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Your data matches 27 different statistics following compositions of up to 3 maps.
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Matching statistic: St001085
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1] => 0
[.,[.,.]]
=> [2,1] => [1,2] => [1,2] => 0
[[.,.],.]
=> [1,2] => [2,1] => [2,1] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [1,3,2] => 0
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [1,3,2] => 0
[[.,.],[.,.]]
=> [3,1,2] => [2,1,3] => [2,1,3] => 1
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => [3,1,2] => 0
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => [3,2,1] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [1,4,3,2] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => [1,4,3,2] => 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => [1,4,3,2] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => [1,4,3,2] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 0
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,3,4] => [2,1,4,3] => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,2,4] => [3,1,4,2] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => [4,1,3,2] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => [4,1,3,2] => 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,2,1,3] => [4,2,1,3] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,5,4,3,2] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => [1,5,4,3,2] => 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,4,3,5] => [1,5,4,3,2] => 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => [1,5,4,3,2] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,5,4,3,2] => 0
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,3,2,4,5] => [1,5,4,3,2] => 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,3,2,5,4] => [1,5,4,3,2] => 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,4,2,3,5] => [1,5,4,3,2] => 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => [1,5,4,3,2] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => [1,5,4,3,2] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => [1,5,4,3,2] => 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,5,3,2,4] => [1,5,4,3,2] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => [1,5,4,3,2] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [2,1,3,4,5] => [2,1,5,4,3] => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,3,5,4] => [2,1,5,4,3] => 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [2,1,4,3,5] => [2,1,5,4,3] => 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [2,1,5,3,4] => [2,1,5,4,3] => 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,5,4,3] => [2,1,5,4,3] => 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [3,1,2,4,5] => [3,1,5,4,2] => 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,2,5,4] => [3,1,5,4,2] => 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,5,4] => 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1,2,3,5] => [4,1,5,3,2] => 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,1,3,2,5] => [4,1,5,3,2] => 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [4,2,1,3,5] => [4,2,1,5,3] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [4,3,1,2,5] => [4,3,1,5,2] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1,5] => 1
Description
The number of occurrences of the vincular pattern |21-3 in a permutation.
This is the number of occurrences of the pattern 213, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive.
In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the permutation.
Matching statistic: St000701
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000701: Binary trees ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000701: Binary trees ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1,0]
=> [.,.]
=> 1 = 0 + 1
[.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,0]
=> [.,[.,.]]
=> 1 = 0 + 1
[[.,.],.]
=> [1,1,0,0]
=> [1,0,1,0]
=> [[.,.],.]
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [.,[[.,.],.]]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> [.,[.,[.,.]]]
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 2 = 1 + 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 1 = 0 + 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [.,[.,[[.,.],.]]]
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[.,[[.,.],.]],.]
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,.]]
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> 2 = 1 + 1
[[[.,.],.],[.,[.,[[.,.],[.,.]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [[.,[.,.]],[[[[.,.],.],[.,.]],.]]
=> ? = 1 + 1
[[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [[.,[.,.]],[[[.,.],.],[.,[.,.]]]]
=> ? = 1 + 1
[[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [[.,[.,.]],[[[.,.],[.,[.,.]]],.]]
=> ? = 1 + 1
[[[.,.],.],[.,[[[.,.],[.,.]],.]]]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [[.,[.,.]],[[.,.],[[.,[.,.]],.]]]
=> ? = 1 + 1
[[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [[.,[.,.]],[[[[.,[.,.]],.],.],.]]
=> ? = 1 + 1
[[[.,.],.],[[.,[.,.]],[.,[.,.]]]]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [[.,[.,.]],[[[.,[[.,.],.]],.],.]]
=> ? = 1 + 1
[[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [[.,[.,.]],[[[.,[.,[.,.]]],.],.]]
=> ? = 1 + 1
[[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [[.,[.,.]],[[.,[.,[.,.]]],[.,.]]]
=> ? = 1 + 1
[[[.,.],.],[[.,[.,[.,.]]],[.,.]]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,[[[.,.],.],.]],.]]
=> ? = 1 + 1
[[[.,.],.],[[[.,[.,.]],.],[.,.]]]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [[.,[.,.]],[[.,[.,[[.,.],.]]],.]]
=> ? = 1 + 1
[[[.,.],.],[[[[.,.],.],.],[.,.]]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[.,[.,.]],[[.,[.,[.,[.,.]]]],.]]
=> ? = 1 + 1
[[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],[[[[.,.],.],.],.]]
=> ? = 1 + 1
[[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,1,0,0]
=> [[.,[.,[.,.]]],[[[.,.],[.,.]],.]]
=> ? = 1 + 1
[[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],[[[.,[.,.]],.],.]]
=> ? = 1 + 1
[[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
=> ? = 1 + 1
[[[[.,.],.],.],[[.,[.,.]],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,[.,.]]],[[.,[[.,.],.]],.]]
=> ? = 1 + 1
[[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> [[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
=> ? = 1 + 1
[[[[.,.],.],.],[[[.,.],[.,.]],.]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,[[.,[.,.]],.]]]
=> ? = 1 + 1
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> ? = 1 + 1
[[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,.]]]],[[[.,.],.],.]]
