- FindStat

Identifier

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

Images

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

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

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

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

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

[[.,.],[.,.]] => [1,3,2] => 10

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

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

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

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

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

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

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

[[.,.],[.,[.,.]]] => [1,4,3,2] => 100

[[.,.],[[.,.],.]] => [1,3,4,2] => 100

[[.,[.,.]],[.,.]] => [2,1,4,3] => 010

[[[.,.],.],[.,.]] => [1,2,4,3] => 110

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

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

[[[.,.],[.,.]],.] => [1,3,2,4] => 101

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 1000

[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => 1000

[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 1000

[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => 1000

[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => 1000

[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 0100

[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => 0100

[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => 1100

[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => 1100

[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => 0010

[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => 0010

[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => 1010

[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => 0110

[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => 1110

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

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

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

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

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

[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => 1001

[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => 1001

[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => 0101

[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => 1101

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

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

[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => 1011

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1671 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

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.