Processing math: 83%

Your data matches 5 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000398
Mapping
St000398: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => 0
[.,[.,.]]
 => 1
[[.,.],.]
 => 1
[.,[.,[.,.]]]
 => 3
[.,[[.,.],.]]
 => 3
[[.,.],[.,.]]
 => 2
[[.,[.,.]],.]
 => 3
[[[.,.],.],.]
 => 3
[.,[.,[.,[.,.]]]]
 => 6
[.,[.,[[.,.],.]]]
 => 6
[.,[[.,.],[.,.]]]
 => 5
[.,[[.,[.,.]],.]]
 => 6
[.,[[[.,.],.],.]]
 => 6
[[.,.],[.,[.,.]]]
 => 4
[[.,.],[[.,.],.]]
 => 4
[[.,[.,.]],[.,.]]
 => 4
[[[.,.],.],[.,.]]
 => 4
[[.,[.,[.,.]]],.]
 => 6
[[.,[[.,.],.]],.]
 => 6
[[[.,.],[.,.]],.]
 => 5
[[[.,[.,.]],.],.]
 => 6
[[[[.,.],.],.],.]
 => 6
[.,[.,[.,[.,[.,.]]]]]
 => 10
[.,[.,[.,[[.,.],.]]]]
 => 10
[.,[.,[[.,.],[.,.]]]]
 => 9
[.,[.,[[.,[.,.]],.]]]
 => 10
[.,[.,[[[.,.],.],.]]]
 => 10
[.,[[.,.],[.,[.,.]]]]
 => 8
[.,[[.,.],[[.,.],.]]]
 => 8
[.,[[.,[.,.]],[.,.]]]
 => 8
[.,[[[.,.],.],[.,.]]]
 => 8
[.,[[.,[.,[.,.]]],.]]
 => 10
[.,[[.,[[.,.],.]],.]]
 => 10
[.,[[[.,.],[.,.]],.]]
 => 9
[.,[[[.,[.,.]],.],.]]
 => 10
[.,[[[[.,.],.],.],.]]
 => 10
[[.,.],[.,[.,[.,.]]]]
 => 7
[[.,.],[.,[[.,.],.]]]
 => 7
[[.,.],[[.,.],[.,.]]]
 => 6
[[.,.],[[.,[.,.]],.]]
 => 7
[[.,.],[[[.,.],.],.]]
 => 7
[[.,[.,.]],[.,[.,.]]]
 => 6
[[.,[.,.]],[[.,.],.]]
 => 6
[[[.,.],.],[.,[.,.]]]
 => 6
[[[.,.],.],[[.,.],.]]
 => 6
[[.,[.,[.,.]]],[.,.]]
 => 7
[[.,[[.,.],.]],[.,.]]
 => 7
[[[.,.],[.,.]],[.,.]]
 => 6
[[[.,[.,.]],.],[.,.]]
 => 7
[[[[.,.],.],.],[.,.]]
 => 7
Description
The sum of the depths of the vertices (or total internal path length) of a binary tree. The depth of a vertex is the number of edges to the tree's root, see Section 2.3.4.5 of [1] and [3]. This statistic is the very first entry of the OEIS, see [2]. Observe that there the term '''height''' is used instead.
Matching statistic: St000081
Mapping
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => ([],1)
 => ([],1)
 => ([],1)
 => 0
[.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 1
[[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 5
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 5
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
[.,[.,[.,[.,[.,.]]]]]
 => ([(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)
 => 10
[.,[.,[.,[[.,.],.]]]]
 => ([(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)
 => 10
[.,[.,[[.,.],[.,.]]]]
 => ([(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)
 => 9
[.,[.,[[.,[.,.]],.]]]
 => ([(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)
 => 10
[.,[.,[[[.,.],.],.]]]
 => ([(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)
 => 10
[.,[[.,.],[.,[.,.]]]]
 => ([(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)
 => 8
[.,[[.,.],[[.,.],.]]]
 => ([(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)
 => 8
[.,[[.,[.,.]],[.,.]]]
 => ([(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)
 => 8
[.,[[[.,.],.],[.,.]]]
 => ([(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)
 => 8
[.,[[.,[.,[.,.]]],.]]
 => ([(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)
 => 10
[.,[[.,[[.,.],.]],.]]
 => ([(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)
 => 10
[.,[[[.,.],[.,.]],.]]
 => ([(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)
 => 9
[.,[[[.,[.,.]],.],.]]
