- FindStat

Identifier

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001866: Signed permutations ⟶ ℤ

Values

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 5 + q$

$F_{4} = 15 + 8\ q + q^{2}$

Description

The nesting alignments of a signed permutation.
A nesting alignment of a signed permutation

$\pi\in\mathfrak H_n$ is a pair

$1\leq i, j \leq n$ such that

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

Map

to signed permutation

Description

The signed permutation with all signs positive.

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

to increasing tree

Description

Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length

$0$ to the empty tree, and sending a permutation

$\sigma$ of length

$n \geq 1$ to a root node with two subtrees

$L$ and

$R$ by splitting

$\sigma$ at the index

$\sigma^{-1}(1)$ , normalizing both sides again to permutations and sending the permutations on the left and on the right of

$\sigma^{-1}(1)$ to the trees

$L$ and

$R$ , respectively.