Identifier
-
Mp00014:
Binary trees
—to 132-avoiding permutation⟶
Permutations
St000092: Permutations ⟶ ℤ (values match St000353The number of inner valleys of a permutation.)
Values
[.,.] => [1] => 1
[.,[.,.]] => [2,1] => 1
[[.,.],.] => [1,2] => 1
[.,[.,[.,.]]] => [3,2,1] => 1
[.,[[.,.],.]] => [2,3,1] => 1
[[.,.],[.,.]] => [3,1,2] => 2
[[.,[.,.]],.] => [2,1,3] => 2
[[[.,.],.],.] => [1,2,3] => 1
[.,[.,[.,[.,.]]]] => [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] => 2
[[.,.],[[.,.],.]] => [3,4,1,2] => 2
[[.,[.,.]],[.,.]] => [4,2,1,3] => 2
[[[.,.],.],[.,.]] => [4,1,2,3] => 2
[[.,[.,[.,.]]],.] => [3,2,1,4] => 2
[[.,[[.,.],.]],.] => [2,3,1,4] => 2
[[[.,.],[.,.]],.] => [3,1,2,4] => 2
[[[.,[.,.]],.],.] => [2,1,3,4] => 2
[[[[.,.],.],.],.] => [1,2,3,4] => 1
[.,[.,[.,[.,[.,.]]]]] => [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] => 2
[[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => 2
[[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => 3
[[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => 3
[[.,.],[[[.,.],.],.]] => [3,4,5,1,2] => 2
[[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => 2
[[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => 2
[[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => 2
[[[.,.],.],[[.,.],.]] => [4,5,1,2,3] => 2
[[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => 2
[[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => 3
[[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => 2
[[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => 2
[[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => 2
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 2
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => 2
[[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => 3
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => 3
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => 2
[[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => 2
[[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => 2
[[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => 2
[[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => 2
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => 2
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => 2
[[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => 2
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => 2
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 1
[.,[.,[.,[.,[.,[.,.]]]]]] => [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 196 entries. <<<
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Description
The number of outer peaks of a permutation.
An 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 $w_1 > w_2$ or $n$ if $w_{n} > w_{n-1}$.
In other words, it is a peak in the word $[0,w_1,..., w_n,0]$.
An 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 $w_1 > w_2$ or $n$ if $w_{n} > w_{n-1}$.
In other words, it is a peak in the word $[0,w_1,..., w_n,0]$.
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.
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.
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