- FindStat

Identifier

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001005: Permutations ⟶ ℤ

Values

[1,0] => [1,1,0,0] => [2,1] => [2,1] => 2
[1,0,1,0] => [1,1,0,1,0,0] => [2,3,1] => [3,2,1] => 2
[1,1,0,0] => [1,1,1,0,0,0] => [3,2,1] => [2,3,1] => 3
[1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => [4,2,3,1] => 2
[1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [2,4,3,1] => [3,2,4,1] => 3
[1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [3,2,4,1] => [2,4,3,1] => 3
[1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [4,2,3,1] => [2,3,4,1] => 4
[1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => [3,4,1,2] => 4
[1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [5,2,3,4,1] => 2
[1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [4,2,3,5,1] => 3
[1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [3,2,5,4,1] => 3
[1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [2,5,3,4,1] => [3,2,4,5,1] => 4
[1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [4,2,5,1,3] => 4
[1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [2,5,3,4,1] => 3
[1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [2,4,3,5,1] => 4
[1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [4,2,3,5,1] => [2,3,5,4,1] => 4
[1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [5,2,3,4,1] => [2,3,4,5,1] => 5
[1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => [2,4,5,1,3] => 5
[1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [3,5,1,4,2] => 4
[1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [5,3,2,4,1] => [3,4,1,5,2] => 5
[1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => [3,5,4,1,2] => 4
[1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [3,4,5,1,2] => 5
[1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => [6,2,3,4,5,1] => 2
[1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => [2,3,4,6,5,1] => [5,2,3,4,6,1] => 3
[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => [2,3,5,4,6,1] => [4,2,3,6,5,1] => 3
[1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => [2,3,6,4,5,1] => [4,2,3,5,6,1] => 4
[1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => [2,3,6,5,4,1] => [5,2,3,6,1,4] => 4
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => [2,4,3,5,6,1] => [3,2,6,4,5,1] => 3
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => [2,4,3,6,5,1] => [3,2,5,4,6,1] => 4
[1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => [2,5,3,4,6,1] => [3,2,4,6,5,1] => 4
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => [2,6,3,4,5,1] => [3,2,4,5,6,1] => 5
[1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => [2,6,3,5,4,1] => [3,2,5,6,1,4] => 5
[1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => [2,5,4,3,6,1] => [4,2,6,1,5,3] => 4
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => [2,6,4,3,5,1] => [4,2,5,1,6,3] => 5
[1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => [2,6,4,5,3,1] => [4,2,6,5,1,3] => 4
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [2,6,5,4,3,1] => [4,2,5,6,1,3] => 5
[1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => [3,2,4,5,6,1] => [2,6,3,4,5,1] => 3
[1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => [3,2,4,6,5,1] => [2,5,3,4,6,1] => 4
[1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => [3,2,5,4,6,1] => [2,4,3,6,5,1] => 4
[1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,6,4,5,1] => [2,4,3,5,6,1] => 5
[1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => [3,2,6,5,4,1] => [2,5,3,6,1,4] => 5
[1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => [4,2,3,5,6,1] => [2,3,6,4,5,1] => 4
[1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => [4,2,3,6,5,1] => [2,3,5,4,6,1] => 5
[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => [5,2,3,4,6,1] => [2,3,4,6,5,1] => 5
[1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => [6,2,3,4,5,1] => [2,3,4,5,6,1] => 6
[1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => [6,2,3,5,4,1] => [2,3,5,6,1,4] => 6
[1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,4,3,6,1] => [2,4,6,1,5,3] => 5
[1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => [6,2,4,3,5,1] => [2,4,5,1,6,3] => 6
[1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => [6,2,4,5,3,1] => [2,4,6,5,1,3] => 5
[1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => [6,2,5,4,3,1] => [2,4,5,6,1,3] => 6
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => [4,3,2,5,6,1] => [3,6,1,4,5,2] => 4
[1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => [4,3,2,6,5,1] => [3,5,1,4,6,2] => 5
[1,1,1,0,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => [3,4,1,6,5,2] => 5
[1,1,1,0,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => [6,3,2,4,5,1] => [3,4,1,5,6,2] => 6
[1,1,1,0,0,1,1,0,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => [6,3,2,5,4,1] => [3,5,1,6,2,4] => 6
[1,1,1,0,1,0,0,0,1,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => [5,3,4,2,6,1] => [3,6,4,1,5,2] => 4
[1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => [6,3,4,2,5,1] => [3,5,4,1,6,2] => 5
[1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => [6,3,4,5,2,1] => [3,6,4,5,1,2] => 4
[1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => [6,3,5,4,2,1] => [3,5,4,6,1,2] => 5
[1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => [5,4,3,2,6,1] => [3,4,6,1,5,2] => 5
[1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => [6,4,3,2,5,1] => [3,4,5,1,6,2] => 6
[1,1,1,1,0,0,1,0,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => [6,4,3,5,2,1] => [3,4,6,5,1,2] => 5
[1,1,1,1,0,1,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => [6,5,3,4,2,1] => [3,4,5,6,1,2] => 6
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [6,5,4,3,2,1] => [4,5,6,1,2,3] => 6

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

$F_{1} = q^{2}$

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

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

$F_{4} = q^{2} + 3\ q^{3} + 6\ q^{4} + 4\ q^{5}$

$F_{5} = q^{2} + 4\ q^{3} + 12\ q^{4} + 16\ q^{5} + 9\ q^{6}$

Description

The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both.

Map

prime Dyck path

Description

Return the Dyck path obtained by adding an initial up and a final down step.

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

Corteel

Description

Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.