=> ? = 1 + 1
[[[[[.,.],.],.],.],[.,[[.,.],.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> ? = 1 + 1
[[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
=> [[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> ? = 1 + 1
[[[[[.,.],.],.],.],[[.,[.,.]],.]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> [[.,[.,[.,[.,.]]]],[.,[[.,.],.]]]
=> ? = 1 + 1
[[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> ? = 1 + 1
[[[[[[.,.],.],.],.],.],[[.,.],.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> ? = 1 + 1
[[.,[.,[.,[[.,[.,.]],.]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [[[[[.,.],.],.],[[.,.],.]],[.,.]]
=> ? = 1 + 1
[[.,[.,[.,[[[.,.],.],.]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [[[[[.,.],.],.],[.,[.,.]]],[.,.]]
=> ? = 1 + 1
[[.,[.,[[.,[.,.]],[.,.]]]],[.,.]]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [[[[[.,.],.],[[.,.],.]],.],[.,.]]
=> ? = 1 + 1
[[.,[.,[[[.,[.,.]],.],.]]],[.,.]]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [[[[.,.],.],[.,[[.,.],.]]],[.,.]]
=> ? = 1 + 1
[[.,[.,[[[[.,.],.],.],.]]],[.,.]]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> ? = 1 + 1
[[.,[[.,.],[[.,.],[.,.]]]],[.,.]]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
=> ? = 1 + 1
[[.,[[.,[.,.]],[.,[.,.]]]],[.,.]]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [[[[[.,.],[[.,.],.]],.],.],[.,.]]
=> ? = 1 + 1
[[.,[[[.,.],.],[.,[.,.]]]],[.,.]]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [[[[[.,.],[.,[.,.]]],.],.],[.,.]]
=> ? = 1 + 1
[[.,[[.,[[.,.],.]],[.,.]]],[.,.]]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [[[[.,.],[[.,.],[.,.]]],.],[.,.]]
=> ? = 1 + 1
[[.,[[[.,.],[.,.]],[.,.]]],[.,.]]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [[[[.,.],[[.,[.,.]],.]],.],[.,.]]
=> ? = 1 + 1
[[.,[[[.,[.,.]],.],[.,.]]],[.,.]]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [[[[.,.],[.,[[.,.],.]]],.],[.,.]]
=> ? = 1 + 1
[[.,[[.,[.,[.,[.,.]]]],.]],[.,.]]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [[[.,.],[[[[.,.],.],.],.]],[.,.]]
=> ? = 1 + 1
[[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [[[.,.],[[.,.],[[.,.],.]]],[.,.]]
=> ? = 1 + 1
[[.,[[.,[[[.,.],.],.]],.]],[.,.]]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [[[.,.],[[.,.],[.,[.,.]]]],[.,.]]
=> ? = 1 + 1
[[.,[[[.,.],[[.,.],.]],.]],[.,.]]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [[[.,.],[[.,[.,.]],[.,.]]],[.,.]]
=> ? = 1 + 1
[[.,[[[.,[.,.]],[.,.]],.]],[.,.]]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [[[.,.],[[.,[[.,.],.]],.]],[.,.]]
=> ? = 1 + 1
[[.,[[[.,[[.,.],.]],.],.]],[.,.]]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [[[.,.],[.,[[.,.],[.,.]]]],[.,.]]
=> ? = 1 + 1
[[.,[[[[.,.],[.,.]],.],.]],[.,.]]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [[[.,.],[.,[[.,[.,.]],.]]],[.,.]]
=> ? = 1 + 1
[[.,[[[[.,[.,.]],.],.],.]],[.,.]]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [[[.,.],[.,[.,[[.,.],.]]]],[.,.]]
=> ? = 1 + 1
[[.,[[[[[.,.],.],.],.],.]],[.,.]]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [[[.,.],[.,[.,[.,[.,.]]]]],[.,.]]
=> ? = 1 + 1
[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]
=> [[[.,[[.,[[.,.],.]],.]],.],[.,.]]
=> ? = 1 + 1
[[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [[.,[.,[[.,[.,[.,.]]],.]]],[.,.]]
=> ? = 1 + 1
[[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> ? = 1 + 1
[[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> ? = 0 + 1
[[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [[.,[.,[[.,[.,[.,[.,.]]]],.]]],.]
=> ? = 0 + 1
Description
The protection number of a binary tree.
This is the minimal distance from the root to a leaf.
Matching statistic: St000553
Values
[.,.]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
[[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(5,4),(6,7),(7,3)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[.,[[.,.],[[.,[.,.]],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[.,[[.,.],[[[.,.],.],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> ([(0,4),(1,3),(3,7),(4,7),(5,2),(6,5),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[.,[[.,[.,.]],[[.,.],.]]]]]
=> ([(0,4),(1,3),(3,7),(4,7),(5,2),(6,5),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[.,[[[.,.],.],[[.,.],.]]]]]
=> ([(0,4),(1,3),(3,7),(4,7),(5,2),(6,5),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[.,[[.,[[.,.],.]],[.,.]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[.,[[[.,.],[.,.]],[.,.]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(5,4),(6,7),(7,3)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[.,[[[.,[.,.]],.],[.,.]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[.,[[[[.,.],.],.],[.,.]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[.,[.,[.,[[.,[.,[[.,.],.]]],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[.,[.,[.,[[.,[[.,.],[.,.]]],.]]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[.,[.,[.,[[.,[[[.,.],.],.]],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[.,[[[.,.],[[.,.],.]],.]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[.,[.,[.,[[[.,[[.,.],.]],.],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[.,[.,[.,[[[[.,.],[.,.]],.],.]]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[.,[[[[.,[.,.]],.],.],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,7),(6,4),(7,3)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,3),(3,7),(4,5),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[[.,.],[[.,.],[[.,.],.]]]]]
