- FindStat

Identifier

Mp00061: Permutations —to increasing tree⟶ Binary trees
St000385: Binary trees ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 12894 entries. <<<

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$F_{2} = 2\ q$

$F_{3} = 2 + 4\ q^{2}$

$F_{4} = 16\ q + 8\ q^{3}$

$F_{5} = 16 + 88\ q^{2} + 16\ q^{4}$

$F_{6} = 272\ q + 416\ q^{3} + 32\ q^{5}$

$F_{7} = 272 + 2880\ q^{2} + 1824\ q^{4} + 64\ q^{6}$

Description

The number of vertices with out-degree 1 in a binary tree.
See the references for several connections of this statistic.
In particular, the number

$T(n,k)$ of binary trees with

$n$ vertices and

$k$ out-degree

$1$ vertices is given by

$T(n,k) = 0$ for

$n-k$ odd and

$T(n,k)=\frac{2^k}{n+1}\binom{n+1}{k}\binom{n+1-k}{(n-k)/2}$
for

$n-k$ is even.

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.