 + 2
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(1,11),(1,13),(2,11),(2,12),(3,4),(3,5),(3,12),(3,13),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,15),(6,17),(7,15),(7,18),(8,16),(8,17),(9,16),(9,18),(10,15),(10,16),(11,14),(12,6),(12,7),(12,14),(13,8),(13,9),(13,14),(14,17),(14,18),(15,19),(16,19),(17,19),(18,19)],20)
=> ? = 2 + 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
=> ? = 2 + 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ? = 2 + 2
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ([(0,1),(0,2),(1,12),(2,3),(2,4),(2,5),(2,12),(3,8),(3,10),(3,11),(4,7),(4,9),(4,11),(5,6),(5,9),(5,10),(6,13),(6,14),(7,13),(7,15),(8,14),(8,15),(9,13),(9,16),(10,14),(10,16),(11,15),(11,16),(12,6),(12,7),(12,8),(13,17),(14,17),(15,17),(16,17)],18)
=> ? = 2 + 2
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1),(0,2),(1,11),(2,3),(2,4),(2,11),(3,8),(3,10),(4,5),(4,9),(4,10),(5,6),(5,7),(6,13),(7,13),(8,12),(9,7),(9,12),(10,6),(10,12),(11,8),(11,9),(12,13)],14)
=> ? = 2 + 2
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,8),(2,6),(2,7),(3,10),(3,11),(4,9),(4,11),(5,9),(5,10),(6,12),(7,12),(8,12),(9,13),(10,13),(11,1),(11,2),(11,13),(13,7),(13,8)],14)
=> ? = 2 + 2
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12)
=> ? = 2 + 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(2,8),(2,9),(3,9),(3,11),(3,12),(4,8),(4,10),(4,12),(5,7),(5,10),(5,11),(7,13),(7,14),(8,13),(8,15),(9,14),(9,15),(10,13),(10,16),(11,14),(11,16),(12,15),(12,16),(13,17),(14,17),(15,17),(16,1),(16,17),(17,6)],18)
=> ? = 2 + 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
=> ? = 2 + 2
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,9),(1,10),(2,8),(2,10),(3,7),(4,6),(5,1),(5,2),(5,6),(6,8),(6,9),(8,11),(9,11),(10,3),(10,11),(11,7)],12)
=> ? = 2 + 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> ? = 2 + 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 4 + 2
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> ? = 2 + 2
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 2 + 2
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 2 + 2
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 2 + 2
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 4 + 2
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
=> ? = 2 + 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? = 4 + 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 2 + 2
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 4 + 2
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
=> ? = 2 + 2
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 4 + 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 4 + 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? = 4 + 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 6 = 4 + 2
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 2 + 2
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 2 + 2
[[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
=> ? = 2 + 2
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 2 + 2
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 4 + 2
[[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 2
[[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 4 + 2
[[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? = 4 + 2
[[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 6 = 4 + 2
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ? = 2 + 2
[[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 4 + 2
[[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 6 = 4 + 2
[[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 6 = 4 + 2
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St000718
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 80%
Mp00046: Ordered trees —to graph⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 80%
Values
[.,[.,.]]
=> [[[]]]
=> ([(0,2),(1,2)],3)
=> 3 = 1 + 2
[[.,.],.]
=> [[],[]]
=> ([(0,2),(1,2)],3)
=> 3 = 1 + 2
[.,[.,[.,.]]]
=> [[[[]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 2
[.,[[.,.],.]]
=> [[[],[]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 4 = 2 + 2
[[.,.],[.,.]]
=> [[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 2
[[.,[.,.]],.]
=> [[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 2
[[[.,.],.],.]
=> [[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 4 = 2 + 2
[.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1 + 2
[.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1 + 2
[.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1 + 2
[.,[[[.,.],.],.]]
=> [[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5 = 3 + 2
[[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[[.,.],[[.,.],.]]
=> [[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 2
[[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 + 2
[[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 2
[[.,[.,[.,.]]],.]
=> [[[[]]],[]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[[.,[[.,.],.]],.]
=> [[[],[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 2
[[[.,.],[.,.]],.]
=> [[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 2
[[[.,[.,.]],.],.]
=> [[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 2
[[[[.,.],.],.],.]
=> [[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5 = 3 + 2
[.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 0 + 2
[.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 0 + 2
[.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 2 + 2
[.,[.,[[.,[.,.]],.]]]
=> [[[[[]],[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 2 + 2
[.,[.,[[[.,.],.],.]]]
=> [[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 2 + 2
[.,[[.,.],[.,[.,.]]]]
=> [[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 2 + 2
[.,[[.,.],[[.,.],.]]]
=> [[[],[[],[]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 + 2
[.,[[.,[.,.]],[.,.]]]
=> [[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 2 + 2
[.,[[[.,.],.],[.,.]]]
