- FindStat

Identifier

Mp00069: Permutations —complement⟶ Permutations
Mp00277: Permutations —catalanization⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions

Images

[1] => [1] => [1] => [1]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 7075 entries. <<<

Map

complement

Description

Sents a permutation to its complement.
The complement of a permutation

$\sigma$ of length

$n$ is the permutation

$\tau$ with

$\tau(i) = n+1-\sigma(i)$

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

cycle type

Description

The cycle type of a permutation as a partition.