- FindStat

Identifier

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000501: Permutations ⟶ ℤ

Values

[1] => [1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 3603 entries. <<<

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$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 6\ q$

$F_{4} = 24\ q$

$F_{5} = 120\ q$

$F_{6} = 720\ q$

Description

The size of the first part in the decomposition of a permutation.
For a permutation

$\pi$ of

$\{1,\ldots,n\}$ , this is defined to be the smallest

$k > 0$ such that

$\{\pi(1),\ldots,\pi(k)\} = \{1,\ldots,k\}$ . This statistic is undefined for the empty permutation.
For the number of parts in the decomposition see St000056The decomposition (or block) number of a permutation..

Map

cycle-as-one-line notation

Description

Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.