=> ([(0,7),(1,6),(2,3),(3,7),(4,5),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> ([(0,7),(1,6),(2,3),(3,7),(4,5),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
[.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 0 + 1
Description
The number of blocks of a graph.
A cut vertex is a vertex whose deletion increases the number of connected components. A block is a maximal connected subgraph which itself has no cut vertices. Two distinct blocks cannot overlap in more than a single cut vertex.
Matching statistic: St000552
Values
[.,.]
=> ([],1)
=> ([],1)
=> ([],1)
=> 0
[.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 0
[[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 0
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[[[[.,.],.],.],.],[.,[.,.]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[[[[.,.],.],.],.],[[.,.],.]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(5,4),(6,7),(7,3)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[.,[[.,.],[[.,[.,.]],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[.,[[.,.],[[[.,.],.],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> ([(0,4),(1,3),(3,7),(4,7),(5,2),(6,5),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[.,[[.,[.,.]],[[.,.],.]]]]]
=> ([(0,4),(1,3),(3,7),(4,7),(5,2),(6,5),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[.,[[[.,.],.],[[.,.],.]]]]]
=> ([(0,4),(1,3),(3,7),(4,7),(5,2),(6,5),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[.,[[.,[[.,.],.]],[.,.]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[.,[[[.,.],[.,.]],[.,.]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(5,4),(6,7),(7,3)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[.,[[[.,[.,.]],.],[.,.]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[.,[[[[.,.],.],.],[.,.]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[.,[[.,[.,[[.,.],.]]],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[.,[[.,[[.,.],[.,.]]],.]]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[.,[[.,[[[.,.],.],.]],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[.,[[[.,.],[[.,.],.]],.]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[.,[[[.,[[.,.],.]],.],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[.,[[[[.,.],[.,.]],.],.]]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[.,[[[[.,[.,.]],.],.],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,7),(6,4),(7,3)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,3),(3,7),(4,5),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0
[.,[.,[[.,.],[[.,.],[[.,.],.]]]]]
=> ([(0,7),(1,6),(2,3),(3,7),(4,5),(6,4),(7,6)],8)
=> ?
=> ?
=> ? = 0
Description
The number of cut vertices of a graph.
A cut vertex is one whose deletion increases the number of connected components.
Matching statistic: St001086
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 100%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1] => 0
[.,[.,.]]
=> [2,1] => [1,2] => [1,2] => 0
[[.,.],.]
=> [1,2] => [2,1] => [2,1] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 0
[.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => [2,3,1] => 0
[[.,.],[.,.]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [3,2,1] => [3,2,1] => 0
[[[.,.],.],.]
=> [1,2,3] => [2,3,1] => [3,1,2] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,1,2,3] => [2,3,4,1] => 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,4,2,3] => [1,3,4,2] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,3,1,2] => [3,4,2,1] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,4,1,2] => [3,4,1,2] => 0
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,2,4,3] => [1,2,4,3] => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [4,1,3,2] => [2,4,3,1] => 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,4,3,2] => [1,4,3,2] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,3,4,2] => [1,4,2,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,3,2,1] => [4,3,2,1] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,4,2,1] => [4,3,1,2] => 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,2,3,1] => [4,2,3,1] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [3,2,4,1] => [4,2,1,3] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,1,2,3,4] => [2,3,4,5,1] => 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,5,2,3,4] => [1,3,4,5,2] => 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,1,2,3] => [3,4,5,2,1] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [4,5,1,2,3] => [3,4,5,1,2] => 0
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,2,5,3,4] => [1,2,4,5,3] => 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [5,1,4,2,3] => [2,4,5,3,1] => 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,5,4,2,3] => [1,4,5,3,2] => 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,5,2,3] => [1,4,5,2,3] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,4,3,1,2] => [4,5,3,2,1] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [4,5,3,1,2] => [4,5,3,1,2] => 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [5,3,4,1,2] => [4,5,2,3,1] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [4,3,5,1,2] => [4,5,2,1,3] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [3,4,5,1,2] => [4,5,1,2,3] => 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [5,1,2,4,3] => [2,3,5,4,1] => 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,5,2,4,3] => [1,3,5,4,2] => 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [5,4,1,3,2] => [3,5,4,2,1] => 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [4,5,1,3,2] => [3,5,4,1,2] => 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [5,1,4,3,2] => [2,5,4,3,1] => 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,4,5,3] => [1,2,5,3,4] => 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [5,1,3,4,2] => [2,5,3,4,1] => 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,4,5,3,2] => [1,5,4,2,3] => 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,5,3,4,2] => [1,5,3,4,2] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,4,3,5,2] => [1,5,3,2,4] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,3,4,5,2] => [1,5,2,3,4] => 1
