Identifier
Values
[1] => [1,0] => [2,1] => [1,2] => 1
[2] => [1,0,1,0] => [3,1,2] => [2,1,3] => 0
[1,1] => [1,1,0,0] => [2,3,1] => [1,3,2] => 1
[3] => [1,0,1,0,1,0] => [4,1,2,3] => [3,2,1,4] => 0
[2,1] => [1,0,1,1,0,0] => [3,1,4,2] => [2,4,1,3] => 1
[1,1,1] => [1,1,0,1,0,0] => [4,3,1,2] => [2,1,3,4] => 1
[4] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [4,3,2,1,5] => 0
[3,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [3,5,2,1,4] => 1
[2,2] => [1,1,1,0,0,0] => [2,3,4,1] => [1,4,3,2] => 1
[2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [3,2,4,1,5] => 0
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [3,2,1,4,5] => 1
[3,2] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [2,5,4,1,3] => 1
[2,2,1] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [3,1,4,5,2] => 1
[3,3] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [2,1,4,3,5] => 0
[2,2,2] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,5,4,3,2] => 1
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Description
The number of augmented double ascents of a permutation.
An augmented double ascent of a permutation $\pi$ is a double ascent of the augmented permutation $\tilde\pi$ obtained from $\pi$ by adding an initial $0$.
A double ascent of $\tilde\pi$ then is a position $i$ such that $\tilde\pi(i) < \tilde\pi(i+1) < \tilde\pi(i+2)$.
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
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.