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Your data matches 37 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000122
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000122: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000122: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [.,.]
=> [.,.]
=> 0
[.,[.,.]]
=> [2,1] => [[.,.],.]
=> [[.,.],.]
=> 0
[[.,.],.]
=> [1,2] => [.,[.,.]]
=> [.,[.,.]]
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => [[[.,.],.],.]
=> [[[.,.],.],.]
=> 0
[.,[[.,.],.]]
=> [2,3,1] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> 0
[[.,.],[.,.]]
=> [1,3,2] => [.,[[.,.],.]]
=> [.,[[.,.],.]]
=> 0
[[.,[.,.]],.]
=> [2,1,3] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> 0
[[[.,.],.],.]
=> [1,2,3] => [.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [[[[[.,.],.],.],.],.]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> 1
Description
The number of occurrences of the contiguous pattern {{{[.,[.,[[.,.],.]]]}}} in a binary tree.
[[oeis:A086581]] counts binary trees avoiding this pattern.
Matching statistic: St000125
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
St000125: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
St000125: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [.,.]
=> [.,.]
=> [.,.]
=> 0
[.,[.,.]]
=> [[.,.],.]
=> [[.,.],.]
=> [.,[.,.]]
=> 0
[[.,.],.]
=> [.,[.,.]]
=> [.,[.,.]]
=> [[.,.],.]
=> 0
[.,[.,[.,.]]]
=> [[[.,.],.],.]
=> [[[.,.],.],.]
=> [.,[.,[.,.]]]
=> 0
[.,[[.,.],.]]
=> [[.,[.,.]],.]
=> [[.,.],[.,.]]
=> [[.,.],[.,.]]
=> 0
[[.,.],[.,.]]
=> [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> [.,[[.,.],.]]
=> 0
[[.,[.,.]],.]
=> [.,[[.,.],.]]
=> [.,[[.,.],.]]
=> [[.,[.,.]],.]
=> 0
[[[.,.],.],.]
=> [.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> [[[.,.],.],.]
=> 0
[.,[.,[.,[.,.]]]]
=> [[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> 0
[.,[.,[[.,.],.]]]
=> [[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> 0
[.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> 0
[.,[[.,[.,.]],.]]
=> [[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> 0
[.,[[[.,.],.],.]]
=> [[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> 0
[[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> [.,[.,[[.,.],.]]]
=> 0
[[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> 0
[[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> [.,[[.,[.,.]],.]]
=> 0
[[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [.,[[[.,.],.],.]]
=> 1
[[.,[.,[.,.]]],.]
=> [.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> [[.,[.,[.,.]]],.]
=> 0
[[.,[[.,.],.]],.]
=> [.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> 0
[[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> [[.,[[.,.],.]],.]
=> 0
[[[.,[.,.]],.],.]
=> [.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> [[[.,[.,.]],.],.]
=> 0
[[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> [[[[.,.],.],.],.]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],.]
=> [[[[[.,.],.],.],.],.]
=> [.,[.,[.,[.,[.,.]]]]]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [[[[.,[.,.]],.],.],.]
=> [[[[.,.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,.]]]]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> [[[[.,.],.],[.,.]],.]
=> [.,[[.,.],[.,[.,.]]]]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],.],.]
=> [[[.,.],.],[[.,.],.]]
=> [[.,[.,.]],[.,[.,.]]]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [[[.,.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,.]]]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> [[[[.,.],[.,.]],.],.]
=> [.,[.,[[.,.],[.,.]]]]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],.]
=> [[[.,.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,.]]]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [[[.,.],[[.,.],.]],.]
=> [.,[[.,[.,.]],[.,.]]]
=> 0
[.,[[[.,.],.],[.,.]]]
=> [[[.,.],[.,[.,.]]],.]
=> [[[.,.],[.,[.,.]]],.]
=> [.,[[[.,.],.],[.,.]]]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],.]
=> [[.,.],[[[.,.],.],.]]
=> [[.,[.,[.,.]]],[.,.]]
=> 0
[.,[[.,[[.,.],.]],.]]
=> [[.,[[.,[.,.]],.]],.]
