- FindStat

Identifier

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words

Images

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

[.,[.,.]] => [2,1] => [2,1] => 0

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

[.,[.,[.,.]]] => [3,2,1] => [3,2,1] => 00

[.,[[.,.],.]] => [2,3,1] => [2,3,1] => 00

[[.,.],[.,.]] => [1,3,2] => [3,1,2] => 00

[[.,[.,.]],.] => [2,1,3] => [2,1,3] => 01

[[[.,.],.],.] => [1,2,3] => [1,2,3] => 11

[.,[.,[.,[.,.]]]] => [4,3,2,1] => [4,3,2,1] => 000

[.,[.,[[.,.],.]]] => [3,4,2,1] => [3,4,2,1] => 000

[.,[[.,.],[.,.]]] => [2,4,3,1] => [4,2,3,1] => 000

[.,[[.,[.,.]],.]] => [3,2,4,1] => [3,2,4,1] => 000

[.,[[[.,.],.],.]] => [2,3,4,1] => [2,3,4,1] => 000

[[.,.],[.,[.,.]]] => [1,4,3,2] => [4,3,1,2] => 000

[[.,.],[[.,.],.]] => [1,3,4,2] => [3,1,4,2] => 000

[[.,[.,.]],[.,.]] => [2,1,4,3] => [4,2,1,3] => 000

[[[.,.],.],[.,.]] => [1,2,4,3] => [4,1,2,3] => 000

[[.,[.,[.,.]]],.] => [3,2,1,4] => [3,2,1,4] => 001

[[.,[[.,.],.]],.] => [2,3,1,4] => [2,3,1,4] => 001

[[[.,.],[.,.]],.] => [1,3,2,4] => [3,1,2,4] => 001

[[[.,[.,.]],.],.] => [2,1,3,4] => [2,1,3,4] => 011

[[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => 111

[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [5,4,3,2,1] => 0000

[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [4,5,3,2,1] => 0000

[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => [5,3,4,2,1] => 0000

[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [4,3,5,2,1] => 0000

[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [3,4,5,2,1] => 0000

[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => [5,4,2,3,1] => 0000

[.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => [4,2,5,3,1] => 0000

[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => [5,3,2,4,1] => 0000

[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => [5,2,3,4,1] => 0000

[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [4,3,2,5,1] => 0000

[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [3,4,2,5,1] => 0000

[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => [4,2,3,5,1] => 0000

[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [3,2,4,5,1] => 0000

[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [2,3,4,5,1] => 0000

[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => [5,4,3,1,2] => 0000

[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => [4,5,1,3,2] => 0000

[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => [5,3,1,4,2] => 0000

[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => [4,3,1,5,2] => 0000

[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => [3,1,4,5,2] => 0000

[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => [5,4,2,1,3] => 0000

[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => [4,2,1,5,3] => 0000

[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => [5,4,1,2,3] => 0000

[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => [4,1,2,5,3] => 0000

[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => [5,3,2,1,4] => 0000

[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => [5,2,3,1,4] => 0000

[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => [5,3,1,2,4] => 0000

[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => [5,2,1,3,4] => 0000

[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => [5,1,2,3,4] => 0000

[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [4,3,2,1,5] => 0001

[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [3,4,2,1,5] => 0001

[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => [4,2,3,1,5] => 0001

[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [3,2,4,1,5] => 0001

[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [2,3,4,1,5] => 0001

[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => [4,3,1,2,5] => 0001

[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => [3,1,4,2,5] => 0001

[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => [4,2,1,3,5] => 0001

[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => [4,1,2,3,5] => 0001

[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [3,2,1,4,5] => 0011

[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [2,3,1,4,5] => 0011

[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => [3,1,2,4,5] => 0011

[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [2,1,3,4,5] => 0111

[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,2,3,4,5] => 1111

[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 00000

[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [5,6,4,3,2,1] => 00000

[.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => [6,4,5,3,2,1] => 00000

[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [5,4,6,3,2,1] => 00000

[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [4,5,6,3,2,1] => 00000

[.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => [6,5,3,4,2,1] => 00000

[.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => [5,3,6,4,2,1] => 00000

[.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => [6,4,3,5,2,1] => 00000

[.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => [6,3,4,5,2,1] => 00000

[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [5,4,3,6,2,1] => 00000

[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [4,5,3,6,2,1] => 00000

[.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => [5,3,4,6,2,1] => 00000

[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [4,3,5,6,2,1] => 00000

[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [3,4,5,6,2,1] => 00000

[.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => [6,5,4,2,3,1] => 00000

[.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => [5,6,2,4,3,1] => 00000

[.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => [6,4,2,5,3,1] => 00000

[.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => [5,4,2,6,3,1] => 00000

[.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => [4,2,5,6,3,1] => 00000

[.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => [6,5,3,2,4,1] => 00000

[.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => [5,3,2,6,4,1] => 00000

[.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => [6,5,2,3,4,1] => 00000

[.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => [5,2,3,6,4,1] => 00000

[.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => [6,4,3,2,5,1] => 00000

[.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => [6,3,4,2,5,1] => 00000

[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => [6,4,2,3,5,1] => 00000

[.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => [6,3,2,4,5,1] => 00000

[.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => [6,2,3,4,5,1] => 00000

[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [5,4,3,2,6,1] => 00000

[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [4,5,3,2,6,1] => 00000

[.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => [5,3,4,2,6,1] => 00000

[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [4,3,5,2,6,1] => 00000

[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [3,4,5,2,6,1] => 00000

[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => [5,4,2,3,6,1] => 00000

[.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => [4,2,5,3,6,1] => 00000

[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => [5,3,2,4,6,1] => 00000

[.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => [5,2,3,4,6,1] => 00000

>>> Load all 1549 entries. <<<

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

Foata bijection

Description

Sends a permutation to its image under the Foata bijection.
The Foata bijection

$\phi$ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word

$w_1 w_2 ... w_n$ , compute the image inductively by starting with

$\phi(w_1) = w_1$ .
At the

$i$ -th step, if

$\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$ , define

$\phi(w_1 w_2 ... w_i w_{i+1})$ by placing

$w_{i+1}$ on the end of the word

$v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.

If $w_{i+1} \geq v_i$ , place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$ .
If $w_{i+1} < v_i$ , place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$ .

In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
To compute

$\phi([1,4,2,5,3])$ , the sequence of words is

$1$
$|1|4 \to 14$
$|14|2 \to 412$
$|4|1|2|5 \to 4125$
$|4|125|3 \to 45123.$

In total, this gives

$\phi([1,4,2,5,3]) = [4,5,1,2,3]$ .
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).

Map

connectivity set

Description

The connectivity set of a permutation as a binary word.
According to [2], also known as the global ascent set.
The connectivity set is

$C(\pi)=\{i\in [n-1] | \forall 1 \leq j \leq i < k \leq n : \pi(j) < \pi(k)\}.$
For

$n > 1$ it can also be described as the set of occurrences of the mesh pattern

$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$
or equivalently

$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$
see [3].
The permutation is connected, when the connectivity set is empty.