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [6,5,7,4,3,2,1] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => ? = 0
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [6,7,4,5,3,2,1] => [7,1,6,2,3,4,5] => [2,4,5,6,7,3,1] => ? = 0
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [6,5,4,7,3,2,1] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => ? = 0
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [6,7,5,1,2,3,4] => [4,5,6,7,3,1,2] => ? = 0
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [6,4,5,7,3,2,1] => [7,5,6,1,2,3,4] => [4,5,6,7,2,3,1] => ? = 0
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [5,4,6,7,3,2,1] => [6,5,7,1,2,3,4] => [4,5,6,7,2,1,3] => ? = 0
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> [4,5,6,7,3,2,1] => [5,6,7,1,2,3,4] => [4,5,6,7,1,2,3] => ? = 0
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [6,7,5,3,4,2,1] => [7,1,2,6,3,4,5] => [2,3,5,6,7,4,1] => ? = 0
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [6,5,7,3,4,2,1] => [7,6,1,5,2,3,4] => [3,5,6,7,4,2,1] => ? = 0
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [5,6,7,3,4,2,1] => [6,7,1,5,2,3,4] => [3,5,6,7,4,1,2] => ? = 0
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [6,7,4,3,5,2,1] => [7,1,6,5,2,3,4] => [2,5,6,7,4,3,1] => ? = 0
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [6,7,3,4,5,2,1] => [7,1,5,6,2,3,4] => [2,5,6,7,3,4,1] => ? = 0
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [7,4,5,3,6,2,1] => [1,6,7,5,2,3,4] => [1,5,6,7,4,2,3] => ? = 0
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [7,5,3,4,6,2,1] => [1,7,5,6,2,3,4] => [1,5,6,7,3,4,2] => ? = 0
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [7,4,3,5,6,2,1] => [1,6,5,7,2,3,4] => [1,5,6,7,3,2,4] => ? = 0
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [6,5,4,3,7,2,1] => [7,6,5,4,1,2,3] => [5,6,7,4,3,2,1] => ? = 0
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [6,7,5,4,1,2,3] => [5,6,7,4,3,1,2] => ? = 0
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [6,4,5,3,7,2,1] => [7,5,6,4,1,2,3] => [5,6,7,4,2,3,1] => ? = 0
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [5,4,6,3,7,2,1] => [6,5,7,4,1,2,3] => [5,6,7,4,2,1,3] => ? = 0
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => [5,6,7,4,1,2,3] => [5,6,7,4,1,2,3] => ? = 0
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [6,5,3,4,7,2,1] => [7,6,4,5,1,2,3] => [5,6,7,3,4,2,1] => ? = 0
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [5,6,3,4,7,2,1] => [6,7,4,5,1,2,3] => [5,6,7,3,4,1,2] => ? = 0
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [6,4,3,5,7,2,1] => [7,5,4,6,1,2,3] => [5,6,7,3,2,4,1] => ? = 0
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [6,3,4,5,7,2,1] => [7,4,5,6,1,2,3] => [5,6,7,2,3,4,1] => ? = 0
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [5,4,3,6,7,2,1] => [6,5,4,7,1,2,3] => [5,6,7,3,2,1,4] => ? = 0
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [4,5,3,6,7,2,1] => [5,6,4,7,1,2,3] => [5,6,7,3,1,2,4] => ? = 0
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [5,3,4,6,7,2,1] => [6,4,5,7,1,2,3] => [5,6,7,2,3,1,4] => ? = 0
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => [5,4,6,7,1,2,3] => [5,6,7,2,1,3,4] => ? = 0
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,2,3,1] => [7,1,2,3,6,4,5] => [2,3,4,6,7,5,1] => ? = 0
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> [6,5,7,4,2,3,1] => [7,6,1,2,5,3,4] => [3,4,6,7,5,2,1] => ? = 0
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [5,6,7,4,2,3,1] => [6,7,1,2,5,3,4] => [3,4,6,7,5,1,2] => ? = 0
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [6,7,4,5,2,3,1] => [7,1,6,2,5,3,4] => [2,4,6,7,5,3,1] => ? = 0
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> [7,4,5,6,2,3,1] => [1,6,7,2,5,3,4] => [1,4,6,7,5,2,3] => ? = 0
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [6,5,4,7,2,3,1] => [7,6,5,1,4,2,3] => [4,6,7,5,3,2,1] => ? = 0
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [5,6,4,7,2,3,1] => [6,7,5,1,4,2,3] => [4,6,7,5,3,1,2] => ? = 0
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> [6,4,5,7,2,3,1] => [7,5,6,1,4,2,3] => [4,6,7,5,2,3,1] => ? = 0
[.,[[.,.],[[[.,[.,.]],.],.]]]
=> [5,4,6,7,2,3,1] => [6,5,7,1,4,2,3] => [4,6,7,5,2,1,3] => ? = 0
[.,[[.,.],[[[[.,.],.],.],.]]]
=> [4,5,6,7,2,3,1] => [5,6,7,1,4,2,3] => [4,6,7,5,1,2,3] => ? = 0
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [6,7,5,3,2,4,1] => [7,1,2,6,5,3,4] => [2,3,6,7,5,4,1] => ? = 0
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [6,5,7,3,2,4,1] => [7,6,1,5,4,2,3] => [3,6,7,5,4,2,1] => ? = 0
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> [5,6,7,3,2,4,1] => [6,7,1,5,4,2,3] => [3,6,7,5,4,1,2] => ? = 0
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> [6,7,5,2,3,4,1] => [7,1,2,5,6,3,4] => [2,3,6,7,4,5,1] => ? = 0
[.,[[[.,.],.],[[.,[.,.]],.]]]