=> [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],.]],[.,.]]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [[.,[.,[[.,.],.]]],.]
=> [[.,.],[.,[[.,.],.]]]
=> [[[.,[.,.]],.],[.,.]]
=> 0
[.,[[[[.,.],.],.],.]]
=> [[.,[.,[.,[.,.]]]],.]
=> [[.,.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,.]]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> [.,[.,[.,[[.,.],.]]]]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> [[.,.],[.,[[.,.],.]]]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> [.,[[.,.],[[.,.],.]]]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> [[.,[.,.]],[[.,.],.]]
=> 0
[[.,.],[[[.,.],.],.]]
=> [[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [[[.,.],.],[[.,.],.]]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> [.,[.,[[.,[.,.]],.]]]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> [[.,.],[[.,[.,.]],.]]
=> 0
[[[.,.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [.,[.,[[[.,.],.],.]]]
=> 1
[[[.,.],.],[[.,.],.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [[.,.],[[[.,.],.],.]]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> [.,[[.,[.,[.,.]]],.]]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [.,[[[.,.],[.,.]],.]]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> [.,[[.,[[.,.],.]],.]]
=> 0
[[[.,[.,.]],.],[.,.]]
=> [[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> [.,[[[.,[.,.]],.],.]]
=> 1
[[[[.,.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [.,[[[[.,.],.],.],.]]
=> 1
Description
The number of occurrences of the contiguous pattern {{{[.,[[[.,.],.],.]]}}} in a binary tree.
[[oeis:A005773]] counts binary trees avoiding this pattern.
Matching statistic: St000871
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000871: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000871: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Values
[.,.]
=> [1] => [.,.]
=> [1] => 0
[.,[.,.]]
=> [2,1] => [[.,.],.]
=> [1,2] => 0
[[.,.],.]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 0
[.,[[.,.],.]]
=> [2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => 0
[[.,.],[.,.]]
=> [3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => 0
[[.,[.,.]],.]
=> [2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => 0
[[[.,.],.],.]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 0
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 0
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 0
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 0
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [7,4,5,6,3,2,1] => [[[[[.,.],.],.],[.,[.,.]]],.]
=> [6,5,1,2,3,4,7] => ? = 1
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [7,5,6,3,4,2,1] => [[[[[.,.],.],[.,.]],[.,.]],.]
=> [6,4,1,2,3,5,7] => ? = 0
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [5,6,7,3,4,2,1] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 0
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [7,6,3,4,5,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
=> [5,4,1,2,3,6,7] => ? = 1
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [6,7,3,4,5,2,1] => [[[[.,.],.],[.,[.,.]]],[.,.]]
=> [7,5,4,1,2,3,6] => ? = 1
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [7,4,5,3,6,2,1] => [[[[[.,.],.],.],[.,[.,.]]],.]
=> [6,5,1,2,3,4,7] => ? = 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [7,5,3,4,6,2,1] => [[[[[.,.],.],[.,.]],[.,.]],.]
=> [6,4,1,2,3,5,7] => ? = 0
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [7,4,3,5,6,2,1] => [[[[[.,.],.],.],[.,[.,.]]],.]
=> [6,5,1,2,3,4,7] => ? = 1
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [7,3,4,5,6,2,1] => [[[[.,.],.],[.,[.,[.,.]]]],.]
=> [6,5,4,1,2,3,7] => ? = 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [5,6,3,4,7,2,1] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 0
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [6,3,4,5,7,2,1] => [[[[.,.],.],[.,[.,.]]],[.,.]]
=> [7,5,4,1,2,3,6] => ? = 1
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [5,3,4,6,7,2,1] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 0
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [7,5,6,4,2,3,1] => [[[[[.,.],[.,.]],.],[.,.]],.]
=> [6,3,1,2,4,5,7] => ? = 0
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [5,6,7,4,2,3,1] => [[[[.,.],[.,.]],.],[.,[.,.]]]
=> [7,6,3,1,2,4,5] => ? = 0
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,2,3,1] => [[[[[.,.],[.,.]],[.,.]],.],.]
