- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 11111

[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [1,2,3,4,6,5] => 11110

[.,[.,[.,[[.,.],[.,.]]]]] => [6,4,5,3,2,1] => [1,2,3,5,4,6] => 11101

[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [1,2,3,6,4,5] => 11100

[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [1,2,3,6,5,4] => 11100

[.,[.,[[.,.],[.,[.,.]]]]] => [6,5,3,4,2,1] => [1,2,4,3,5,6] => 11011

[.,[.,[[.,.],[[.,.],.]]]] => [5,6,3,4,2,1] => [1,2,4,3,6,5] => 11010

[.,[.,[[.,[.,.]],[.,.]]]] => [6,4,3,5,2,1] => [1,2,5,3,4,6] => 11001

[.,[.,[[[.,.],.],[.,.]]]] => [6,3,4,5,2,1] => [1,2,5,4,3,6] => 11001

[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [1,2,6,3,4,5] => 11000

[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [1,2,6,3,5,4] => 11000

[.,[.,[[[.,.],[.,.]],.]]] => [5,3,4,6,2,1] => [1,2,6,4,3,5] => 11000

[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [1,2,6,5,3,4] => 11000

[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [1,2,6,5,4,3] => 11000

[.,[[.,.],[.,[.,[.,.]]]]] => [6,5,4,2,3,1] => [1,3,2,4,5,6] => 10111

[.,[[.,.],[.,[[.,.],.]]]] => [5,6,4,2,3,1] => [1,3,2,4,6,5] => 10110

[.,[[.,.],[[.,.],[.,.]]]] => [6,4,5,2,3,1] => [1,3,2,5,4,6] => 10101

[.,[[.,.],[[.,[.,.]],.]]] => [5,4,6,2,3,1] => [1,3,2,6,4,5] => 10100

[.,[[.,.],[[[.,.],.],.]]] => [4,5,6,2,3,1] => [1,3,2,6,5,4] => 10100

[.,[[.,[.,.]],[.,[.,.]]]] => [6,5,3,2,4,1] => [1,4,2,3,5,6] => 10011

[.,[[.,[.,.]],[[.,.],.]]] => [5,6,3,2,4,1] => [1,4,2,3,6,5] => 10010

[.,[[[.,.],.],[.,[.,.]]]] => [6,5,2,3,4,1] => [1,4,3,2,5,6] => 10011

[.,[[[.,.],.],[[.,.],.]]] => [5,6,2,3,4,1] => [1,4,3,2,6,5] => 10010

[.,[[.,[.,[.,.]]],[.,.]]] => [6,4,3,2,5,1] => [1,5,2,3,4,6] => 10001

[.,[[.,[[.,.],.]],[.,.]]] => [6,3,4,2,5,1] => [1,5,2,4,3,6] => 10001

[.,[[[.,.],[.,.]],[.,.]]] => [6,4,2,3,5,1] => [1,5,3,2,4,6] => 10001

[.,[[[.,[.,.]],.],[.,.]]] => [6,3,2,4,5,1] => [1,5,4,2,3,6] => 10001

[.,[[[[.,.],.],.],[.,.]]] => [6,2,3,4,5,1] => [1,5,4,3,2,6] => 10001

[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [1,6,2,3,4,5] => 10000

[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [1,6,2,3,5,4] => 10000

[.,[[.,[[.,.],[.,.]]],.]] => [5,3,4,2,6,1] => [1,6,2,4,3,5] => 10000

[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [1,6,2,5,3,4] => 10000

[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [1,6,2,5,4,3] => 10000

[.,[[[.,.],[.,[.,.]]],.]] => [5,4,2,3,6,1] => [1,6,3,2,4,5] => 10000

[.,[[[.,.],[[.,.],.]],.]] => [4,5,2,3,6,1] => [1,6,3,2,5,4] => 10000

[.,[[[.,[.,.]],[.,.]],.]] => [5,3,2,4,6,1] => [1,6,4,2,3,5] => 10000

[.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => [1,6,4,3,2,5] => 10000

>>> Load all 1722 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

reverse

Description

Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.

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.