- FindStat

Identifier

Mp00252: Permutations —restriction⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000201: Binary trees ⟶ ℤ (values match St000196The number of occurrences of the contiguous pattern [[.,.],[.,.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 12803 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The number of leaf nodes in a binary tree.
Equivalently, the number of cherries [1] in the complete binary tree.
The number of binary trees of size $n$, at least $1$, with exactly one leaf node for is $2^{n-1}$, see [2].
The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].

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.

Map

restriction

Description

The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.