=> [5,3,1,2,4,6,7] => ? = 0
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [6,7,4,5,2,3,1] => [[[[.,.],[.,.]],[.,.]],[.,.]]
=> [7,5,3,1,2,4,6] => ? = 0
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [7,5,4,6,2,3,1] => [[[[[.,.],[.,.]],.],[.,.]],.]
=> [6,3,1,2,4,5,7] => ? = 0
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> [7,4,5,6,2,3,1] => [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [6,5,3,1,2,4,7] => ? = 1
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [5,6,4,7,2,3,1] => [[[[.,.],[.,.]],.],[.,[.,.]]]
=> [7,6,3,1,2,4,5] => ? = 0
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> [6,4,5,7,2,3,1] => [[[[.,.],[.,.]],[.,.]],[.,.]]
=> [7,5,3,1,2,4,6] => ? = 0
[.,[[.,.],[[[.,[.,.]],.],.]]]
=> [5,4,6,7,2,3,1] => [[[[.,.],[.,.]],.],[.,[.,.]]]
=> [7,6,3,1,2,4,5] => ? = 0
[.,[[.,.],[[[[.,.],.],.],.]]]
=> [4,5,6,7,2,3,1] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ? = 0
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [7,5,6,3,2,4,1] => [[[[[.,.],.],[.,.]],[.,.]],.]
=> [6,4,1,2,3,5,7] => ? = 0
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> [5,6,7,3,2,4,1] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 0
[.,[[[.,.],.],[.,[.,[.,.]]]]]
=> [7,6,5,2,3,4,1] => [[[[[.,.],[.,[.,.]]],.],.],.]
=> [4,3,1,2,5,6,7] => ? = 1
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> [6,7,5,2,3,4,1] => [[[[.,.],[.,[.,.]]],.],[.,.]]
=> [7,4,3,1,2,5,6] => ? = 1
[.,[[[.,.],.],[[.,.],[.,.]]]]
=> [7,5,6,2,3,4,1] => [[[[.,.],[.,[.,.]]],[.,.]],.]
=> [6,4,3,1,2,5,7] => ? = 1
[.,[[[.,.],.],[[.,[.,.]],.]]]
=> [6,5,7,2,3,4,1] => [[[[.,.],[.,[.,.]]],.],[.,.]]
=> [7,4,3,1,2,5,6] => ? = 1
[.,[[[.,.],.],[[[.,.],.],.]]]
=> [5,6,7,2,3,4,1] => [[[.,.],[.,[.,.]]],[.,[.,.]]]
=> [7,6,4,3,1,2,5] => ? = 1
[.,[[.,[[.,.],.]],[.,[.,.]]]]
=> [7,6,3,4,2,5,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
=> [5,4,1,2,3,6,7] => ? = 1
[.,[[.,[[.,.],.]],[[.,.],.]]]
=> [6,7,3,4,2,5,1] => [[[[.,.],.],[.,[.,.]]],[.,.]]
=> [7,5,4,1,2,3,6] => ? = 1
[.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [7,6,4,2,3,5,1] => [[[[[.,.],[.,.]],[.,.]],.],.]
=> [5,3,1,2,4,6,7] => ? = 0
[.,[[[.,.],[.,.]],[[.,.],.]]]
=> [6,7,4,2,3,5,1] => [[[[.,.],[.,.]],[.,.]],[.,.]]
=> [7,5,3,1,2,4,6] => ? = 0
[.,[[[.,[.,.]],.],[.,[.,.]]]]
=> [7,6,3,2,4,5,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
=> [5,4,1,2,3,6,7] => ? = 1
[.,[[[.,[.,.]],.],[[.,.],.]]]
=> [6,7,3,2,4,5,1] => [[[[.,.],.],[.,[.,.]]],[.,.]]
=> [7,5,4,1,2,3,6] => ? = 1
[.,[[[[.,.],.],.],[.,[.,.]]]]
=> [7,6,2,3,4,5,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [5,4,3,1,2,6,7] => ? = 1
[.,[[[[.,.],.],.],[[.,.],.]]]
=> [6,7,2,3,4,5,1] => [[[.,.],[.,[.,[.,.]]]],[.,.]]