=> [6,5,7,2,3,4,1] => [7,6,1,4,5,2,3] => [3,6,7,4,5,2,1] => ? = 0
[.,[[[.,.],.],[[[.,.],.],.]]]
=> [5,6,7,2,3,4,1] => [6,7,1,4,5,2,3] => [3,6,7,4,5,1,2] => ? = 0
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [6,7,4,3,2,5,1] => [7,1,6,5,4,2,3] => [2,6,7,5,4,3,1] => ? = 0
[.,[[.,[[.,.],.]],[[.,.],.]]]
=> [6,7,3,4,2,5,1] => [7,1,5,6,4,2,3] => [2,6,7,5,3,4,1] => ? = 0
[.,[[[.,.],[.,.]],[[.,.],.]]]
=> [6,7,4,2,3,5,1] => [7,1,6,4,5,2,3] => [2,6,7,4,5,3,1] => ? = 0
[.,[[[.,[.,.]],.],[[.,.],.]]]
=> [6,7,3,2,4,5,1] => [7,1,5,4,6,2,3] => [2,6,7,4,3,5,1] => ? = 0
[.,[[[[.,.],.],.],[[.,.],.]]]
=> [6,7,2,3,4,5,1] => [7,1,4,5,6,2,3] => [2,6,7,3,4,5,1] => ? = 0
[.,[[.,[.,[[.,.],.]]],[.,.]]]
=> [7,4,5,3,2,6,1] => [1,6,7,5,4,2,3] => [1,6,7,5,4,2,3] => ? = 0
Description
The number of occurrences of the consecutive pattern 132 in a permutation.
This is the number of occurrences of the pattern 132, where the matched entries are all adjacent.
Matching statistic: St001878
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
[.,.]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? = 0 + 1
[.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0 + 1
[[.,.],.]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0 + 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 2 = 1 + 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1 = 0 + 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
=> ([(0,5),(1,7),(2,7),(3,4),(4,6),(5,3),(6,1),(6,2)],8)
=> ? = 0 + 1
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
=> ([(0,5),(1,7),(2,7),(3,6),(4,1),(5,3),(6,2),(6,4)],8)
=> ? = 0 + 1
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
=> ([(0,5),(1,7),(2,7),(3,6),(4,1),(5,3),(6,2),(6,4)],8)
=> ? = 0 + 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
=> ([(0,5),(1,7),(2,7),(3,6),(4,1),(5,3),(6,2),(6,4)],8)
=> ? = 0 + 1
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
=> ([(0,5),(1,7),(2,7),(3,6),(4,1),(5,3),(6,2),(6,4)],8)
=> ? = 0 + 1
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
=> ([(0,5),(1,7),(2,7),(3,4),(4,6),(5,3),(6,1),(6,2)],8)
=> ? = 0 + 1
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,5),(3,4),(4,1),(5,6),(6,2),(6,3)],7)
=> ([(0,5),(1,7),(2,7),(3,4),(4,1),(5,6),(6,2),(6,3)],8)
=> ? = 0 + 1
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,5),(3,4),(4,1),(5,6),(6,2),(6,3)],7)
=> ([(0,5),(1,7),(2,7),(3,4),(4,1),(5,6),(6,2),(6,3)],8)
=> ? = 0 + 1
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(0,4),(4,6),(5,2),(5,3),(6,1),(6,5)],7)
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,2),(5,3),(6,1),(6,5)],8)
=> ? = 0 + 1
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,5),(3,4),(4,1),(5,6),(6,2),(6,3)],7)
=> ([(0,5),(1,7),(2,7),(3,4),(4,1),(5,6),(6,2),(6,3)],8)
=> ? = 0 + 1
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,5),(3,4),(4,1),(5,6),(6,2),(6,3)],7)
=> ([(0,5),(1,7),(2,7),(3,4),(4,1),(5,6),(6,2),(6,3)],8)
=> ? = 0 + 1
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(0,5),(3,2),(4,1),(5,6),(6,3),(6,4)],7)
=> ([(0,5),(1,7),(2,7),(3,2),(4,1),(5,6),(6,3),(6,4)],8)
=> ? = 0 + 1
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(0,5),(3,2),(4,1),(5,6),(6,3),(6,4)],7)
=> ([(0,5),(1,7),(2,7),(3,2),(4,1),(5,6),(6,3),(6,4)],8)
=> ? = 0 + 1
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(0,5),(3,2),(4,1),(5,6),(6,3),(6,4)],7)
=> ([(0,5),(1,7),(2,7),(3,2),(4,1),(5,6),(6,3),(6,4)],8)
=> ? = 0 + 1
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(0,5),(3,2),(4,1),(5,6),(6,3),(6,4)],7)
=> ([(0,5),(1,7),(2,7),(3,2),(4,1),(5,6),(6,3),(6,4)],8)
=> ? = 0 + 1
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,5),(3,4),(4,1),(5,6),(6,2),(6,3)],7)
=> ([(0,5),(1,7),(2,7),(3,4),(4,1),(5,6),(6,2),(6,3)],8)
=> ? = 0 + 1
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,5),(3,4),(4,1),(5,6),(6,2),(6,3)],7)
=> ([(0,5),(1,7),(2,7),(3,4),(4,1),(5,6),(6,2),(6,3)],8)
=> ? = 0 + 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(0,4),(4,6),(5,2),(5,3),(6,1),(6,5)],7)
=> ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,2),(5,3),(6,1),(6,5)],8)
=> ? = 0 + 1
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,5),(3,4),(4,1),(5,6),(6,2),(6,3)],7)
=> ([(0,5),(1,7),(2,7),(3,4),(4,1),(5,6),(6,2),(6,3)],8)
=> ? = 0 + 1
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(0,5),(3,4),(4,1),(5,6),(6,2),(6,3)],7)
=> ([(0,5),(1,7),(2,7),(3,4),(4,1),(5,6),(6,2),(6,3)],8)
=> ? = 0 + 1
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
=> ([(0,5),(1,7),(2,7),(3,4),(4,6),(5,3),(6,1),(6,2)],8)
=> ? = 0 + 1
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
=> ([(0,5),(1,7),(2,7),(3,6),(4,1),(5,3),(6,2),(6,4)],8)
=> ? = 0 + 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
=> ([(0,5),(1,7),(2,7),(3,6),(4,1),(5,3),(6,2),(6,4)],8)
=> ? = 0 + 1
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
