- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000035: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => 1

[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => 1

[.,[.,[.,[[.,.],[.,.]]]]] => [6,4,5,3,2,1] => 2

[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => 2

[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => 1

[.,[.,[[.,.],[.,[.,.]]]]] => [6,5,3,4,2,1] => 2

[.,[.,[[.,.],[[.,.],.]]]] => [5,6,3,4,2,1] => 2

[.,[.,[[.,[.,.]],[.,.]]]] => [6,4,3,5,2,1] => 2

[.,[.,[[[.,.],.],[.,.]]]] => [6,3,4,5,2,1] => 2

[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => 2

[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => 2

[.,[.,[[[.,.],[.,.]],.]]] => [5,3,4,6,2,1] => 2

[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => 2

[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => 1

[.,[[.,.],[.,[.,[.,.]]]]] => [6,5,4,2,3,1] => 2

[.,[[.,.],[.,[[.,.],.]]]] => [5,6,4,2,3,1] => 2

[.,[[.,.],[[.,.],[.,.]]]] => [6,4,5,2,3,1] => 3

[.,[[.,.],[[.,[.,.]],.]]] => [5,4,6,2,3,1] => 3

[.,[[.,.],[[[.,.],.],.]]] => [4,5,6,2,3,1] => 2

[.,[[.,[.,.]],[.,[.,.]]]] => [6,5,3,2,4,1] => 2

[.,[[.,[.,.]],[[.,.],.]]] => [5,6,3,2,4,1] => 2

[.,[[[.,.],.],[.,[.,.]]]] => [6,5,2,3,4,1] => 2

[.,[[[.,.],.],[[.,.],.]]] => [5,6,2,3,4,1] => 2

[.,[[.,[.,[.,.]]],[.,.]]] => [6,4,3,2,5,1] => 2

[.,[[.,[[.,.],.]],[.,.]]] => [6,3,4,2,5,1] => 3

[.,[[[.,.],[.,.]],[.,.]]] => [6,4,2,3,5,1] => 2

[.,[[[.,[.,.]],.],[.,.]]] => [6,3,2,4,5,1] => 2

[.,[[[[.,.],.],.],[.,.]]] => [6,2,3,4,5,1] => 2

[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => 2

[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => 2

[.,[[.,[[.,.],[.,.]]],.]] => [5,3,4,2,6,1] => 3

[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => 3

[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => 2

[.,[[[.,.],[.,[.,.]]],.]] => [5,4,2,3,6,1] => 2

[.,[[[.,.],[[.,.],.]],.]] => [4,5,2,3,6,1] => 2

[.,[[[.,[.,.]],[.,.]],.]] => [5,3,2,4,6,1] => 2

[.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => 2

>>> Load all 445 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 left outer peaks of a permutation.
A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$.
In other words, it is a peak in the word $[0,w_1,..., w_n]$.
This appears in [1, def.3.1]. The joint distribution with St000366The number of double descents of a permutation. is studied in [3], where left outer peaks are called exterior peaks.

Map

to 132-avoiding permutation

Description

Return a 132-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the maximal element of the Sylvester class.