- FindStat

Identifier

Mp00069: Permutations —complement⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000196: Binary trees ⟶ ℤ (values match St000201The number of leaf nodes in a binary tree.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 5913 entries. <<<

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$F_{1} = 1$

$F_{2} = 2$

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

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

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

$F_{6} = 32 + 416\ q + 272\ q^{2}$

$F_{7} = 64 + 1824\ q + 2880\ q^{2} + 272\ q^{3}$

Description

The number of occurrences of the contiguous pattern [[.,.],[.,.]] in a binary tree.
Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by

$2^{n-2}$ .

Map

complement

Description

Sents a permutation to its complement.
The complement of a permutation

$\sigma$ of length

$n$ is the permutation

$\tau$ with

$\tau(i) = n+1-\sigma(i)$

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.