- FindStat

Identifier

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001557: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

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

$F_{4} = 6 + 15\ q + 3\ q^{2}$

Description

The number of inversions of the second entry of a permutation.
This is, for a permutation

$\pi$ of length

$n$ ,

$\# \{2 < k \leq n \mid \pi(2) > \pi(k)\}.$
The number of inversions of the first entry is St000054The first entry of the permutation. and the number of inversions of the third entry is St001556The number of inversions of the third entry of a permutation.. The sequence of inversions of all the entries define the Lehmer code of a permutation.

Map

to 132-avoiding permutation

Description

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

Map

cycle type

Description

The cycle type of a permutation as a partition.

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.