- FindStat

Identifier

Mp00064: Permutations —reverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000366: Permutations ⟶ ℤ

Values

[1] => [1] => [1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 5 + q$

$F_{4} = 17 + 6\ q + q^{2}$

$F_{5} = 70 + 41\ q + 8\ q^{2} + q^{3}$

$F_{6} = 349 + 274\ q + 86\ q^{2} + 10\ q^{3} + q^{4}$

Description

The number of double descents of a permutation.
A double descent of a permutation

$\pi$ is a position

$i$ such that

$\pi(i) > \pi(i+1) > \pi(i+2)$ .

Map

Inverse fireworks map

Description

Sends a permutation to an inverse fireworks permutation.
A permutation

$\sigma$ is inverse fireworks if its inverse avoids the vincular pattern

$3-12$ . The inverse fireworks map sends any permutation

$\sigma$ to an inverse fireworks permutation that is below

$\sigma$ in left weak order and has the same Rajchgot index St001759The Rajchgot index of a permutation..

Map

reverse

Description

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

$\sigma$ of length

$n$ is given by

$\tau$ with

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