=> [7,5,4,3,1,2,6] => ? = 1
[.,[[.,[.,[[.,.],.]]],[.,.]]]
=> [7,4,5,3,2,6,1] => [[[[[.,.],.],.],[.,[.,.]]],.]
=> [6,5,1,2,3,4,7] => ? = 1
[.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [7,5,3,4,2,6,1] => [[[[[.,.],.],[.,.]],[.,.]],.]
=> [6,4,1,2,3,5,7] => ? = 0
[.,[[.,[[.,[.,.]],.]],[.,.]]]
=> [7,4,3,5,2,6,1] => [[[[[.,.],.],.],[.,[.,.]]],.]
=> [6,5,1,2,3,4,7] => ? = 1
[.,[[.,[[[.,.],.],.]],[.,.]]]
=> [7,3,4,5,2,6,1] => [[[[.,.],.],[.,[.,[.,.]]]],.]
=> [6,5,4,1,2,3,7] => ? = 1
[.,[[[.,.],[.,[.,.]]],[.,.]]]
=> [7,5,4,2,3,6,1] => [[[[[.,.],[.,.]],.],[.,.]],.]
=> [6,3,1,2,4,5,7] => ? = 0
[.,[[[.,.],[[.,.],.]],[.,.]]]
=> [7,4,5,2,3,6,1] => [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [6,5,3,1,2,4,7] => ? = 1
[.,[[[.,[.,.]],[.,.]],[.,.]]]
=> [7,5,3,2,4,6,1] => [[[[[.,.],.],[.,.]],[.,.]],.]
=> [6,4,1,2,3,5,7] => ? = 0
[.,[[[[.,.],.],[.,.]],[.,.]]]
=> [7,5,2,3,4,6,1] => [[[[.,.],[.,[.,.]]],[.,.]],.]
=> [6,4,3,1,2,5,7] => ? = 1
[.,[[[.,[.,[.,.]]],.],[.,.]]]
=> [7,4,3,2,5,6,1] => [[[[[.,.],.],.],[.,[.,.]]],.]
=> [6,5,1,2,3,4,7] => ? = 1
[.,[[[.,[[.,.],.]],.],[.,.]]]
=> [7,3,4,2,5,6,1] => [[[[.,.],.],[.,[.,[.,.]]]],.]
=> [6,5,4,1,2,3,7] => ? = 1
[.,[[[[.,.],[.,.]],.],[.,.]]]
=> [7,4,2,3,5,6,1] => [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [6,5,3,1,2,4,7] => ? = 1
[.,[[[[.,[.,.]],.],.],[.,.]]]
=> [7,3,2,4,5,6,1] => [[[[.,.],.],[.,[.,[.,.]]]],.]
=> [6,5,4,1,2,3,7] => ? = 1
[.,[[[[[.,.],.],.],.],[.,.]]]
=> [7,2,3,4,5,6,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [6,5,4,3,1,2,7] => ? = 1
Description
The number of very big ascents of a permutation.
A very big ascent of a permutation π is an index i such that πi+1−πi>2.
For the number of ascents, see [[St000245]] and for the number of big ascents, see [[St000646]]. General r-ascents were for example be studied in [1, Section 2].
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)
=> ? = 0
[[.,[.,.]],.]
=> ([(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)
=> ? = 0
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[[[.,.],.],[.,.]]
=> ([(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)
=> ? = 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),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,.],[[.,.],[.,.]]]
=> ([(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)
=> ? = 0
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[[.,.],.],[.,[.,.]]]
=> ([(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)
=> ? = 0
[[.,[[.,.],.]],[.,.]]
=> ([(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)
=> ? = 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),(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)
=> ? = 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
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(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)
=> ? = 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(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)
=> ? = 1
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(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)
=> ? = 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(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)
=> ? = 0
[[.,[.,.]],.]
=> ([(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)
=> ? = 0
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[[[.,.],.],[.,.]]
=> ([(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)
=> ? = 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),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,.],[[.,.],[.,.]]]
=> ([(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)
=> ? = 0
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[[.,.],.],[.,[.,.]]]