=> ([(0,5),(1,7),(2,7),(3,6),(4,1),(5,3),(6,2),(6,4)],8)
=> ? = 0 + 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
=> ([(0,5),(1,7),(2,7),(3,6),(4,1),(5,3),(6,2),(6,4)],8)
=> ? = 0 + 1
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
=> ([(0,5),(1,7),(2,7),(3,4),(4,6),(5,3),(6,1),(6,2)],8)
=> ? = 0 + 1
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,6),(3,5),(4,3),(5,1),(6,2),(6,4)],7)
=> ([(0,6),(1,7),(2,7),(3,5),(4,3),(5,1),(6,2),(6,4)],8)
=> ? = 0 + 1
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,6),(3,5),(4,3),(5,1),(6,2),(6,4)],7)
=> ([(0,6),(1,7),(2,7),(3,5),(4,3),(5,1),(6,2),(6,4)],8)
=> ? = 0 + 1
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(0,6),(4,5),(5,2),(5,3),(6,1),(6,4)],7)
=> ([(0,6),(1,7),(2,7),(3,7),(4,5),(5,2),(5,3),(6,1),(6,4)],8)
=> ? = 0 + 1
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,6),(3,5),(4,3),(5,1),(6,2),(6,4)],7)
=> ([(0,6),(1,7),(2,7),(3,5),(4,3),(5,1),(6,2),(6,4)],8)
=> ? = 0 + 1
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,6),(3,5),(4,3),(5,1),(6,2),(6,4)],7)
=> ([(0,6),(1,7),(2,7),(3,5),(4,3),(5,1),(6,2),(6,4)],8)
=> ? = 0 + 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(0,6),(4,3),(5,2),(5,4),(6,1),(6,5)],7)
=> ([(0,6),(1,7),(2,7),(3,7),(4,3),(5,2),(5,4),(6,1),(6,5)],8)
=> ? = 0 + 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(0,6),(4,3),(5,2),(5,4),(6,1),(6,5)],7)
=> ([(0,6),(1,7),(2,7),(3,7),(4,3),(5,2),(5,4),(6,1),(6,5)],8)
=> ? = 0 + 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(0,6),(4,3),(5,2),(5,4),(6,1),(6,5)],7)
=> ([(0,6),(1,7),(2,7),(3,7),(4,3),(5,2),(5,4),(6,1),(6,5)],8)
=> ? = 0 + 1
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(0,6),(4,3),(5,2),(5,4),(6,1),(6,5)],7)
=> ([(0,6),(1,7),(2,7),(3,7),(4,3),(5,2),(5,4),(6,1),(6,5)],8)
=> ? = 0 + 1
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,6),(3,5),(4,3),(5,1),(6,2),(6,4)],7)
=> ([(0,6),(1,7),(2,7),(3,5),(4,3),(5,1),(6,2),(6,4)],8)
=> ? = 0 + 1
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,6),(3,5),(4,3),(5,1),(6,2),(6,4)],7)
=> ([(0,6),(1,7),(2,7),(3,5),(4,3),(5,1),(6,2),(6,4)],8)
=> ? = 0 + 1
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(0,6),(4,5),(5,2),(5,3),(6,1),(6,4)],7)
=> ([(0,6),(1,7),(2,7),(3,7),(4,5),(5,2),(5,3),(6,1),(6,4)],8)
=> ? = 0 + 1
[.,[[.,.],[[[.,[.,.]],.],.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,6),(3,5),(4,3),(5,1),(6,2),(6,4)],7)
=> ([(0,6),(1,7),(2,7),(3,5),(4,3),(5,1),(6,2),(6,4)],8)
=> ? = 0 + 1
[.,[[.,.],[[[[.,.],.],.],.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(0,6),(3,5),(4,3),(5,1),(6,2),(6,4)],7)
=> ([(0,6),(1,7),(2,7),(3,5),(4,3),(5,1),(6,2),(6,4)],8)
=> ? = 0 + 1
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,6),(3,4),(4,1),(5,2),(6,3),(6,5)],7)
=> ([(0,6),(1,7),(2,7),(3,4),(4,1),(5,2),(6,3),(6,5)],8)
=> ? = 0 + 1
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,6),(3,4),(4,1),(5,2),(6,3),(6,5)],7)
=> ([(0,6),(1,7),(2,7),(3,4),(4,1),(5,2),(6,3),(6,5)],8)
=> ? = 0 + 1
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(0,6),(4,3),(5,1),(5,2),(6,4),(6,5)],7)
=> ([(0,6),(1,7),(2,7),(3,7),(4,3),(5,1),(5,2),(6,4),(6,5)],8)
=> ? = 0 + 1
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,6),(3,4),(4,1),(5,2),(6,3),(6,5)],7)
=> ([(0,6),(1,7),(2,7),(3,4),(4,1),(5,2),(6,3),(6,5)],8)
=> ? = 0 + 1
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,6),(3,4),(4,1),(5,2),(6,3),(6,5)],7)
=> ([(0,6),(1,7),(2,7),(3,4),(4,1),(5,2),(6,3),(6,5)],8)
=> ? = 0 + 1
[.,[[[.,.],.],[.,[.,[.,.]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,6),(3,4),(4,1),(5,2),(6,3),(6,5)],7)
=> ([(0,6),(1,7),(2,7),(3,4),(4,1),(5,2),(6,3),(6,5)],8)
=> ? = 0 + 1
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(0,6),(3,4),(4,1),(5,2),(6,3),(6,5)],7)
=> ([(0,6),(1,7),(2,7),(3,4),(4,1),(5,2),(6,3),(6,5)],8)
=> ? = 0 + 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St000700
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000700: Ordered trees ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
St000700: Ordered trees ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [[],[]]
=> 1 = 0 + 1
[.,[.,.]]
=> [[],[[],[]]]
=> 1 = 0 + 1
[[.,.],.]
=> [[[],[]],[]]
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [[],[[[],[]],[]]]
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 1 + 1
[[.,[.,.]],.]
=> [[[],[[],[]]],[]]
=> 1 = 0 + 1
[[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [[],[[],[[[],[]],[]]]]
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [[],[[[],[[],[]]],[]]]
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> [[[],[[],[[],[]]]],[]]
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [[[],[[[],[]],[]]],[]]
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [[[[],[]],[[],[]]],[]]
