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Identifier

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000542: Permutations ⟶ ℤ (values match St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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

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

[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

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

[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

[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

[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

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

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$F_{1} = q$

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

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

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

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

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

Description

The number of left-to-right-minima of a permutation.
An integer

$\sigma_i$ in the one-line notation of a permutation

$\sigma$ is a left-to-right-minimum if there does not exist a j < i such that

$\sigma_j < \sigma_i$ .

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].