=> [[[],[],[[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 2 + 2
[.,[[.,[.,[.,.]]],.]]
=> [[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 2 + 2
[.,[[.,[[.,.],.]],.]]
=> [[[[],[]],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 + 2
[.,[[[.,.],[.,.]],.]]
=> [[[],[[]],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 2 + 2
[.,[[[.,[.,.]],.],.]]
=> [[[[]],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 2 + 2
[.,[[[[.,.],.],.],.]]
=> [[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 6 = 4 + 2
[[.,.],[.,[.,[.,.]]]]
=> [[],[[[[]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 + 2
[[.,.],[.,[[.,.],.]]]
=> [[],[[[],[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 2 + 2
[[.,.],[[.,.],[.,.]]]
=> [[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 2 + 2
[[.,.],[[.,[.,.]],.]]
=> [[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 2 + 2
[[.,.],[[[.,.],.],.]]
=> [[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 4 + 2
[[.,[.,.]],[.,[.,.]]]
=> [[[]],[[[]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 + 2
[[.,[.,.]],[[.,.],.]]
=> [[[]],[[],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 4 + 2
[[[.,.],.],[.,[.,.]]]
=> [[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 2 + 2
[[[.,.],.],[[.,.],.]]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 4 + 2
[[.,[.,[.,.]]],[.,.]]
=> [[[[]]],[[]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 + 2
[[.,[[.,.],.]],[.,.]]
=> [[[],[]],[[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 4 + 2
[[[.,.],[.,.]],[.,.]]
=> [[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 4 + 2
[[[.,[.,.]],.],[.,.]]
=> [[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 4 + 2
[[[[.,.],.],.],[.,.]]
=> [[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 4 + 2
[[.,[.,[.,[.,.]]]],.]
=> [[[[[]]]],[]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 + 2
[[.,[.,[[.,.],.]]],.]
=> [[[[],[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 2 + 2
[[.,[[.,.],[.,.]]],.]
=> [[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 2 + 2
[[.,[[.,[.,.]],.]],.]
=> [[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 2 + 2
[[.,[[[.,.],.],.]],.]
=> [[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 4 + 2
[[[.,.],[.,[.,.]]],.]
=> [[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 2 + 2
[[[.,.],[[.,.],.]],.]
=> [[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 4 + 2
[[[.,[.,.]],[.,.]],.]
=> [[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 4 + 2
[[[[[.,.],.],.],.],.]
=> [[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 6 = 4 + 2
Description
The largest Laplacian eigenvalue of a graph if it is integral.
This statistic is undefined if the largest Laplacian eigenvalue of the graph is not integral.
Various results are collected in Section 3.9 of [1]
Matching statistic: St000454
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 80%
Mp00046: Ordered trees —to graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 80%
Values
[.,[.,.]]
=> [[[]]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
[[.,.],.]
=> [[],[]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
[.,[.,[.,.]]]
=> [[[[]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0
[.,[[.,.],.]]
=> [[[],[]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[.,.],[.,.]]
=> [[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[[.,[.,.]],.]
=> [[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[[[.,.],.],.]
=> [[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1
[.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1
[.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1
[.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1
[.,[[[.,.],.],.]]
=> [[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[.,.],[[.,.],.]]
=> [[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3
[[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3
[[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3
[[.,[.,[.,.]]],.]
=> [[[[]]],[]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[.,[[.,.],.]],.]
=> [[[],[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3
[[[.,.],[.,.]],.]
=> [[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3
[[[.,[.,.]],.],.]
=> [[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3
[[[[.,.],.],.],.]
=> [[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 0
[.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2
[.,[.,[[.,[.,.]],.]]]
=> [[[[[]],[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2
[.,[.,[[[.,.],.],.]]]
=> [[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[.,[[.,.],[.,[.,.]]]]
=> [[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2
[.,[[.,.],[[.,.],.]]]
=> [[[],[[],[]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[.,[[.,[.,.]],[.,.]]]
=> [[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2
[.,[[[.,.],.],[.,.]]]
=> [[[],[],[[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[.,[[.,[.,[.,.]]],.]]
=> [[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2
[.,[[.,[[.,.],.]],.]]
=> [[[[],[]],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[.,[[[.,.],[.,.]],.]]
=> [[[],[[]],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[.,[[[.,[.,.]],.],.]]
=> [[[[]],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[.,[[[[.,.],.],.],.]]
=> [[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[.,.],[.,[.,[.,.]]]]
=> [[],[[[[]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[.,.],[.,[[.,.],.]]]
=> [[],[[[],[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[.,.],[[.,.],[.,.]]]
=> [[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[.,.],[[.,[.,.]],.]]
=> [[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[.,.],[[[.,.],.],.]]
=> [[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[[.,[.,.]],[.,[.,.]]]
=> [[[]],[[[]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[.,[.,.]],[[.,.],.]]