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> [[[[],[[],[]]],[]],[]]
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> [[[[[],[]],[]],[]],[]]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [[],[[],[[],[[],[[],[]]]]]]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [[],[[],[[],[[[],[]],[]]]]]
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [[],[[],[[[],[]],[[],[]]]]]
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [[],[[],[[[],[[],[]]],[]]]]
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [[],[[],[[[[],[]],[]],[]]]]
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [[],[[[],[]],[[],[[],[]]]]]
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [[],[[[],[]],[[[],[]],[]]]]
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [[],[[[],[[],[]]],[[],[]]]]
=> 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> [[],[[[[],[]],[]],[[],[]]]]
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [[],[[[],[[],[[],[]]]],[]]]
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> [[],[[[],[[[],[]],[]]],[]]]
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [[],[[[[],[]],[[],[]]],[]]]
=> 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> [[],[[[[],[[],[]]],[]],[]]]
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> [[],[[[[[],[]],[]],[]],[]]]
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [[[],[]],[[],[[[],[]],[]]]]
=> 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [[[],[]],[[[],[]],[[],[]]]]
=> 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [[[],[]],[[[],[[],[]]],[]]]
=> 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [[[],[]],[[[[],[]],[]],[]]]
=> 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [[[],[[],[]]],[[],[[],[]]]]
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [[[],[[],[]]],[[[],[]],[]]]
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [[[[],[]],[]],[[],[[],[]]]]
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [[[[],[]],[]],[[[],[]],[]]]
=> 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [[[],[[],[[],[]]]],[[],[]]]
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [[[],[[[],[]],[]]],[[],[]]]
=> 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [[[[],[]],[[],[]]],[[],[]]]
=> 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> [[[[],[[],[]]],[]],[[],[]]]
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [[[[[],[]],[]],[]],[[],[]]]
=> 2 = 1 + 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[],[[],[[],[[],[[],[[],[[],[]]]]]]]]
=> ? = 0 + 1
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [[],[[],[[],[[],[[],[[[],[]],[]]]]]]]
=> ? = 0 + 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [[],[[],[[],[[],[[[],[]],[[],[]]]]]]]
=> ? = 0 + 1
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [[],[[],[[],[[],[[[],[[],[]]],[]]]]]]
=> ? = 0 + 1
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [[],[[],[[],[[],[[[[],[]],[]],[]]]]]]
=> ? = 0 + 1
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [[],[[],[[],[[[],[]],[[],[[],[]]]]]]]
=> ? = 0 + 1
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [[],[[],[[],[[[],[]],[[[],[]],[]]]]]]
=> ? = 0 + 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [[],[[],[[],[[[],[[],[]]],[[],[]]]]]]
=> ? = 0 + 1
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [[],[[],[[],[[[[],[]],[]],[[],[]]]]]]
=> ? = 0 + 1
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [[],[[],[[],[[[],[[],[[],[]]]],[]]]]]
=> ? = 0 + 1
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [[],[[],[[],[[[],[[[],[]],[]]],[]]]]]
=> ? = 0 + 1
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [[],[[],[[],[[[[],[]],[[],[]]],[]]]]]
=> ? = 0 + 1
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [[],[[],[[],[[[[],[[],[]]],[]],[]]]]]
=> ? = 0 + 1
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> [[],[[],[[],[[[[[],[]],[]],[]],[]]]]]
=> ? = 0 + 1
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [[],[[],[[[],[]],[[],[[],[[],[]]]]]]]
=> ? = 0 + 1
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [[],[[],[[[],[]],[[],[[[],[]],[]]]]]]
=> ? = 0 + 1
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [[],[[],[[[],[]],[[[],[]],[[],[]]]]]]
=> ? = 0 + 1
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [[],[[],[[[],[]],[[[],[[],[]]],[]]]]]
=> ? = 0 + 1
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [[],[[],[[[],[]],[[[[],[]],[]],[]]]]]
=> ? = 0 + 1
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [[],[[],[[[],[[],[]]],[[],[[],[]]]]]]
=> ? = 0 + 1
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [[],[[],[[[],[[],[]]],[[[],[]],[]]]]]
=> ? = 0 + 1
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [[],[[],[[[[],[]],[]],[[],[[],[]]]]]]
=> ? = 0 + 1
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [[],[[],[[[[],[]],[]],[[[],[]],[]]]]]
=> ? = 0 + 1
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [[],[[],[[[],[[],[[],[]]]],[[],[]]]]]
