Identifier
-
Mp00132:
Dyck paths
—switch returns and last double rise⟶
Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001683: Permutations ⟶ ℤ
Values
[1,0] => [1,0] => [1] => 0
[1,0,1,0] => [1,0,1,0] => [1,2] => 0
[1,1,0,0] => [1,1,0,0] => [2,1] => 0
[1,0,1,0,1,0] => [1,0,1,0,1,0] => [1,2,3] => 0
[1,0,1,1,0,0] => [1,1,0,1,0,0] => [2,3,1] => 0
[1,1,0,0,1,0] => [1,1,0,0,1,0] => [2,1,3] => 0
[1,1,0,1,0,0] => [1,0,1,1,0,0] => [1,3,2] => 1
[1,1,1,0,0,0] => [1,1,1,0,0,0] => [3,2,1] => 0
[1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => 0
[1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => 0
[1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => 2
[1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => [3,4,2,1] => 0
[1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 0
[1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,0,0] => [3,2,4,1] => 0
[1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 1
[1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 1
[1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,0,0] => [2,4,3,1] => 1
[1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => 0
[1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 1
[1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => 2
[1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0] => [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] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 0
[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 0
[1,0,1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 3
[1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => 0
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => 0
[1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => 0
[1,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => 2
[1,0,1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => 2
[1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => 2
[1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => 0
[1,0,1,1,1,0,0,1,0,0] => [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] => [1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => 3
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => 0
[1,1,0,0,1,0,1,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => 0
[1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => 0
[1,1,0,0,1,1,0,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => 2
[1,1,0,0,1,1,1,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => 0
[1,1,0,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => 1
[1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 1
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 1
[1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => 1
[1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => 1
[1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => 1
[1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => 3
[1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => 2
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 0
[1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => 0
[1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 1
[1,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 1
[1,1,1,0,0,1,1,0,0,0] => [1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => 1
[1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 2
[1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 2
[1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => 2
[1,1,1,0,1,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => 1
[1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 0
[1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 1
[1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 2
[1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 3
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [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] => [1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,1,6] => 0
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => 4
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => [3,4,5,6,2,1] => 0
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => [2,3,4,1,5,6] => 0
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => [3,4,5,2,6,1] => 0
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => 3
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => 3
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => [2,4,5,6,3,1] => 3
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,1,0,0,0,1,0] => [3,4,5,2,1,6] => 0
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => 4
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => 4
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => [4,5,6,3,2,1] => 0
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => [2,3,1,4,5,6] => 0
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => [3,4,2,5,6,1] => 0
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0,1,0] => [3,4,2,5,1,6] => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => 4
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => [4,5,3,6,2,1] => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => [2,4,5,3,6,1] => 2
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => [2,3,5,6,4,1] => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,1,0,0,0,1,0] => [2,4,5,3,1,6] => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => 4
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => 4
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => [3,5,6,4,2,1] => 2
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,0,0,0,1,0,1,0] => [3,4,2,1,5,6] => 0
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => [4,5,3,2,6,1] => 0
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => 3
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => 3
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => [2,5,4,6,3,1] => 3
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,3,2,6] => 3
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => 3
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => 3
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => [2,5,6,4,3,1] => 3
>>> Load all 304 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 distinct positions of the pattern letter 3 in occurrences of 132 in a permutation.
Map
switch returns and last double rise
Description
An alternative to the Adin-Bagno-Roichman transformation of a Dyck path.
This is a bijection preserving the number of up steps before each peak and exchanging the number of components with the position of the last double rise.
This is a bijection preserving the number of up steps before each peak and exchanging the number of components with the position of the last double rise.
Map
to 312-avoiding permutation
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!