=> [[[]],[[],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 4
[[[.,.],.],[.,[.,.]]]
=> [[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[[.,.],.],[[.,.],.]]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[[.,[.,[.,.]]],[.,.]]
=> [[[[]]],[[]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[.,[[.,.],.]],[.,.]]
=> [[[],[]],[[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 4
[[[.,.],[.,.]],[.,.]]
=> [[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 4
[[[.,[.,.]],.],[.,.]]
=> [[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 4
[[[[.,.],.],.],[.,.]]
=> [[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[[.,[.,[.,[.,.]]]],.]
=> [[[[[]]]],[]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[.,[.,[[.,.],.]]],.]
=> [[[[],[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[.,[[.,.],[.,.]]],.]
=> [[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[.,[[.,[.,.]],.]],.]
=> [[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[.,[[[.,.],.],.]],.]
=> [[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[[[.,.],[.,[.,.]]],.]
=> [[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[[.,.],[[.,.],.]],.]
=> [[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[[[.,[.,.]],[.,.]],.]
=> [[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 4
[[[[[.,.],.],.],.],.]
=> [[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
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: St000363
Mp00009: Binary trees —left rotate⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000363: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 60%
Mp00008: Binary trees —to complete tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000363: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 60%
Values
[.,[.,.]]
=> [[.,.],.]
=> [[[],[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[[.,.],.]
=> [.,[.,.]]
=> [[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[.,[.,[.,.]]]
=> [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [[.,[.,.]],.]
=> [[[],[[],[]]],[]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[.,.],[.,.]]
=> [[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[.,[.,.]],.]
=> [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[[.,.],.],.]
=> [.,[[.,.],.]]
=> [[],[[[],[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 3 = 2 + 1
[.,[.,[.,[.,.]]]]
=> [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9)
=> ? = 1 + 1
[.,[.,[[.,.],.]]]
=> [[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> ([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9)
=> ? = 1 + 1
[.,[[.,.],[.,.]]]
=> [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9)
=> ? = 1 + 1
[.,[[.,[.,.]],.]]
=> [[.,[.,[.,.]]],.]
=> [[[],[[],[[],[]]]],[]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ? = 1 + 1
[.,[[[.,.],.],.]]
=> [[.,[[.,.],.]],.]
=> [[[],[[[],[]],[]]],[]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ? = 3 + 1
[[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> ([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9)
=> ? = 1 + 1
[[.,.],[[.,.],.]]
=> [[[.,.],[.,.]],.]
=> [[[[],[]],[[],[]]],[]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,8),(6,8),(7,8)],9)
=> ? = 3 + 1
[[.,[.,.]],[.,.]]
=> [[[.,[.,.]],.],.]
=> [[[[],[[],[]]],[]],[]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ? = 3 + 1
[[[.,.],.],[.,.]]
=> [[[[.,.],.],.],.]
=> [[[[[],[]],[]],[]],[]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ? = 3 + 1
[[.,[.,[.,.]]],.]
=> [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ? = 1 + 1
[[.,[[.,.],.]],.]
=> [.,[.,[[.,.],.]]]
=> [[],[[],[[[],[]],[]]]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ? = 3 + 1
[[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,8),(6,8),(7,8)],9)
=> ? = 3 + 1
[[[.,[.,.]],.],.]
=> [.,[[.,[.,.]],.]]
=> [[],[[[],[[],[]]],[]]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ? = 3 + 1
[[[[.,.],.],.],.]
=> [.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ? = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ? = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [[[],[]],[[],[[[],[]],[]]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ? = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [[[],[]],[[[],[]],[[],[]]]]
=> ([(0,9),(1,9),(2,8),(3,8),(4,7),(5,7),(6,9),(6,10),(7,10),(8,10)],11)
=> ? = 2 + 1
[.,[.,[[.,[.,.]],.]]]
=> [[.,.],[[.,[.,.]],.]]
=> [[[],[]],[[[],[[],[]]],[]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ? = 2 + 1
[.,[.,[[[.,.],.],.]]]
=> [[.,.],[[[.,.],.],.]]
=> [[[],[]],[[[[],[]],[]],[]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ? = 2 + 1
[.,[[.,.],[.,[.,.]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [[[],[[],[]]],[[],[[],[]]]]
=> ([(0,10),(1,9),(2,7),(3,7),(4,8),(5,8),(6,9),(6,10),(7,9),(8,10)],11)
=> ? = 2 + 1
[.,[[.,.],[[.,.],.]]]
=> [[.,[.,.]],[[.,.],.]]
=> [[[],[[],[]]],[[[],[]],[]]]
=> ([(0,10),(1,9),(2,7),(3,7),(4,8),(5,8),(6,9),(6,10),(7,9),(8,10)],11)
=> ? = 2 + 1
[.,[[.,[.,.]],[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [[[],[[],[[],[]]]],[[],[]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ? = 2 + 1
[.,[[[.,.],.],[.,.]]]
