- FindStat

Identifier

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000868: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The aid statistic in the sense of Shareshian-Wachs.
This is the number of admissible inversions St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. plus the number of descents St000021The number of descents of a permutation.. This statistic was introduced by John Shareshian and Michelle L. Wachs in [1]. Theorem 4.1 states that the aid statistic together with the descent statistic is Euler-Mahonian.

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.).