Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000204
(load all 4 compositions to match this statistic)
Mapping
St000204: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => 0
[.,[.,.]]
 => 0
[[.,.],.]
 => 0
[.,[.,[.,.]]]
 => 0
[.,[[.,.],.]]
 => 1
[[.,.],[.,.]]
 => 0
[[.,[.,.]],.]
 => 1
[[[.,.],.],.]
 => 0
[.,[.,[.,[.,.]]]]
 => 0
[.,[.,[[.,.],.]]]
 => 1
[.,[[.,.],[.,.]]]
 => 1
[.,[[.,[.,.]],.]]
 => 2
[.,[[[.,.],.],.]]
 => 2
[[.,.],[.,[.,.]]]
 => 0
[[.,.],[[.,.],.]]
 => 1
[[.,[.,.]],[.,.]]
 => 1
[[[.,.],.],[.,.]]
 => 0
[[.,[.,[.,.]]],.]
 => 2
[[.,[[.,.],.]],.]
 => 2
[[[.,.],[.,.]],.]
 => 1
[[[.,[.,.]],.],.]
 => 1
[[[[.,.],.],.],.]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => 0
[.,[.,[.,[[.,.],.]]]]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => 2
[.,[.,[[[.,.],.],.]]]
 => 2
[.,[[.,.],[.,[.,.]]]]
 => 1
[.,[[.,.],[[.,.],.]]]
 => 2
[.,[[.,[.,.]],[.,.]]]
 => 2
[.,[[[.,.],.],[.,.]]]
 => 2
[.,[[.,[.,[.,.]]],.]]
 => 3
[.,[[.,[[.,.],.]],.]]
 => 3
[.,[[[.,.],[.,.]],.]]
 => 3
[.,[[[.,[.,.]],.],.]]
 => 3
[.,[[[[.,.],.],.],.]]
 => 3
[[.,.],[.,[.,[.,.]]]]
 => 0
[[.,.],[.,[[.,.],.]]]
 => 1
[[.,.],[[.,.],[.,.]]]
 => 1
[[.,.],[[.,[.,.]],.]]
 => 2
[[.,.],[[[.,.],.],.]]
 => 2
[[.,[.,.]],[.,[.,.]]]
 => 1
[[.,[.,.]],[[.,.],.]]
 => 2
[[[.,.],.],[.,[.,.]]]
 => 0
[[[.,.],.],[[.,.],.]]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => 2
[[.,[[.,.],.]],[.,.]]
 => 2
[[[.,.],[.,.]],[.,.]]
 => 1
[[[.,[.,.]],.],[.,.]]
 => 1
[[[[.,.],.],.],[.,.]]
 => 0
Description
The number of internal nodes of a binary tree. That is, the total number of nodes of the tree minus [[St000203]]. A counting formula for the total number of internal nodes across all binary trees of size $n$ is given in [1]. This is equivalent to the number of internal triangles in all triangulations of an $(n+1)$-gon.
Matching statistic: St000019
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => [] => ? = 0
[.,[.,.]]
 => [2,1] => [2,1] => [1] => 0
[[.,.],.]
 => [1,2] => [1,2] => [1] => 0
[.,[.,[.,.]]]
 => [3,2,1] => [3,1,2] => [1,2] => 0
[.,[[.,.],.]]
 => [2,3,1] => [3,2,1] => [2,1] => 1
[[.,.],[.,.]]
 => [1,3,2] => [1,3,2] => [1,2] => 0
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => [2,1] => 1
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [1,2] => 0
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,1,2,3] => [1,2,3] => 0
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,1,3,2] => [1,3,2] => 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [4,2,1,3] => [2,1,3] => 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,3,2,1] => [3,2,1] => 2
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [4,2,3,1] => [2,3,1] => 2
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,4,2,3] => [1,2,3] => 0
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,4,3,2] => [1,3,2] => 1
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,1,4,3] => [2,1,3] => 1
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,4,3] => [1,2,3] => 0
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,1,2,4] => [3,1,2] => 2
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,2,1,4] => [3,2,1] => 2
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,3,2,4] => [1,3,2] => 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3] => 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3] => 0
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [5,1,2,4,3] => [1,2,4,3] => 1
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [5,1,3,2,4] => [1,3,2,4] => 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,1,4,3,2] => [1,4,3,2] => 2
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [5,1,3,4,2] => [1,3,4,2] => 2
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [5,2,1,3,4] => [2,1,3,4] => 1
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [5,2,1,4,3] => [2,1,4,3] => 2
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [5,3,2,1,4] => [3,2,1,4] => 2
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [5,2,3,1,4] => [2,3,1,4] => 2
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,4,2,3,1] => [4,2,3,1] => 3
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [5,4,3,2,1] => [4,3,2,1] => 3
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [5,2,4,3,1] => [2,4,3,1] => 3
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [5,3,2,4,1] => [3,2,4,1] => 3
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,1] => 3
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,5,2,3,4] => [1,2,3,4] => 0
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,5,2,4,3] => [1,2,4,3] => 1
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,5,3,2,4] => [1,3,2,4] => 1
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,5,4,3,2] => [1,4,3,2] => 2
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,2] => 2
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [2,1,5,3,4] => [2,1,3,4] => 1
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,3] => 2
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,2,5,3,4] => [1,2,3,4] => 0
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,3] => 1
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [3,1,2,5,4] => [3,1,2,4] => 2
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,4] => 2
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,4] => 1
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,4] => 1
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4] => 0
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [4,1,2,3,5] => [4,1,2,3] => 3
Description
The cardinality of the support of a permutation. A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$. The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product. See [2], Definition 1 and Proposition 10. The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$. Thus, the connectivity set is the complement of the support.