Identifier
Values
[1] => [1,0] => [1,0] => [2,1] => 2
[2] => [1,0,1,0] => [1,0,1,0] => [3,1,2] => 2
[1,1] => [1,1,0,0] => [1,1,0,0] => [2,3,1] => 2
[3] => [1,0,1,0,1,0] => [1,0,1,0,1,0] => [4,1,2,3] => 3
[2,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => [4,3,1,2] => 3
[1,1,1] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => [3,1,4,2] => 2
[4] => [1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 4
[3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => 3
[2,2] => [1,1,1,0,0,0] => [1,1,1,0,0,0] => [2,3,4,1] => 3
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 3
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 3
[5] => [1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 5
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => 4
[3,2] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 3
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => 4
[2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 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] => [6,1,2,5,3,4] => 4
[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] => [5,1,2,3,6,4] => 4
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => 4
[3,3] => [1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 3
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => 3
[2,2,2] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 4
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 3
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => 3
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 2
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => 3
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 3
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 3
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => 3
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 4
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 5
[] => [] => [] => [1] => 1
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Description
The length of the longest arithmetic progression in a permutation.
For a permutation $\pi$ of length $n$, this is the biggest $k$ such that there exist $1 \leq i_1 < \dots < i_k \leq n$ with
$$\pi(i_2) - \pi(i_1) = \pi(i_3) - \pi(i_2) = \dots = \pi(i_k) - \pi(i_{k-1}).$$
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
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
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.