- FindStat

Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001772: Signed permutations ⟶ ℤ

Values

[1] => [1,0,1,0] => [1,2] => [1,2] => 1
[2] => [1,1,0,0,1,0] => [2,1,3] => [2,1,3] => 2
[1,1] => [1,0,1,1,0,0] => [1,3,2] => [1,3,2] => 2
[3] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => [3,2,1,4] => 3
[2,1] => [1,0,1,0,1,0] => [1,2,3] => [1,2,3] => 3
[1,1,1] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => [1,4,3,2] => 3
[3,1] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => [2,3,1,4] => 4
[2,2] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,4,3] => 4
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,3,4,2] => 4
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [1,5,4,3,2] => 4
[3,2] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => 5
[3,1,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3,2,4] => 5
[2,2,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,4,3] => 5
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => [1,4,5,3,2] => 5
[3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => 6
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [1,4,3,5,2] => 6
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [1,3,5,4,2] => 6
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [1,4,3,2,5] => 7
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,3,4,5,2] => 7
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [1,2,5,4,3] => 7
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,3,4,2,5] => 8
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [1,3,2,5,4] => 8
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,4,5,3] => 8
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,3,2,4,5] => 9
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,2,4,3,5] => 9
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => 9
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => 10

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = q$

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

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

Description

The number of occurrences of the signed pattern 12 in a signed permutation.
This is the number of pairs

$1\leq i < j\leq n$ such that

$0 < \pi(i) < \pi(j)$ .

Map

to signed permutation

Description

The signed permutation with all signs positive.

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

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.