=> [[.,[[.,.],.]],[.,.]]
=> [[[],[[[],[]],[]]],[[],[]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ? = 2 + 1
[.,[[.,[.,[.,.]]],.]]
=> [[.,[.,[.,[.,.]]]],.]
=> [[[],[[],[[],[[],[]]]]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ? = 2 + 1
[.,[[.,[[.,.],.]],.]]
=> [[.,[.,[[.,.],.]]],.]
=> [[[],[[],[[[],[]],[]]]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ? = 2 + 1
[.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [[[],[[[],[]],[[],[]]]],[]]
=> ([(0,10),(1,8),(2,8),(3,7),(4,7),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 2 + 1
[.,[[[.,[.,.]],.],.]]
=> [[.,[[.,[.,.]],.]],.]
=> [[[],[[[],[[],[]]],[]]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ? = 2 + 1
[.,[[[[.,.],.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> [[[],[[[[],[]],[]],[]]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ? = 4 + 1
[[.,.],[.,[.,[.,.]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [[[[],[]],[]],[[],[[],[]]]]
=> ([(0,10),(1,9),(2,7),(3,7),(4,8),(5,8),(6,9),(6,10),(7,9),(8,10)],11)
=> ? = 2 + 1
[[.,.],[.,[[.,.],.]]]
=> [[[.,.],.],[[.,.],.]]
=> [[[[],[]],[]],[[[],[]],[]]]
=> ([(0,10),(1,9),(2,7),(3,7),(4,8),(5,8),(6,9),(6,10),(7,9),(8,10)],11)
=> ? = 2 + 1
[[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> [[[[],[]],[[],[]]],[[],[]]]
=> ([(0,9),(1,9),(2,8),(3,8),(4,7),(5,7),(6,9),(6,10),(7,10),(8,10)],11)
=> ? = 2 + 1
[[.,.],[[.,[.,.]],.]]
=> [[[.,.],[.,[.,.]]],.]
=> [[[[],[]],[[],[[],[]]]],[]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ? = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [[[.,.],[[.,.],.]],.]
=> [[[[],[]],[[[],[]],[]]],[]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ? = 4 + 1
[[.,[.,.]],[.,[.,.]]]
=> [[[.,[.,.]],.],[.,.]]
=> [[[[],[[],[]]],[]],[[],[]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ? = 2 + 1
[[.,[.,.]],[[.,.],.]]
=> [[[.,[.,.]],[.,.]],.]
=> [[[[],[[],[]]],[[],[]]],[]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ? = 4 + 1
[[[.,.],.],[.,[.,.]]]
=> [[[[.,.],.],.],[.,.]]
=> [[[[[],[]],[]],[]],[[],[]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ? = 2 + 1
[[[.,.],.],[[.,.],.]]
=> [[[[.,.],.],[.,.]],.]
=> [[[[[],[]],[]],[[],[]]],[]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ? = 4 + 1
[[.,[.,[.,.]]],[.,.]]
=> [[[.,[.,[.,.]]],.],.]
=> [[[[],[[],[[],[]]]],[]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ? = 2 + 1
[[.,[[.,.],.]],[.,.]]
=> [[[.,[[.,.],.]],.],.]
=> [[[[],[[[],[]],[]]],[]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ? = 4 + 1
[[[.,.],[.,.]],[.,.]]
=> [[[[.,.],[.,.]],.],.]
=> [[[[[],[]],[[],[]]],[]],[]]
=> ([(0,10),(1,8),(2,8),(3,7),(4,7),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 4 + 1
[[[.,[.,.]],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> [[[[[],[[],[]]],[]],[]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ? = 4 + 1
[[[[.,.],.],.],[.,.]]
=> [[[[[.,.],.],.],.],.]
=> [[[[[[],[]],[]],[]],[]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ? = 4 + 1
[[.,[.,[.,[.,.]]]],.]
=> [.,[.,[.,[.,[.,.]]]]]
=> [[],[[],[[],[[],[[],[]]]]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ? = 2 + 1
[[.,[.,[[.,.],.]]],.]
=> [.,[.,[.,[[.,.],.]]]]
=> [[],[[],[[],[[[],[]],[]]]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ? = 2 + 1
[[.,[[.,.],[.,.]]],.]
=> [.,[.,[[.,.],[.,.]]]]
=> [[],[[],[[[],[]],[[],[]]]]]
=> ([(0,10),(1,8),(2,8),(3,7),(4,7),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 2 + 1
[[.,[[.,[.,.]],.]],.]
=> [.,[.,[[.,[.,.]],.]]]
=> [[],[[],[[[],[[],[]]],[]]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ? = 2 + 1
[[.,[[[.,.],.],.]],.]
=> [.,[.,[[[.,.],.],.]]]
