- FindStat

Identifier

Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000007: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].

Map

to Dyck path

Description

Sends a parking function to its support Dyck word.

Map

to 132-avoiding permutation

Description

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