Identifier
Values
[.,[.,.]] => [2,1] => [1,2] => [1,2] => 1
[[.,.],.] => [1,2] => [1,2] => [1,2] => 1
[.,[.,[.,.]]] => [3,2,1] => [1,3,2] => [3,1,2] => 0
[.,[[.,.],.]] => [2,3,1] => [1,2,3] => [1,2,3] => 2
[[.,.],[.,.]] => [1,3,2] => [1,2,3] => [1,2,3] => 2
[[.,[.,.]],.] => [2,1,3] => [1,2,3] => [1,2,3] => 2
[[[.,.],.],.] => [1,2,3] => [1,2,3] => [1,2,3] => 2
[.,[.,[.,[.,.]]]] => [4,3,2,1] => [1,4,2,3] => [1,4,2,3] => 1
[.,[.,[[.,.],.]]] => [3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 1
[.,[[.,.],[.,.]]] => [2,4,3,1] => [1,2,4,3] => [4,1,2,3] => 1
[.,[[.,[.,.]],.]] => [3,2,4,1] => [1,3,4,2] => [3,1,2,4] => 1
[.,[[[.,.],.],.]] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 3
[[.,.],[.,[.,.]]] => [1,4,3,2] => [1,2,4,3] => [4,1,2,3] => 1
[[.,.],[[.,.],.]] => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 3
[[.,[.,.]],[.,.]] => [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 3
[[[.,.],.],[.,.]] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 3
[[.,[.,[.,.]]],.] => [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 1
[[.,[[.,.],.]],.] => [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 3
[[[.,.],[.,.]],.] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 3
[[[.,[.,.]],.],.] => [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 3
[[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 3
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [1,5,2,4,3] => [5,1,4,2,3] => 0
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [1,4,2,5,3] => [4,1,5,2,3] => 0
[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => [1,3,4,2,5] => [1,3,2,4,5] => 2
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [1,4,2,3,5] => [1,2,4,3,5] => 2
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [1,3,5,2,4] => [3,5,1,2,4] => 2
[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => [1,2,5,3,4] => [1,5,2,3,4] => 2
[.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => [1,2,4,3,5] => [1,4,2,3,5] => 2
[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => [1,3,5,2,4] => [3,5,1,2,4] => 2
[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => 2
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [1,4,5,2,3] => [4,5,1,2,3] => 2
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [1,3,2,4,5] => [1,3,4,2,5] => 2
[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => [1,2,4,5,3] => [4,1,2,3,5] => 2
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [1,3,4,5,2] => [3,1,2,4,5] => 2
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => [1,2,5,3,4] => [1,5,2,3,4] => 2
[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => [1,2,4,3,5] => [1,4,2,3,5] => 2
[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => 2
[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => [1,2,4,5,3] => [4,1,2,3,5] => 2
[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => 2
[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => 2
[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => [1,3,2,4,5] => [1,3,4,2,5] => 2
[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [1,4,2,3,5] => [1,2,4,3,5] => 2
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [1,3,2,4,5] => [1,3,4,2,5] => 2
[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => [1,2,4,3,5] => [1,4,2,3,5] => 2
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [1,3,4,2,5] => [1,3,2,4,5] => 2
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => [1,2,4,3,5] => [1,4,2,3,5] => 2
[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [1,3,2,4,5] => [1,3,4,2,5] => 2
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 4
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Description
The number of augmented double ascents of a permutation.
An augmented double ascent of a permutation $\pi$ is a double ascent of the augmented permutation $\tilde\pi$ obtained from $\pi$ by adding an initial $0$.
A double ascent of $\tilde\pi$ then is a position $i$ such that $\tilde\pi(i) < \tilde\pi(i+1) < \tilde\pi(i+2)$.
Map
to 312-avoiding permutation
Description
Return a 312-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 minimal element of this Sylvester class.
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
Map
cactus evacuation
Description
The cactus evacuation of a permutation.
This is the involution obtained by applying evacuation to the recording tableau, while preserving the insertion tableau.