- FindStat

Identifier

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000724: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$F_{5} = 5\ q^{2} + 9\ q^{3} + 12\ q^{4} + 16\ q^{5}$

$F_{6} = 14\ q^{2} + 24\ q^{3} + 30\ q^{4} + 32\ q^{5} + 32\ q^{6}$

Description

The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation.
Associate an increasing binary tree to the permutation using Mp00061to increasing tree. Then follow the path starting at the root which always selects the child with the smaller label. This statistic is the label of the leaf in the path, see [1].
Han [2] showed that this statistic is (up to a shift) equidistributed on zigzag permutations (permutations

$\pi$ such that

$\pi(1) < \pi(2) > \pi(3) \cdots$ ) with the greater neighbor of the maximum (St000060The greater neighbor of the maximum.), see also [3].

Map

reverse

Description

Sends a permutation to its reverse.
The reverse of a permutation

$\sigma$ of length

$n$ is given by

$\tau$ with

$\tau(i) = \sigma(n+1-i)$ .

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.).