- FindStat

Identifier

Mp00252: Permutations —restriction⟶ Permutations
St001074: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,3,4,2,5] => [1,3,4,2] => 8

[1,3,4,5,2] => [1,3,4,2] => 8

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

[1,3,5,4,2] => [1,3,4,2] => 8

[1,4,2,3,5] => [1,4,2,3] => 8

[1,4,2,5,3] => [1,4,2,3] => 8

[1,4,3,2,5] => [1,4,3,2] => 8

[1,4,3,5,2] => [1,4,3,2] => 8

[1,4,5,2,3] => [1,4,2,3] => 8

[1,4,5,3,2] => [1,4,3,2] => 8

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

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

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

[1,5,3,4,2] => [1,3,4,2] => 8

[1,5,4,2,3] => [1,4,2,3] => 8

[1,5,4,3,2] => [1,4,3,2] => 8

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

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

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

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

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

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

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

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

[2,3,4,1,5] => [2,3,4,1] => 8

[2,3,4,5,1] => [2,3,4,1] => 8

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

[2,3,5,4,1] => [2,3,4,1] => 8

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

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

[2,4,3,1,5] => [2,4,3,1] => 8

[2,4,3,5,1] => [2,4,3,1] => 8

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

[2,4,5,3,1] => [2,4,3,1] => 8

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

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

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

[2,5,3,4,1] => [2,3,4,1] => 8

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

[2,5,4,3,1] => [2,4,3,1] => 8

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

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

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

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

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

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

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

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

[3,2,4,1,5] => [3,2,4,1] => 8

[3,2,4,5,1] => [3,2,4,1] => 8

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

[3,2,5,4,1] => [3,2,4,1] => 8

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

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

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

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

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

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

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

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

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

[3,5,2,4,1] => [3,2,4,1] => 8

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

>>> Load all 6341 entries. <<<

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Description

The number of inversions of the cyclic embedding of a permutation.
The cyclic embedding of a permutation $\pi$ of length $n$ is given by the permutation of length $n+1$ represented in cycle notation by $(\pi_1,\ldots,\pi_n,n+1)$.
This reflects in particular the fact that the number of long cycles of length $n+1$ equals $n!$.
This statistic counts the number of inversions of this embedding, see [1]. As shown in [2], the sum of this statistic on all permutations of length $n$ equals $n!\cdot(3n-1)/12$.

Map

restriction

Description

The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.