- FindStat

Identifier

Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000354: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1961 entries. <<<

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$F_{2} = 1 + q$

$F_{3} = 1 + 4\ q + q^{2}$

$F_{4} = 1 + 9\ q + 13\ q^{2} + q^{3}$

$F_{5} = 1 + 16\ q + 61\ q^{2} + 41\ q^{3} + q^{4}$

$F_{6} = 1 + 25\ q + 181\ q^{2} + 381\ q^{3} + 131\ q^{4} + q^{5}$

Description

The number of recoils of a permutation.
A recoil, or inverse descent of a permutation

$\pi$ is a value

$i$ such that

$i+1$ appears to the left of

$i$ in

$\pi_1,\pi_2,\dots,\pi_n$ .
In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern

$([2,1], {(0,1),(1,1),(2,1)})$ , i.e., the middle row is shaded.

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..