Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000398
(load all 2 compositions to match this statistic)
Mapping
Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000398: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [.,.]
 => 0
[.,[.,.]]
 => [.,[.,.]]
 => 1
[[.,.],.]
 => [[.,.],.]
 => 1
[.,[.,[.,.]]]
 => [.,[.,[.,.]]]
 => 3
[.,[[.,.],.]]
 => [.,[[.,.],.]]
 => 3
[[.,.],[.,.]]
 => [[.,[.,.]],.]
 => 3
[[.,[.,.]],.]
 => [[.,.],[.,.]]
 => 2
[[[.,.],.],.]
 => [[[.,.],.],.]
 => 3
[.,[.,[.,[.,.]]]]
 => [.,[.,[.,[.,.]]]]
 => 6
[.,[.,[[.,.],.]]]
 => [.,[.,[[.,.],.]]]
 => 6
[.,[[.,.],[.,.]]]
 => [.,[[.,[.,.]],.]]
 => 6
[.,[[.,[.,.]],.]]
 => [.,[[.,.],[.,.]]]
 => 5
[.,[[[.,.],.],.]]
 => [.,[[[.,.],.],.]]
 => 6
[[.,.],[.,[.,.]]]
 => [[.,[.,[.,.]]],.]
 => 6
[[.,.],[[.,.],.]]
 => [[.,[[.,.],.]],.]
 => 6
[[.,[.,.]],[.,.]]
 => [[.,[.,.]],[.,.]]
 => 4
[[[.,.],.],[.,.]]
 => [[[.,[.,.]],.],.]
 => 6
[[.,[.,[.,.]]],.]
 => [[.,.],[.,[.,.]]]
 => 4
[[.,[[.,.],.]],.]
 => [[.,.],[[.,.],.]]
 => 4
[[[.,.],[.,.]],.]
 => [[[.,.],[.,.]],.]
 => 5
[[[.,[.,.]],.],.]
 => [[[.,.],.],[.,.]]
 => 4
[[[[.,.],.],.],.]
 => [[[[.,.],.],.],.]
 => 6
[.,[.,[.,[.,[.,.]]]]]
 => [.,[.,[.,[.,[.,.]]]]]
 => 10
[.,[.,[.,[[.,.],.]]]]
 => [.,[.,[.,[[.,.],.]]]]
 => 10
[.,[.,[[.,.],[.,.]]]]
 => [.,[.,[[.,[.,.]],.]]]
 => 10
[.,[.,[[.,[.,.]],.]]]
 => [.,[.,[[.,.],[.,.]]]]
 => 9
[.,[.,[[[.,.],.],.]]]
 => [.,[.,[[[.,.],.],.]]]
 => 10
[.,[[.,.],[.,[.,.]]]]
 => [.,[[.,[.,[.,.]]],.]]
 => 10
[.,[[.,.],[[.,.],.]]]
 => [.,[[.,[[.,.],.]],.]]
 => 10
[.,[[.,[.,.]],[.,.]]]
 => [.,[[.,[.,.]],[.,.]]]
 => 8
[.,[[[.,.],.],[.,.]]]
 => [.,[[[.,[.,.]],.],.]]
 => 10
[.,[[.,[.,[.,.]]],.]]
 => [.,[[.,.],[.,[.,.]]]]
 => 8
[.,[[.,[[.,.],.]],.]]
 => [.,[[.,.],[[.,.],.]]]
 => 8
[.,[[[.,.],[.,.]],.]]
 => [.,[[[.,.],[.,.]],.]]
 => 9
[.,[[[.,[.,.]],.],.]]
 => [.,[[[.,.],.],[.,.]]]
 => 8
[.,[[[[.,.],.],.],.]]
 => [.,[[[[.,.],.],.],.]]
 => 10
[[.,.],[.,[.,[.,.]]]]
 => [[.,[.,[.,[.,.]]]],.]
 => 10
[[.,.],[.,[[.,.],.]]]
 => [[.,[.,[[.,.],.]]],.]
 => 10
[[.,.],[[.,.],[.,.]]]
 => [[.,[[.,[.,.]],.]],.]
 => 10
[[.,.],[[.,[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => 9
[[.,.],[[[.,.],.],.]]
 => [[.,[[[.,.],.],.]],.]
 => 10
[[.,[.,.]],[.,[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => 7
[[.,[.,.]],[[.,.],.]]
 => [[.,[[.,.],.]],[.,.]]
 => 7
[[[.,.],.],[.,[.,.]]]
 => [[[.,[.,[.,.]]],.],.]
 => 10
[[[.,.],.],[[.,.],.]]
 => [[[.,[[.,.],.]],.],.]
 => 10
[[.,[.,[.,.]]],[.,.]]
 => [[.,[.,.]],[.,[.,.]]]
 => 6
[[.,[[.,.],.]],[.,.]]
 => [[.,[.,.]],[[.,.],.]]
 => 6
[[[.,.],[.,.]],[.,.]]
 => [[[.,[.,.]],[.,.]],.]
 => 8
[[[.,[.,.]],.],[.,.]]
 => [[[.,[.,.]],.],[.,.]]
 => 7
[[[[.,.],.],.],[.,.]]
 => [[[[.,[.,.]],.],.],.]
 => 10
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: 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.