- FindStat

Identifier

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000991: Permutations ⟶ ℤ

Values

[1,0] => [1] => [1] => [1] => 1

[1,0,1,0] => [2,1] => [2,1] => [1,2] => 2

[1,1,0,0] => [1,2] => [1,2] => [2,1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 196 entries. <<<

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

$F_{1} = q$

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

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

$F_{4} = q + 8\ q^{2} + 4\ q^{3} + q^{4}$

$F_{5} = q + 21\ q^{2} + 14\ q^{3} + 5\ q^{4} + q^{5}$

$F_{6} = q + 58\ q^{2} + 46\ q^{3} + 20\ q^{4} + 6\ q^{5} + q^{6}$

Description

The number of right-to-left minima of a permutation.
For the number of left-to-right maxima, see St000314The number of left-to-right-maxima of a permutation..

Map

reverse

Description

Sends a permutation to its reverse.
The reverse of a permutation

$\sigma$ of length

$n$ is given by

$\tau$ with

$\tau(i) = \sigma(n+1-i)$ .

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.

Map

to 132-avoiding permutation

Description

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