- FindStat

Identifier

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000883: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

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

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

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

$F_{5} = 23\ q + 12\ q^{2} + 5\ q^{3} + q^{4} + q^{5}$

$F_{6} = 71\ q + 36\ q^{2} + 17\ q^{3} + 6\ q^{4} + q^{5} + q^{6}$

Description

The number of longest increasing subsequences 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].