- FindStat

Identifier

Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001863: Signed permutations ⟶ ℤ

Values

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

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = q$

$F_{2} = q + q^{2}$

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

$F_{4} = 2\ q^{2} + 11\ q^{3} + q^{4}$

Description

The number of weak excedances of a signed permutation.
For a signed permutation

$\pi\in\mathfrak H_n$ , this is

$\lvert\{i\in[n] \mid \pi(i) \geq i\}\rvert$ .

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

Elizalde-Deutsch bijection

Description

The Elizalde-Deutsch bijection on Dyck paths.
.Let

$n$ be the length of the Dyck path. Consider the steps

$1,n,2,n-1,\dots$ of

$D$ . When considering the

$i$ -th step its corresponding matching step has not yet been read, let the

$i$ -th step of the image of

$D$ be an up step, otherwise let it be a down step.

Map

to signed permutation

Description

The signed permutation with all signs positive.