- FindStat

Identifier

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001862: Signed permutations ⟶ ℤ

Values

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 5$

$F_{4} = 13 + q$

Description

The number of crossings of a signed permutation.
A crossing of a signed permutation

$\pi$ is a pair

$(i, j)$ of indices such that

$i < j \leq \pi(i) < \pi(j)$ , or
$-i < j \leq -\pi(i) < \pi(j)$ , or
$i > j > \pi(i) > \pi(j)$ .

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

inverse

Description

Sends a permutation to its inverse.

Map

to signed permutation

Description

The signed permutation with all signs positive.