- FindStat

Identifier

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 216 entries. <<<

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$F_{1} = q$

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

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

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

$F_{5} = 14\ q + 9\ q^{2} + 12\ q^{3} + 6\ q^{4} + q^{5}$

$F_{6} = 42\ q + 23\ q^{2} + 33\ q^{3} + 25\ q^{4} + 8\ q^{5} + q^{6}$

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

Kreweras complement

Description

Sends the permutation

$\pi \in \mathfrak{S}_n$ to the permutation

$\pi^{-1}c$ where

$c = (1,\ldots,n)$ is the long cycle.

Map

to 312-avoiding permutation

Description

Sends a Dyck path to the 312-avoiding permutation according to Bandlow-Killpatrick.
This map is defined in [1] and sends the area (St000012The area of a Dyck path.) to the inversion number (St000018The number of inversions of a permutation.).

Map

inverse Foata bijection

Description

The inverse of Foata's bijection.
See Mp00067Foata bijection.