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Identifier

Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001866: Signed permutations ⟶ ℤ

Values

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 5 + q$

$F_{4} = 15 + 8\ q + q^{2}$

Description

The nesting alignments of a signed permutation.
A nesting alignment of a signed permutation

$\pi\in\mathfrak H_n$ is a pair

$1\leq i, j \leq n$ such that

$-i < -j < -\pi(j) < -\pi(i)$ , or
$-i < j \leq \pi(j) < -\pi(i)$ , or
$i < j \leq \pi(j) < \pi(i)$ .

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..

Map

inverse first fundamental transformation

Description

Let

$\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps

$\sigma$ to the permutation

$[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.

Map

to signed permutation

Description

The signed permutation with all signs positive.