- FindStat

Identifier

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000446: Permutations ⟶ ℤ

Values

[1] => [1] => 0

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

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

[1,2,3] => [1,2,3] => 0

[1,3,2] => [1,2,3] => 0

[2,1,3] => [1,2,3] => 0

[2,3,1] => [1,2,3] => 0

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

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

[1,2,3,4] => [1,2,3,4] => 0

[1,2,4,3] => [1,2,3,4] => 0

[1,3,2,4] => [1,2,3,4] => 0

[1,3,4,2] => [1,2,3,4] => 0

[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] => 0

[2,1,4,3] => [1,2,3,4] => 0

[2,3,1,4] => [1,2,3,4] => 0

[2,3,4,1] => [1,2,3,4] => 0

[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] => 2

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

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

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

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

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

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

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

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

[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] => 0

[1,2,3,5,4] => [1,2,3,4,5] => 0

[1,2,4,3,5] => [1,2,3,4,5] => 0

[1,2,4,5,3] => [1,2,3,4,5] => 0

[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] => 0

[1,3,2,5,4] => [1,2,3,4,5] => 0

[1,3,4,2,5] => [1,2,3,4,5] => 0

[1,3,4,5,2] => [1,2,3,4,5] => 0

[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] => 2

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

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

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

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

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

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

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

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

[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] => 0

[2,1,3,5,4] => [1,2,3,4,5] => 0

[2,1,4,3,5] => [1,2,3,4,5] => 0

[2,1,4,5,3] => [1,2,3,4,5] => 0

[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] => 0

[2,3,1,5,4] => [1,2,3,4,5] => 0

[2,3,4,1,5] => [1,2,3,4,5] => 0

[2,3,4,5,1] => [1,2,3,4,5] => 0

[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] => 2

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

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

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

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

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

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

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

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

[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] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The disorder of a permutation.
Consider a permutation $\pi = [\pi_1,\ldots,\pi_n]$ and cyclically scanning $\pi$ from left to right and remove the elements $1$ through $n$ on this order one after the other. The disorder of $\pi$ is defined to be the number of times a position was not removed in this process.
For example, the disorder of $[3,5,2,1,4]$ is $8$ since on the first scan, 3,5,2 and 4 are not removed, on the second, 3,5 and 4, and on the third and last scan, 5 is once again not removed.

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.