- FindStat

Identifier

Mp00064: Permutations —reverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000470: Permutations ⟶ ℤ (values match St000021The number of descents of a permutation., St000325The width of the tree associated to a permutation.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1233 entries. <<<

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$F_{1} = q$

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

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

$F_{4} = q + 13\ q^{2} + 9\ q^{3} + q^{4}$

$F_{5} = q + 46\ q^{2} + 56\ q^{3} + 16\ q^{4} + q^{5}$

$F_{6} = q + 199\ q^{2} + 334\ q^{3} + 160\ q^{4} + 25\ q^{5} + q^{6}$

Description

The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.

Map

reverse

Description

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

$\sigma$ of length

$n$ is given by

$\tau$ with

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

Map

Simion-Schmidt map

Description

The Simion-Schmidt map sends any permutation to a

$123$ -avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between

$132$ -avoiding permutations and

$123$ -avoiding permutations, see [1, Proposition 19].

Map

Lehmer code rotation

Description

Sends a permutation

$\pi$ to the unique permutation

$\tau$ (of the same length) such that every entry in the Lehmer code of

$\tau$ is cyclically one larger than the Lehmer code of

$\pi$ .