- FindStat

Identifier

Mp00277: Permutations —catalanization⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words

Images

[1] => [1] =>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 6850 entries. <<<

Map

catalanization

Description

The catalanization of a permutation.
For a permutation $\sigma$, this is the product of the reflections corresponding to the inversions of $\sigma$ in lex-order.
A permutation is $231$-avoiding if and only if it is a fixpoint of this map. Also, for every permutation there exists an index $k$ such that the $k$-fold application of this map is $231$-avoiding.

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.