Identifier
Values
[1,0] => [2,1] => [1,1,0,0] => [1,2] => 0
[1,0,1,0] => [3,1,2] => [1,1,1,0,0,0] => [1,2,3] => 0
[1,1,0,0] => [2,3,1] => [1,1,0,1,0,0] => [3,1,2] => 0
[1,0,1,0,1,0] => [4,1,2,3] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
[1,0,1,1,0,0] => [3,1,4,2] => [1,1,1,0,0,1,0,0] => [1,4,2,3] => 0
[1,1,0,0,1,0] => [2,4,1,3] => [1,1,0,1,1,0,0,0] => [3,1,2,4] => 0
[1,1,0,1,0,0] => [4,3,1,2] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
[1,1,1,0,0,0] => [2,3,4,1] => [1,1,0,1,0,1,0,0] => [3,4,1,2] => 1
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => 0
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => 0
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => 1
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => 0
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4] => 1
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => 0
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => 1
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => 0
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => 2
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       
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 $\pi(i) > \pi(i+3)$.
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.