=> [[],[[],[[[[],[]],[]],[]]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ? = 4 + 1
[[[.,.],[.,[.,.]]],.]
=> [.,[[.,.],[.,[.,.]]]]
=> [[],[[[],[]],[[],[[],[]]]]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ? = 2 + 1
[[[.,.],[[.,.],.]],.]
=> [.,[[.,.],[[.,.],.]]]
=> [[],[[[],[]],[[[],[]],[]]]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ? = 4 + 1
[[[.,[.,.]],[.,.]],.]
=> [.,[[.,[.,.]],[.,.]]]
=> [[],[[[],[[],[]]],[[],[]]]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ? = 4 + 1
Description
The number of minimal vertex covers of a graph.
A '''vertex cover''' of a graph $G$ is a subset $S$ of the vertices of $G$ such that each edge of $G$ contains at least one vertex of $S$. A vertex cover is minimal if it contains the least possible number of vertices.
This is also the leading coefficient of the clique polynomial of the complement of $G$.
This is also the number of independent sets of maximal cardinality of $G$.
Matching statistic: St001199
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00008: Binary trees —to complete tree⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 60%
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 60%
Values
[.,[.,.]]
=> [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[[.,.],.]
=> [[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [[],[[[],[]],[]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 3 = 2 + 1
[[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[.,[.,.]],.]
=> [[[],[[],[]]],[]]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 3 = 2 + 1
[.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
[.,[.,[[.,.],.]]]
=> [[],[[],[[[],[]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 + 1
[.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 1 + 1
[.,[[.,[.,.]],.]]
=> [[],[[[],[[],[]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 + 1
[.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 3 + 1
[[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 1 + 1
[[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> ? = 3 + 1
[[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 3 + 1
[[.,[.,[.,.]]],.]
=> [[[],[[],[[],[]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[[.,[[.,.],.]],.]
=> [[[],[[[],[]],[]]],[]]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> ? = 3 + 1
[[[.,.],[.,.]],.]
=> [[[[],[]],[[],[]]],[]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 3 + 1
[[[.,[.,.]],.],.]
=> [[[[],[[],[]]],[]],[]]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 3 + 1
[[[[.,.],.],.],.]
=> [[[[[],[]],[]],[]],[]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [[],[[],[[],[[],[[],[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [[],[[],[[],[[[],[]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [[],[[],[[[],[]],[[],[]]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 2 + 1
[.,[.,[[.,[.,.]],.]]]
=> [[],[[],[[[],[[],[]]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2 + 1
[.,[.,[[[.,.],.],.]]]
=> [[],[[],[[[[],[]],[]],[]]]]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> ? = 2 + 1
[.,[[.,.],[.,[.,.]]]]
=> [[],[[[],[]],[[],[[],[]]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 1
[.,[[.,.],[[.,.],.]]]
=> [[],[[[],[]],[[[],[]],[]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 + 1
[.,[[.,[.,.]],[.,.]]]
=> [[],[[[],[[],[]]],[[],[]]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 2 + 1
[.,[[[.,.],.],[.,.]]]
=> [[],[[[[],[]],[]],[[],[]]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2 + 1
[.,[[.,[.,[.,.]]],.]]
=> [[],[[[],[[],[[],[]]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 2 + 1
[.,[[.,[[.,.],.]],.]]
=> [[],[[[],[[[],[]],[]]],[]]]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> ? = 2 + 1
[.,[[[.,.],[.,.]],.]]
=> [[],[[[[],[]],[[],[]]],[]]]
=> [1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 + 1
[.,[[[.,[.,.]],.],.]]
=> [[],[[[[],[[],[]]],[]],[]]]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> ? = 2 + 1
[.,[[[[.,.],.],.],.]]
=> [[],[[[[[],[]],[]],[]],[]]]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 4 + 1
[[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,1,0,1,1,0,0,0,0]
=> ? = 2 + 1
[[.,.],[.,[[.,.],.]]]
=> [[[],[]],[[],[[[],[]],[]]]]
=> [1,1,0,1,0,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,1,1,0,0,0,0,1,0]
=> ? = 2 + 1
[[.,.],[[.,.],[.,.]]]
=> [[[],[]],[[[],[]],[[],[]]]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 2 + 1
[[.,.],[[.,[.,.]],.]]
=> [[[],[]],[[[],[[],[]]],[]]]
=> [1,1,0,1,0,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [[[],[]],[[[[],[]],[]],[]]]
=> [1,1,0,1,0,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,1,0,0,1,0]
=> ? = 4 + 1
[[.,[.,.]],[.,[.,.]]]
=> [[[],[[],[]]],[[],[[],[]]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 2 + 1
[[.,[.,.]],[[.,.],.]]
=> [[[],[[],[]]],[[[],[]],[]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 4 + 1
[[[.,.],.],[.,[.,.]]]
