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Identifier

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001821: Signed permutations ⟶ ℤ

Values

[1,0] => [1] => [1] => [1] => 0
[1,0,1,0] => [1,2] => [1,2] => [1,2] => 0
[1,1,0,0] => [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0] => [2,3,1] => [3,1,2] => [3,1,2] => 3
[1,1,1,0,0,0] => [3,2,1] => [2,3,1] => [2,3,1] => 2
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 3
[1,0,1,1,1,0,0,0] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 2
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 3
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 6
[1,1,0,1,1,0,0,0] => [2,4,3,1] => [3,4,1,2] => [3,4,1,2] => 4
[1,1,1,0,0,0,1,0] => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[1,1,1,0,0,1,0,0] => [3,2,4,1] => [2,4,1,3] => [2,4,1,3] => 4
[1,1,1,0,1,0,0,0] => [4,2,3,1] => [2,3,4,1] => [2,3,4,1] => 3
[1,1,1,1,0,0,0,0] => [4,3,2,1] => [3,2,4,1] => [3,2,4,1] => 3

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$F_{1} = 1$

$F_{2} = 1 + q$

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

$F_{4} = 1 + 3\ q + 3\ q^{2} + 4\ q^{3} + 2\ q^{4} + q^{6}$

Description

The sorting index of a signed permutation.
A signed permutation

$\sigma = [\sigma(1),\ldots,\sigma(n)]$ can be sorted

$[1,\ldots,n]$ by signed transpositions in the following way:
First move

$\pm n$ to its position and swap the sign if needed, then

$\pm (n-1), \pm (n-2)$ and so on.
For example for

$[2,-4,5,-1,-3]$ we have the swaps

$[2,-4,5,-1,-3] \rightarrow [2,-4,-3,-1,5] \rightarrow [2,1,-3,4,5] \rightarrow [2,1,3,4,5] \rightarrow [1,2,3,4,5]$
given by the signed transpositions

$(3,5), (-2,4), (-3,3), (1,2)$ .
If

$(i_1,j_1),\ldots,(i_n,j_n)$ is the decomposition of

$\sigma$ obtained this way (including trivial transpositions) then the sorting index of

$\sigma$ is defined as

$\operatorname{sor}_B(\sigma) = \sum_{k=1}^{n-1} j_k - i_k - \chi(i_k < 0),$
where

$\chi(i_k < 0)$ is 1 if

$i_k$ is negative and 0 otherwise.
For

$\sigma = [2,-4,5,-1,-3]$ we have

$\operatorname{sor}_B(\sigma) = (5-3) + (4-(-2)-1) + (3-(-3)-1) + (2-1) = 13.$

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 non-crossing permutation

Description

Sends a Dyck path

$D$ with valley at positions

$\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation

$\pi$ having descents

$\{i_1,\ldots,i_k\}$ and whose inverse has descents

$\{j_1,\ldots,j_k\}$ .
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to

$n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..

Map

to signed permutation

Description

The signed permutation with all signs positive.