Identifier
Values
[.,[.,.]] => [2,1] => [2,1] => 0
[[.,.],.] => [1,2] => [1,2] => 0
[.,[.,[.,.]]] => [3,2,1] => [3,2,1] => 0
[.,[[.,.],.]] => [2,3,1] => [1,3,2] => 0
[[.,.],[.,.]] => [1,3,2] => [1,3,2] => 0
[[.,[.,.]],.] => [2,1,3] => [2,1,3] => 0
[[[.,.],.],.] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]] => [4,3,2,1] => [4,3,2,1] => 1
[.,[.,[[.,.],.]]] => [3,4,2,1] => [1,4,3,2] => 0
[.,[[.,.],[.,.]]] => [2,4,3,1] => [1,4,3,2] => 0
[.,[[.,[.,.]],.]] => [3,2,4,1] => [2,1,4,3] => 0
[.,[[[.,.],.],.]] => [2,3,4,1] => [1,2,4,3] => 0
[[.,.],[.,[.,.]]] => [1,4,3,2] => [1,4,3,2] => 0
[[.,.],[[.,.],.]] => [1,3,4,2] => [1,2,4,3] => 0
[[.,[.,.]],[.,.]] => [2,1,4,3] => [2,1,4,3] => 0
[[[.,.],.],[.,.]] => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,[.,.]]],.] => [3,2,1,4] => [3,2,1,4] => 0
[[.,[[.,.],.]],.] => [2,3,1,4] => [1,3,2,4] => 0
[[[.,.],[.,.]],.] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,[.,.]],.],.] => [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [5,4,3,2,1] => 2
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [1,5,4,3,2] => 1
[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => [1,5,4,3,2] => 1
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [2,1,5,4,3] => 0
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [1,2,5,4,3] => 0
[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => [1,5,4,3,2] => 1
[.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => [1,2,5,4,3] => 0
[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => [2,1,5,4,3] => 0
[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => [1,2,5,4,3] => 0
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [3,2,1,5,4] => 0
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [1,3,2,5,4] => 0
[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => [1,3,2,5,4] => 0
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [2,1,3,5,4] => 0
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [1,2,3,5,4] => 0
[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => [1,2,5,4,3] => 0
[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => [1,2,5,4,3] => 0
[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => [1,3,2,5,4] => 0
[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => [1,2,3,5,4] => 0
[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => [2,1,5,4,3] => 0
[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => [2,1,3,5,4] => 0
[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => [1,2,3,5,4] => 0
[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => [3,2,1,5,4] => 0
[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => [1,3,2,5,4] => 0
[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [1,4,3,2,5] => 0
[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => [1,4,3,2,5] => 0
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [2,1,4,3,5] => 0
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [1,2,4,3,5] => 0
[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => [1,2,4,3,5] => 0
[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => [2,1,4,3,5] => 0
[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [3,2,1,4,5] => 0
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [1,3,2,4,5] => 0
[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,2,3,4,5] => 0
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Description
The number of strict 3-descents.
A strict 3-descent of a permutation $\pi$ of $\{1,2, \dots ,n \}$ is a pair $(i,i+3)$ with $ i+3 \leq n$ and $\pi(i) > \pi(i+3)$.
Map
Inverse fireworks map
Description
Sends a permutation to an inverse fireworks permutation.
A permutation $\sigma$ is inverse fireworks if its inverse avoids the vincular pattern $3-12$. The inverse fireworks map sends any permutation $\sigma$ to an inverse fireworks permutation that is below $\sigma$ in left weak order and has the same Rajchgot index St001759The Rajchgot index of a permutation..
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.