=> [[[[],[]],[]],[[],[[],[]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 + 1
[[[.,.],.],[[.,.],.]]
=> [[[[],[]],[]],[[[],[]],[]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 4 + 1
[[.,[.,[.,.]]],[.,.]]
=> [[[],[[],[[],[]]]],[[],[]]]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,1,0,1,1,0,0,0]
=> ? = 2 + 1
[[.,[[.,.],.]],[.,.]]
=> [[[],[[[],[]],[]]],[[],[]]]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,1,0,0,0,1,0]
=> ? = 4 + 1
[[[.,.],[.,.]],[.,.]]
=> [[[[],[]],[[],[]]],[[],[]]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 4 + 1
[[[.,[.,.]],.],[.,.]]
=> [[[[],[[],[]]],[]],[[],[]]]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 4 + 1
[[[[.,.],.],.],[.,.]]
=> [[[[[],[]],[]],[]],[[],[]]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 4 + 1
[[.,[.,[.,[.,.]]]],.]
=> [[[],[[],[[],[[],[]]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0,0]
=> ? = 2 + 1
[[.,[.,[[.,.],.]]],.]
=> [[[],[[],[[[],[]],[]]]],[]]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 2 + 1
[[.,[[.,.],[.,.]]],.]
=> [[[],[[[],[]],[[],[]]]],[]]
=> [1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 2 + 1
[[.,[[.,[.,.]],.]],.]
=> [[[],[[[],[[],[]]],[]]],[]]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 2 + 1
[[.,[[[.,.],.],.]],.]
=> [[[],[[[[],[]],[]],[]]],[]]
=> [1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> ? = 4 + 1
[[[.,.],[.,[.,.]]],.]
=> [[[[],[]],[[],[[],[]]]],[]]
=> [1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 2 + 1
[[[.,.],[[.,.],.]],.]
=> [[[[],[]],[[[],[]],[]]],[]]
=> [1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 4 + 1
[[[.,[.,.]],[.,.]],.]
=> [[[[],[[],[]]],[[],[]]],[]]
=> [1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,1,0,0]
=> ? = 4 + 1
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001461
Mp00008: Binary trees —to complete tree⟶ Ordered trees
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St001461: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 60%
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St001461: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 60%
Values
[.,[.,.]]
=> [[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2 = 1 + 1
[[.,.],.]
=> [[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2 = 1 + 1
[.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> [.,[[.,[[.,[.,.]],.]],.]]
=> [4,3,5,2,6,1] => 1 = 0 + 1
[.,[[.,.],.]]
=> [[],[[[],[]],[]]]
=> [.,[[[.,[.,.]],[.,.]],.]]
=> [3,2,5,4,6,1] => 3 = 2 + 1
[[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> [[.,[.,.]],[[.,[.,.]],.]]
=> [2,1,5,4,6,3] => 3 = 2 + 1
[[.,[.,.]],.]
=> [[[],[[],[]]],[]]
=> [[.,[[.,[.,.]],.]],[.,.]]
=> [3,2,4,1,6,5] => 3 = 2 + 1
[[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> [[[.,[.,.]],[.,.]],[.,.]]
=> [2,1,4,3,6,5] => 3 = 2 + 1
[.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [5,4,6,3,7,2,8,1] => ? = 1 + 1
[.,[.,[[.,.],.]]]
=> [[],[[],[[[],[]],[]]]]
=> [.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [4,3,6,5,7,2,8,1] => ? = 1 + 1
[.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> [.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [3,2,6,5,7,4,8,1] => ? = 1 + 1
[.,[[.,[.,.]],.]]
=> [[],[[[],[[],[]]],[]]]
=> [.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [4,3,5,2,7,6,8,1] => ? = 1 + 1
[.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> [.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [3,2,5,4,7,6,8,1] => ? = 3 + 1
[[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> [[.,[.,.]],[[.,[[.,[.,.]],.]],.]]
=> [2,1,6,5,7,4,8,3] => ? = 1 + 1
[[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> [[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [2,1,5,4,7,6,8,3] => ? = 3 + 1
[[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> [[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [3,2,4,1,7,6,8,5] => ? = 3 + 1
[[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> [[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> [2,1,4,3,7,6,8,5] => ? = 3 + 1
[[.,[.,[.,.]]],.]
=> [[[],[[],[[],[]]]],[]]
=> [[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
=> [4,3,5,2,6,1,8,7] => ? = 1 + 1
[[.,[[.,.],.]],.]
=> [[[],[[[],[]],[]]],[]]
=> [[.,[[[.,[.,.]],[.,.]],.]],[.,.]]
=> [3,2,5,4,6,1,8,7] => ? = 3 + 1
[[[.,.],[.,.]],.]
=> [[[[],[]],[[],[]]],[]]
=> [[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
=> [2,1,5,4,6,3,8,7] => ? = 3 + 1
[[[.,[.,.]],.],.]
