- FindStat

Identifier

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000355: Permutations ⟶ ℤ

Values

[1] => [1,0] => [1,0] => [1] => 0
[2] => [1,0,1,0] => [1,1,0,0] => [1,2] => 0
[1,1] => [1,1,0,0] => [1,0,1,0] => [2,1] => 0
[3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => [1,2,3] => 0
[2,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => [2,1,3] => 1
[1,1,1] => [1,1,0,1,0,0] => [1,1,0,0,1,0] => [3,1,2] => 0
[4] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
[3,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => [2,1,3,4] => 2
[2,2] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => [3,2,1] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,0] => [3,1,2,4] => 1
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 0
[5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => 3
[3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => [3,2,1,4] => 2
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => 2
[2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 1
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => 1
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 0
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => 0
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => [2,1,3,4,5,6] => 4
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => 4
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => [3,1,2,4,5,6] => 3
[3,3] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => 0
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => 3
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => [4,1,2,3,5,6] => 2
[2,2,2] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => 0
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => 1
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => [5,1,2,3,4,6] => 1
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => 0
[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7] => 0
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => [3,2,1,4,5,6] => 6
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => 2
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => [4,2,1,3,5,6] => 5
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => 1
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => 3
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,3,1,4,6] => 3
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => 2
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => [6,2,3,4,1,5] => 1
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => [4,2,3,1,5,6] => 4
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 0
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => [5,3,1,2,4,6] => 3
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => [4,3,2,1,5,6] => 6
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => 0
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [1,1,1,0,0,1,1,0,0,0,1,0] => [6,3,4,1,2,5] => 1
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => [5,3,2,1,4,6] => 5
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => 0
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,1,0,1,0,1,1,0,0,0,1,0] => [6,3,4,2,1,5] => 2
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => [5,2,3,4,1,6] => 2
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0,1,0] => [6,4,1,2,3,5] => 1
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => [5,4,2,3,1,6] => 3
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 0
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [1,1,1,0,0,1,0,1,0,0,1,0] => [6,4,3,1,2,5] => 2
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => [5,3,4,2,1,6] => 3
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0,1,0] => [6,4,2,3,1,5] => 2
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => 0
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => 0
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [5,4,3,2,1,6] => 4
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => [6,4,3,2,1,5] => 3
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => 0
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => 0
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => 0
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => 1
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => 0
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => 0

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 2 + q$

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

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

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

Description

The number of occurrences of the pattern 21-3.
See Permutations/#Pattern-avoiding_permutations for the definition of the pattern

$21\!\!-\!\!3$ .

Map

parallelogram polyomino

Description

Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.

Map

to 132-avoiding permutation

Description

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

Map

inverse Kreweras complement

Description

Return the inverse of the Kreweras complement of a Dyck path, regarded as a noncrossing set partition.
To identify Dyck paths and noncrossing set partitions, this maps uses the following classical bijection. The number of down steps after the

$i$ -th up step of the Dyck path is the size of the block of the set partition whose maximal element is

$i$ . If

$i$ is not a maximal element of a block, the

$(i+1)$ -st step is also an up step.