- FindStat

Identifier

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St001735: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 8

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

$F_{2} = 2\ q$

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

$F_{4} = 9\ q + 4\ q^{2} + q^{4}$

$F_{5} = 23\ q + 11\ q^{2} + 5\ q^{4} + 2\ q^{6} + q^{8}$

$F_{6} = 65\ q + 30\ q^{2} + 14\ q^{4} + 9\ q^{6} + 8\ q^{8} + 3\ q^{12} + q^{16} + 2\ q^{18}$

Description

The number of permutations with the same set of runs.
For example, the set of runs of

$4132$ is

$\{(13), (2), (4)\}$ . The only other permutation with this set of runs is

$4213$ , so the statistic equals

$2$ for these two permutations.

Map

Simion-Schmidt map

Description

The Simion-Schmidt map sends any permutation to a

$123$ -avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between

$132$ -avoiding permutations and

$123$ -avoiding permutations, see [1, Proposition 19].

Map

inverse Foata bijection

Description

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

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.).