=> [[[[],[[],[]]],[]],[]]
=> [[[.,[[.,[.,.]],.]],[.,.]],[.,.]]
=> [3,2,4,1,6,5,8,7] => ? = 3 + 1
[[[[.,.],.],.],.]
=> [[[[[],[]],[]],[]],[]]
=> [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [2,1,4,3,6,5,8,7] => ? = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [[],[[],[[],[[],[[],[]]]]]]
=> [.,[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],.]]
=> [6,5,7,4,8,3,9,2,10,1] => ? = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [[],[[],[[],[[[],[]],[]]]]]
=> [.,[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],.]]
=> [5,4,7,6,8,3,9,2,10,1] => ? = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [[],[[],[[[],[]],[[],[]]]]]
=> [.,[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],.]]
=> [4,3,7,6,8,5,9,2,10,1] => ? = 2 + 1
[.,[.,[[.,[.,.]],.]]]
=> [[],[[],[[[],[[],[]]],[]]]]
=> [.,[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],.]]
=> [5,4,6,3,8,7,9,2,10,1] => ? = 2 + 1
[.,[.,[[[.,.],.],.]]]
=> [[],[[],[[[[],[]],[]],[]]]]
=> [.,[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],.]]
=> [4,3,6,5,8,7,9,2,10,1] => ? = 2 + 1
[.,[[.,.],[.,[.,.]]]]
=> [[],[[[],[]],[[],[[],[]]]]]
=> [.,[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],.]]
=> [3,2,7,6,8,5,9,4,10,1] => ? = 2 + 1
[.,[[.,.],[[.,.],.]]]
=> [[],[[[],[]],[[[],[]],[]]]]
=> [.,[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],.]]
=> [3,2,6,5,8,7,9,4,10,1] => ? = 2 + 1
[.,[[.,[.,.]],[.,.]]]
=> [[],[[[],[[],[]]],[[],[]]]]
=> [.,[[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],.]]
=> [4,3,5,2,8,7,9,6,10,1] => ? = 2 + 1
[.,[[[.,.],.],[.,.]]]
=> [[],[[[[],[]],[]],[[],[]]]]
=> [.,[[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],.]]
=> [3,2,5,4,8,7,9,6,10,1] => ? = 2 + 1
[.,[[.,[.,[.,.]]],.]]
=> [[],[[[],[[],[[],[]]]],[]]]
=> [.,[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],.]]
=> [5,4,6,3,7,2,9,8,10,1] => ? = 2 + 1
[.,[[.,[[.,.],.]],.]]
=> [[],[[[],[[[],[]],[]]],[]]]
=> [.,[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],.]]
=> [4,3,6,5,7,2,9,8,10,1] => ? = 2 + 1
[.,[[[.,.],[.,.]],.]]
=> [[],[[[[],[]],[[],[]]],[]]]
=> [.,[[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],.]]
=> [3,2,6,5,7,4,9,8,10,1] => ? = 2 + 1
[.,[[[.,[.,.]],.],.]]
=> [[],[[[[],[[],[]]],[]],[]]]
=> [.,[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],.]]
=> [4,3,5,2,7,6,9,8,10,1] => ? = 2 + 1
[.,[[[[.,.],.],.],.]]
=> [[],[[[[[],[]],[]],[]],[]]]
=> [.,[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],.]]
=> [3,2,5,4,7,6,9,8,10,1] => ? = 4 + 1
[[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> [[.,[.,.]],[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [2,1,7,6,8,5,9,4,10,3] => ? = 2 + 1
[[.,.],[.,[[.,.],.]]]
=> [[[],[]],[[],[[[],[]],[]]]]
=> [[.,[.,.]],[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [2,1,6,5,8,7,9,4,10,3] => ? = 2 + 1
[[.,.],[[.,.],[.,.]]]
=> [[[],[]],[[[],[]],[[],[]]]]
=> [[.,[.,.]],[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [2,1,5,4,8,7,9,6,10,3] => ? = 2 + 1
[[.,.],[[.,[.,.]],.]]
=> [[[],[]],[[[],[[],[]]],[]]]
=> [[.,[.,.]],[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [2,1,6,5,7,4,9,8,10,3] => ? = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [[[],[]],[[[[],[]],[]],[]]]
=> [[.,[.,.]],[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [2,1,5,4,7,6,9,8,10,3] => ? = 4 + 1
[[.,[.,.]],[.,[.,.]]]
=> [[[],[[],[]]],[[],[[],[]]]]
=> [[.,[[.,[.,.]],.]],[[.,[[.,[.,.]],.]],.]]
=> [3,2,4,1,8,7,9,6,10,5] => ? = 2 + 1
[[.,[.,.]],[[.,.],.]]
=> [[[],[[],[]]],[[[],[]],[]]]
=> [[.,[[.,[.,.]],.]],[[[.,[.,.]],[.,.]],.]]