 => ([(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)
 => 10
[.,[[[[.,.],.],.],.]]
 => ([(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)
 => 10
[[.,.],[.,[.,[.,.]]]]
 => ([(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)
 => 7
[[.,.],[.,[[.,.],.]]]
 => ([(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)
 => 7
[[.,.],[[.,.],[.,.]]]
 => ([(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)
 => 6
[[.,.],[[.,[.,.]],.]]
 => ([(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)
 => 7
[[.,.],[[[.,.],.],.]]
 => ([(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)
 => 7
[[.,[.,.]],[.,[.,.]]]
 => ([(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)
 => 6
[[.,[.,.]],[[.,.],.]]
 => ([(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)
 => 6
[[[.,.],.],[.,[.,.]]]
 => ([(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)
 => 6
[[[.,.],.],[[.,.],.]]
 => ([(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)
 => 6
[[.,[.,[.,.]]],[.,.]]
 => ([(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)
 => 7
[[.,[[.,.],.]],[.,.]]
 => ([(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)
 => 7
[[[.,.],[.,.]],[.,.]]
 => ([(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)
 => 6
[[[.,[.,.]],.],[.,.]]
 => ([(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)
 => 7
[[[[.,.],.],.],[.,.]]
 => ([(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)
 => 7
Description
The number of edges of a graph.
Matching statistic: St000161
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000161: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => [.,.]
 => 0
[.,[.,.]]
 => [2,1] => [1,2] => [.,[.,.]]
 => 1
[[.,.],.]
 => [1,2] => [1,2] => [.,[.,.]]
 => 1
[.,[.,[.,.]]]
 => [3,2,1] => [1,2,3] => [.,[.,[.,.]]]
 => 3
[.,[[.,.],.]]
 => [2,3,1] => [1,2,3] => [.,[.,[.,.]]]
 => 3
[[.,.],[.,.]]
 => [3,1,2] => [1,2,3] => [.,[.,[.,.]]]
 => 3
[[.,[.,.]],.]
 => [2,1,3] => [1,3,2] => [.,[[.,.],.]]
 => 2
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [.,[.,[.,.]]]
 => 3
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 6
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 6
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 6
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => 5
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 6
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 6
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 6
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => 4
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 6
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [1,4,2,3] => [.,[[.,[.,.]],.]]
 => 4
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [1,4,2,3] => [.,[[.,[.,.]],.]]
 => 4
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => 5
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => 4
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 6
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 10
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 10
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 10
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => 9
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 10
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 10
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 10
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => 8
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 10
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => 8
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => 8
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => 9
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => 8
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 10
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 10
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 10
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 10
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => 9
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 10
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => 7
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => 7
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 10
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 10
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => 6
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => 6
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => 8
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => 7
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 10
Description
The sum of the sizes of the right subtrees of a binary tree. This statistic corresponds to [[St000012]] under the Tamari Dyck path-binary tree bijection, and to [[St000018]] of the 312-avoiding permutation corresponding to the binary tree. It is also the sum of all heights j of the coordinates (i,j) of the Dyck path corresponding to the binary tree.
Matching statistic: St000448
Mapping
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000448: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => ([],1)
 => ([],1)
 => ([(0,1)],2)
 => 0
[.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,2),(1,2)],3)
 => 1
[[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,2),(1,2)],3)
 => 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 6
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 6
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 6
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 6
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 6
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 6
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 6
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 6
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 10
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 10
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 9
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 10
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 10
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 8
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 8
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 8
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 8
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 10
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 10
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 9
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 10
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 10
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 6
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 6
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 6
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 6
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
Description
The number of pairs of vertices of a graph with distance 2. This is the coefficient of the quadratic term of the Wiener polynomial.
Matching statistic: St001646
Mapping
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001646: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => ([],1)
 => ([],1)
 => ([(0,1)],2)
 => 0
[.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,2),(1,2)],3)
 => 1
[[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,2),(1,2)],3)
 => 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 6
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 6
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 6
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 6
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 6
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 6
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 6
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 6
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 10
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 10
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 9
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 10
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 10
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 8
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 8
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 8
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 8
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 10
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 10
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 9
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 10
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 10
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 6
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 6
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 6
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 6
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
Description
The number of edges that can be added without increasing the maximal degree of a graph. This statistic is (except for the degenerate case of two vertices) maximized by the star-graph on n vertices, which has maximal degree n−1 and therefore has statistic \binom{n-1}{2}.