=> ([(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)
=> ? = 0
[[.,[[.,.],.]],[.,.]]
=> ([(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)
=> ? = 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),(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)
=> ? = 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
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(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)
=> ? = 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(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)
=> ? = 1
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(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)
=> ? = 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(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.
Matching statistic: St000302
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)
=> ? = 0
[[.,[.,.]],.]
=> ([(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)
=> ? = 0
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[[[.,.],.],[.,.]]
=> ([(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)
=> ? = 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),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,.],[[.,.],[.,.]]]
=> ([(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)
=> ? = 0
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[[.,.],.],[.,[.,.]]]
=> ([(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)
=> ? = 0
[[.,[[.,.],.]],[.,.]]
=> ([(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)
=> ? = 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),(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)
=> ? = 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
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(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)
=> ? = 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(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)
=> ? = 1
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(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)
=> ? = 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(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 determinant of the distance matrix of a connected graph.
Matching statistic: St000466
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)
=> ? = 0
[[.,[.,.]],.]
=> ([(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)
=> ? = 0
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[[[.,.],.],[.,.]]
=> ([(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)
=> ? = 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),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,.],[[.,.],[.,.]]]
=> ([(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)
=> ? = 0
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[[.,.],.],[.,[.,.]]]
=> ([(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)
=> ? = 0
[[.,[[.,.],.]],[.,.]]
=> ([(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)
=> ? = 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),(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)
=> ? = 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
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(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)
=> ? = 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(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)
=> ? = 1
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(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)
=> ? = 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(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 Gutman (or modified Schultz) index of a connected graph.
This is
∑{u,v}⊆Vd(u)d(v)d(u,v)
where d(u) is the degree of vertex u and d(u,v) is the distance between vertices u and v.
For trees on n vertices, the modified Schultz index is related to the Wiener index via S∗(T)=4W(T)−(n−1)(2n−1) [1].
Matching statistic: St000467
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)
=> ? = 0
[[.,[.,.]],.]
=> ([(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)
=> ? = 0
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[[[.,.],.],[.,.]]
=> ([(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)
=> ? = 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),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,.],[[.,.],[.,.]]]
=> ([(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)
=> ? = 0
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0
[[[.,.],.],[.,[.,.]]]
=> ([(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)
=> ? = 0
[[.,[[.,.],.]],[.,.]]
=> ([(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)
=> ? = 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),(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)
=> ? = 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
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(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)
=> ? = 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(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)
=> ? = 1
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(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)
=> ? = 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(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 hyper-Wiener index of a connected graph.
This is
∑{u,v}⊆Vd(u,v)+d(u,v)2.
Matching statistic: St000771
Values
[.,.]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 1 = 0 + 1
[[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 1 = 0 + 1
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 1 = 0 + 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 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)
=> ? = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 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)
=> ? = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[[.,.],[[.,.],[.,.]]]]
=> ([(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 + 1
[.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums 0, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
(4−1−2−1−14−1−2−2−14−1−1−2−14).
Its eigenvalues are 0,4,4,6, so the statistic is 2.
The path on four vertices has eigenvalues 0,4.7…,6,9.2… and therefore statistic 1.
Matching statistic: St000772
Values
[.,.]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 1 = 0 + 1
[[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 1 = 0 + 1
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 1 = 0 + 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 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)
=> ? = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 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)
=> ? = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[[.,.],[[.,.],[.,.]]]]
=> ([(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 + 1
[.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 1
[.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
[[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],1)
=> 1 = 0 + 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums 0, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
(4−1−2−1−14−1−2−2−14−1−1−2−14).
Its eigenvalues are 0,4,4,6, so the statistic is 1.
The path on four vertices has eigenvalues 0,4.7…,6,9.2… and therefore also statistic 1.
The graphs with statistic n−1, n−2 and n−3 have been characterised, see [1].
The following 27 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001490The number of connected components of a skew partition. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000629The defect of a binary word. St001141The number of occurrences of hills of size 3 in a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St001584The area statistic between a Dyck path and its bounce path. St000655The length of the minimal rise of a Dyck path. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices.
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