=> [3,2,4,1,7,6,9,8,10,5] => ? = 4 + 1
[[[.,.],.],[.,[.,.]]]
=> [[[[],[]],[]],[[],[[],[]]]]
=> [[[.,[.,.]],[.,.]],[[.,[[.,[.,.]],.]],.]]
=> [2,1,4,3,8,7,9,6,10,5] => ? = 2 + 1
[[[.,.],.],[[.,.],.]]
=> [[[[],[]],[]],[[[],[]],[]]]
=> [[[.,[.,.]],[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [2,1,4,3,7,6,9,8,10,5] => ? = 4 + 1
[[.,[.,[.,.]]],[.,.]]
=> [[[],[[],[[],[]]]],[[],[]]]
=> [[.,[[.,[[.,[.,.]],.]],.]],[[.,[.,.]],.]]
=> [4,3,5,2,6,1,9,8,10,7] => ? = 2 + 1
[[.,[[.,.],.]],[.,.]]
=> [[[],[[[],[]],[]]],[[],[]]]
=> [[.,[[[.,[.,.]],[.,.]],.]],[[.,[.,.]],.]]
=> [3,2,5,4,6,1,9,8,10,7] => ? = 4 + 1
[[[.,.],[.,.]],[.,.]]
=> [[[[],[]],[[],[]]],[[],[]]]
=> [[[.,[.,.]],[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [2,1,5,4,6,3,9,8,10,7] => ? = 4 + 1
[[[.,[.,.]],.],[.,.]]
=> [[[[],[[],[]]],[]],[[],[]]]
=> [[[.,[[.,[.,.]],.]],[.,.]],[[.,[.,.]],.]]
=> [3,2,4,1,6,5,9,8,10,7] => ? = 4 + 1
[[[[.,.],.],.],[.,.]]
=> [[[[[],[]],[]],[]],[[],[]]]
=> [[[[.,[.,.]],[.,.]],[.,.]],[[.,[.,.]],.]]
=> [2,1,4,3,6,5,9,8,10,7] => ? = 4 + 1
[[.,[.,[.,[.,.]]]],.]
=> [[[],[[],[[],[[],[]]]]],[]]
=> [[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],[.,.]]
=> [5,4,6,3,7,2,8,1,10,9] => ? = 2 + 1
[[.,[.,[[.,.],.]]],.]
=> [[[],[[],[[[],[]],[]]]],[]]
=> [[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],[.,.]]
=> [4,3,6,5,7,2,8,1,10,9] => ? = 2 + 1
[[.,[[.,.],[.,.]]],.]
=> [[[],[[[],[]],[[],[]]]],[]]
=> [[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],[.,.]]
=> [3,2,6,5,7,4,8,1,10,9] => ? = 2 + 1
[[.,[[.,[.,.]],.]],.]
=> [[[],[[[],[[],[]]],[]]],[]]
=> [[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],[.,.]]
=> [4,3,5,2,7,6,8,1,10,9] => ? = 2 + 1
[[.,[[[.,.],.],.]],.]
=> [[[],[[[[],[]],[]],[]]],[]]
=> [[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],[.,.]]
=> [3,2,5,4,7,6,8,1,10,9] => ? = 4 + 1
[[[.,.],[.,[.,.]]],.]
=> [[[[],[]],[[],[[],[]]]],[]]
=> [[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],[.,.]]
=> [2,1,6,5,7,4,8,3,10,9] => ? = 2 + 1
[[[.,.],[[.,.],.]],.]
=> [[[[],[]],[[[],[]],[]]],[]]
=> [[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],[.,.]]
=> [2,1,5,4,7,6,8,3,10,9] => ? = 4 + 1
[[[.,[.,.]],[.,.]],.]
=> [[[[],[[],[]]],[[],[]]],[]]
=> [[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],[.,.]]
=> [3,2,4,1,7,6,8,5,10,9] => ? = 4 + 1
Description
The number of topologically connected components of the chord diagram of a permutation.
The chord diagram of a permutation $\pi\in\mathfrak S_n$ is obtained by placing labels $1,\dots,n$ in cyclic order on a cycle and drawing a (straight) arc from $i$ to $\pi(i)$ for every label $i$.
This statistic records the number of topologically connected components in the chord diagram. In particular, if two arcs cross, all four labels connected by the two arcs are in the same component.
The permutation $\pi\in\mathfrak S_n$ stabilizes an interval $I=\{a,a+1,\dots,b\}$ if $\pi(I)=I$. It is stabilized-interval-free, if the only interval $\pi$ stablizes is $\{1,\dots,n\}$. Thus, this statistic is $1$ if $\pi$ is stabilized-interval-free.
The following 8 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000893The number of distinct diagonal sums of an alternating sign matrix. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice.
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