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Identifier

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001867: Signed permutations ⟶ ℤ

Values

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 5$

$F_{4} = 13 + q$

Description

The number of alignments of type EN of a signed permutation.
An alignment of type EN of a signed permutation π∈Hn is a pair −n≤i≤j≤n, i,j≠0, such that one of the following conditions hold:

$-i < 0 < -\pi(i) < \pi(j) < j$
$i \leq\pi(i) < \pi(j) < j$ .

Map

to signed permutation

Description

The signed permutation with all signs positive.

Map

first fundamental transformation

Description

Return the permutation whose cycles are the subsequences between successive left to right maxima.

Map

to 132-avoiding permutation

Description

Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].