Identifier
-
Mp00108:
Permutations
—cycle type⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001520: Permutations ⟶ ℤ
Values
[1] => [1] => [1,0,1,0] => [3,1,2] => 0
[1,2] => [1,1] => [1,0,1,1,0,0] => [3,1,4,2] => 1
[2,1] => [2] => [1,1,0,0,1,0] => [2,4,1,3] => 0
[1,2,3] => [1,1,1] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 0
[1,3,2] => [2,1] => [1,0,1,0,1,0] => [4,1,2,3] => 1
[2,1,3] => [2,1] => [1,0,1,0,1,0] => [4,1,2,3] => 1
[2,3,1] => [3] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 1
[3,1,2] => [3] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 1
[3,2,1] => [2,1] => [1,0,1,0,1,0] => [4,1,2,3] => 1
[1,2,4,3] => [2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 1
[1,3,2,4] => [2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 1
[1,3,4,2] => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 1
[1,4,2,3] => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 1
[1,4,3,2] => [2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 1
[2,1,3,4] => [2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 1
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 1
[2,3,1,4] => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 1
[2,4,3,1] => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 1
[3,1,2,4] => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 1
[3,2,1,4] => [2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 1
[3,2,4,1] => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 1
[4,1,3,2] => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 1
[4,2,1,3] => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 1
[4,2,3,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 1
[4,3,2,1] => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 1
[1,2,4,5,3] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[1,2,5,3,4] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[1,3,2,5,4] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[1,3,4,2,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[1,3,5,4,2] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[1,4,2,3,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[1,4,3,5,2] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[1,4,5,2,3] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[1,5,2,4,3] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[1,5,3,2,4] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[1,5,4,3,2] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[2,1,3,5,4] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[2,1,4,3,5] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[2,1,4,5,3] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[2,1,5,3,4] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[2,1,5,4,3] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[2,3,1,4,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[2,3,1,5,4] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[2,4,3,1,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[2,4,5,1,3] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[2,5,3,4,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[2,5,4,3,1] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[3,1,2,4,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[3,1,2,5,4] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[3,2,1,5,4] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[3,2,4,1,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[3,2,5,4,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[3,4,1,2,5] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[3,4,1,5,2] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[3,4,5,2,1] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[3,5,1,2,4] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[3,5,1,4,2] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[3,5,4,1,2] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[4,1,3,2,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[4,1,5,2,3] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[4,2,1,3,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[4,2,3,5,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[4,2,5,1,3] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[4,3,2,1,5] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[4,3,2,5,1] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[4,3,5,1,2] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[4,5,1,3,2] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[4,5,2,1,3] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[4,5,3,1,2] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[5,1,3,4,2] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[5,1,4,3,2] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[5,2,1,4,3] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[5,2,3,1,4] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[5,2,4,3,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[5,3,2,1,4] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[5,3,2,4,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[5,3,4,2,1] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[5,4,1,2,3] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[5,4,2,3,1] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[5,4,3,2,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[1,3,2,5,6,4] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[1,3,2,6,4,5] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[1,3,4,2,6,5] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[1,3,5,6,2,4] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[1,3,6,5,4,2] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[1,4,2,3,6,5] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[1,4,5,2,6,3] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[1,4,5,6,3,2] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[1,4,6,2,3,5] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[1,4,6,5,2,3] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[1,5,2,6,3,4] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[1,5,4,3,6,2] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[1,5,4,6,2,3] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[1,5,6,2,4,3] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[1,5,6,3,2,4] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[1,6,2,5,4,3] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[1,6,4,3,2,5] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[1,6,4,5,3,2] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[1,6,5,2,3,4] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[1,6,5,3,4,2] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
>>> Load all 201 entries. <<<
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
Description
The number of strict 3-descents.
A strict 3-descent of a permutation $\pi$ of $\{1,2, \dots ,n \}$ is a pair $(i,i+3)$ with $ i+3 \leq n$ and $\pi(i) > \pi(i+3)$.
A strict 3-descent of a permutation $\pi$ of $\{1,2, \dots ,n \}$ is a pair $(i,i+3)$ with $ i+3 \leq n$ and $\pi(i) > \pi(i+3)$.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
cycle type
Description
The cycle type of a permutation as a partition.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!