- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1656 entries. <<<

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.

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.

Map

to composition

Description

The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.