- FindStat

Identifier

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St001737: Permutations ⟶ ℤ

Values

[1] => [1] => [.,.] => [1] => 0

[1,2] => [1,2] => [.,[.,.]] => [2,1] => 1

[2,1] => [1,2] => [.,[.,.]] => [2,1] => 1

[1,2,3] => [1,2,3] => [.,[.,[.,.]]] => [3,2,1] => 1

[1,3,2] => [1,2,3] => [.,[.,[.,.]]] => [3,2,1] => 1

[2,1,3] => [1,2,3] => [.,[.,[.,.]]] => [3,2,1] => 1

[2,3,1] => [1,2,3] => [.,[.,[.,.]]] => [3,2,1] => 1

[3,1,2] => [1,3,2] => [.,[[.,.],.]] => [2,3,1] => 1

[3,2,1] => [1,3,2] => [.,[[.,.],.]] => [2,3,1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2\ q$

$F_{3} = 6\ q$

$F_{4} = 22\ q + 2\ q^{2}$

$F_{5} = 106\ q + 14\ q^{2}$

$F_{6} = 630\ q + 88\ q^{2} + 2\ q^{3}$

Description

The number of descents of type 2 in a permutation.
A position

$i\in[1,n-1]$ is a descent of type 2 of a permutation

$\pi$ of

$n$ letters, if it is a descent and if

$\pi(j) < \pi(i)$ for all

$j < i$ .

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.

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

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.