- FindStat

Identifier

Mp00223: Permutations —runsort⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000204: Binary trees ⟶ ℤ

Values

[1] => [1] => [.,.] => 0

[1,2] => [1,2] => [.,[.,.]] => 0

[2,1] => [1,2] => [.,[.,.]] => 0

[1,2,3] => [1,2,3] => [.,[.,[.,.]]] => 0

[1,3,2] => [1,3,2] => [.,[[.,.],.]] => 1

[2,1,3] => [1,3,2] => [.,[[.,.],.]] => 1

[2,3,1] => [1,2,3] => [.,[.,[.,.]]] => 0

[3,1,2] => [1,2,3] => [.,[.,[.,.]]] => 0

[3,2,1] => [1,2,3] => [.,[.,[.,.]]] => 0

[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 0

[1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]] => 1

[1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]] => 1

[1,3,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]] => 2

[1,4,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]] => 1

[1,4,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]] => 1

[2,1,3,4] => [1,3,4,2] => [.,[[.,[.,.]],.]] => 2

[2,1,4,3] => [1,4,2,3] => [.,[[.,.],[.,.]]] => 1

[2,3,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]] => 1

[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 0

[2,4,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]] => 1

[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]] => 1

[3,1,2,4] => [1,2,4,3] => [.,[.,[[.,.],.]]] => 1

[3,1,4,2] => [1,4,2,3] => [.,[[.,.],[.,.]]] => 1

[3,2,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]] => 1

[3,2,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]] => 1

[3,4,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 0

[3,4,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 0

[4,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 0

[4,1,3,2] => [1,3,2,4] => [.,[[.,.],[.,.]]] => 1

[4,2,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]] => 1

[4,2,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 0

[4,3,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 0

[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 0

[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 0

[1,2,3,5,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => 1

[1,2,4,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]] => 1

[1,2,4,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]] => 2

[1,2,5,3,4] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]] => 1

[1,2,5,4,3] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]] => 1

[1,3,2,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]] => 1

[1,3,2,5,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]] => 2

[1,3,4,2,5] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]] => 2

[1,3,4,5,2] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]] => 3

[1,3,5,2,4] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]] => 2

[1,3,5,4,2] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]] => 2

[1,4,2,3,5] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]] => 1

[1,4,2,5,3] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]] => 2

[1,4,3,2,5] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]] => 2

[1,4,3,5,2] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]] => 1

[1,4,5,2,3] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]] => 2

[1,4,5,3,2] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]] => 2

[1,5,2,3,4] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]] => 1

[1,5,2,4,3] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]] => 2

[1,5,3,2,4] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]] => 2

[1,5,3,4,2] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]] => 1

[1,5,4,2,3] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]] => 1

[1,5,4,3,2] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]] => 1

[2,1,3,4,5] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]] => 3

[2,1,3,5,4] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]] => 2

[2,1,4,3,5] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]] => 1

[2,1,4,5,3] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]] => 2

[2,1,5,3,4] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]] => 1

[2,1,5,4,3] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]] => 1

[2,3,1,4,5] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]] => 2

[2,3,1,5,4] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]] => 1

[2,3,4,1,5] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]] => 1

[2,3,4,5,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 0

[2,3,5,1,4] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]] => 1

[2,3,5,4,1] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => 1

[2,4,1,3,5] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]] => 2

[2,4,1,5,3] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]] => 2

[2,4,3,1,5] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]] => 2

[2,4,3,5,1] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]] => 1

[2,4,5,1,3] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]] => 1

[2,4,5,3,1] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]] => 2

[2,5,1,3,4] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]] => 2

[2,5,1,4,3] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]] => 2

[2,5,3,1,4] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]] => 2

[2,5,3,4,1] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]] => 1

[2,5,4,1,3] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]] => 2

[2,5,4,3,1] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]] => 1

[3,1,2,4,5] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]] => 2

[3,1,2,5,4] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]] => 1

[3,1,4,2,5] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]] => 2

[3,1,4,5,2] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]] => 2

[3,1,5,2,4] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]] => 2

[3,1,5,4,2] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]] => 1

[3,2,1,4,5] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]] => 2

[3,2,1,5,4] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]] => 1

[3,2,4,1,5] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]] => 2

[3,2,4,5,1] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]] => 2

[3,2,5,1,4] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]] => 2

[3,2,5,4,1] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]] => 1

[3,4,1,2,5] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]] => 1

[3,4,1,5,2] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]] => 1

[3,4,2,1,5] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]] => 1

[3,4,2,5,1] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]] => 1

[3,4,5,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 0

[3,4,5,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 0

[3,5,1,2,4] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]] => 1

[3,5,1,4,2] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]] => 1

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Description

The number of internal nodes of a binary tree.
That is, the total number of nodes of the tree minus St000203The number of external nodes of a binary tree.. 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.

Map

runsort

Description

The permutation obtained by sorting the increasing runs lexicographically.

Map

to increasing tree

Description

Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.