Identifier
Values
[2] => [1,0,1,0] => [1,0,1,0] => [2,1] => 1
[1,1] => [1,1,0,0] => [1,1,0,0] => [1,2] => 0
[3] => [1,0,1,0,1,0] => [1,0,1,0,1,0] => [3,2,1] => 1
[2,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => [2,1,3] => 1
[1,1,1] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => [2,3,1] => 1
[4] => [1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => 1
[3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0] => [3,2,1,4] => 2
[2,2] => [1,1,1,0,0,0] => [1,1,1,0,0,0] => [1,2,3] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => 2
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => 2
[5] => [1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 2
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => 2
[3,2] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => [2,1,3,4] => 1
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => 2
[2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 2
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => 2
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => 2
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => 2
[3,3] => [1,1,1,0,1,0,0,0] => [1,1,0,1,1,0,0,0] => [2,3,1,4] => 2
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => 2
[2,2,2] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => 2
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => 2
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => 2
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => 1
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 2
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => 2
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => 2
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => 2
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
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 maximal modular displacement of a permutation.
This is $\max_{1\leq i \leq n} \left(\min(\pi(i)-i\pmod n, i-\pi(i)\pmod n)\right)$ for a permutation $\pi$ of $\{1,\dots,n\}$.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
Adin-Bagno-Roichman transformation
Description
The Adin-Bagno-Roichman transformation of a Dyck path.
This is a bijection preserving the number of up steps before each peak and sending the number of returns to the number of up steps after the last double up step.
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].