- FindStat

Identifier

Mp00064: Permutations —reverse⟶ Permutations
St001806: Permutations ⟶ ℤ

Values

[1] => [1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 873 entries. <<<

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$F_{1} = q$

$F_{2} = q + q^{2}$

$F_{3} = 2\ q + 2\ q^{2} + 2\ q^{3}$

$F_{4} = 6\ q + 6\ q^{2} + 6\ q^{3} + 6\ q^{4}$

$F_{5} = 24\ q + 24\ q^{2} + 24\ q^{3} + 24\ q^{4} + 24\ q^{5}$

$F_{6} = 120\ q + 120\ q^{2} + 120\ q^{3} + 120\ q^{4} + 120\ q^{5} + 120\ q^{6}$

Description

The upper middle entry of a permutation.
This is the entry

$\sigma(\frac{n+1}{2})$ when

$n$ is odd, and

$\sigma(\frac{n}{2}+1)$ when

$n$ is even, where

$n$ is the size of the permutation

$\sigma$ .

Map

reverse

Description

Sends a permutation to its reverse.
The reverse of a permutation

$\sigma$ of length

$n$ is given by

$\tau$ with

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