=> ? = 0 + 1
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [[],[[],[[[],[[[],[]],[]]],[[],[]]]]]
=> ? = 0 + 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [[],[[],[[[[],[]],[[],[]]],[[],[]]]]]
=> ? = 0 + 1
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [[],[[],[[[[],[[],[]]],[]],[[],[]]]]]
=> ? = 0 + 1
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [[],[[],[[[[[],[]],[]],[]],[[],[]]]]]
=> ? = 0 + 1
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [[],[[],[[[],[[],[[],[[],[]]]]],[]]]]
=> ? = 0 + 1
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [[],[[],[[[],[[],[[[],[]],[]]]],[]]]]
=> ? = 0 + 1
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [[],[[],[[[],[[[],[]],[[],[]]]],[]]]]
=> ? = 0 + 1
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [[],[[],[[[],[[[],[[],[]]],[]]],[]]]]
=> ? = 0 + 1
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [[],[[],[[[],[[[[],[]],[]],[]]],[]]]]
=> ? = 0 + 1
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [[],[[],[[[[],[]],[[],[[],[]]]],[]]]]
=> ? = 0 + 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [[],[[],[[[[],[]],[[[],[]],[]]],[]]]]
=> ? = 0 + 1
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [[],[[],[[[[],[[],[]]],[[],[]]],[]]]]
=> ? = 0 + 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [[],[[],[[[[[],[]],[]],[[],[]]],[]]]]
=> ? = 0 + 1
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [[],[[],[[[[],[[],[[],[]]]],[]],[]]]]
=> ? = 0 + 1
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [[],[[],[[[[],[[[],[]],[]]],[]],[]]]]
=> ? = 0 + 1
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [[],[[],[[[[[],[]],[[],[]]],[]],[]]]]
=> ? = 0 + 1
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [[],[[],[[[[[],[[],[]]],[]],[]],[]]]]
=> ? = 0 + 1
[.,[.,[[[[[.,.],.],.],.],.]]]
=> [[],[[],[[[[[[],[]],[]],[]],[]],[]]]]
=> ? = 0 + 1
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [[],[[[],[]],[[],[[],[[],[[],[]]]]]]]
=> ? = 0 + 1
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [[],[[[],[]],[[],[[],[[[],[]],[]]]]]]
=> ? = 0 + 1
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [[],[[[],[]],[[],[[[],[]],[[],[]]]]]]
=> ? = 0 + 1
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> [[],[[[],[]],[[],[[[],[[],[]]],[]]]]]
=> ? = 0 + 1
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [[],[[[],[]],[[],[[[[],[]],[]],[]]]]]
=> ? = 0 + 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [[],[[[],[]],[[[],[]],[[],[[],[]]]]]]
=> ? = 0 + 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [[],[[[],[]],[[[],[]],[[[],[]],[]]]]]
=> ? = 0 + 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [[],[[[],[]],[[[],[[],[]]],[[],[]]]]]
=> ? = 0 + 1
Description
The protection number of an ordered tree.
This is the minimal distance from the root to a leaf.
Matching statistic: St001570
Values
[.,.]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? = 0
[.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> ? = 0
[[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> ? = 0
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[.,[[[[[.,.],.],.],.],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St000259
Values
[.,.]
=> ([],1)
=> ([],1)
=> ([],1)
=> 0
[.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 0
[[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 0
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 0
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 0
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 0
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 0
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000260
Values
[.,.]
=> ([],1)
=> ([],1)
=> ([],1)
=> 0
[.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 0
[[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 0
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 0
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 0
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 0
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 0
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
[[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 0
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
The following 17 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St000456The monochromatic index of a connected graph. St001198The number of simple modules in the algebra eAe with projective dimension at most 1 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001206The maximal dimension of an indecomposable projective eAe-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module eA. St000768The number of peaks in an integer composition. St001964The interval resolution global dimension of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1.
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