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Identifier

Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000461: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The rix statistic of a permutation.
This statistic is defined recursively as follows: $rix([]) = 0$, and if $w_i = \max\{w_1, w_2,\dots, w_k\}$, then
$rix(w) := 0$ if $i = 1 < k$,
$rix(w) := 1 + rix(w_1,w_2,\dots,w_{k−1})$ if $i = k$ and
$rix(w) := rix(w_{i+1},w_{i+2},\dots,w_k)$ if $1 < i < k$.

Map

complement

Description

Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$

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

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..