Identifier
-
Mp00199:
Dyck paths
—prime Dyck path⟶
Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St001729: Permutations ⟶ ℤ
Values
[1,0] => [1,1,0,0] => [2,3,1] => [3,2,1] => 1
[1,0,1,0] => [1,1,0,1,0,0] => [4,3,1,2] => [2,4,1,3] => 1
[1,1,0,0] => [1,1,1,0,0,0] => [2,3,4,1] => [4,2,3,1] => 1
[1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [2,3,5,1,4] => 1
[1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [4,5,1,3,2] => 2
[1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [3,2,5,1,4] => 1
[1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [2,5,4,1,3] => 1
[1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [2,3,4,6,5,1] => 1
[1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [2,5,6,1,4,3] => 2
[1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [5,4,1,6,2,3] => 2
[1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [5,3,6,1,2,4] => 1
[1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [4,6,1,3,5,2] => 2
[1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [3,2,4,6,1,5] => 1
[1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [5,2,6,1,4,3] => 2
[1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [2,4,5,6,1,3] => 1
[1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [2,3,6,1,4,5] => 1
[1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [5,6,4,1,3,2] => 2
[1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [4,2,3,6,1,5] => 1
[1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [3,2,6,5,1,4] => 1
[1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [2,6,4,5,1,3] => 1
[1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,2,3,4,5,1] => 1
[] => [1,0] => [2,1] => [2,1] => 1
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Description
The number of visible descents of a permutation.
A visible descent of a permutation $\pi$ is a position $i$ such that $\pi(i+1) \leq \min(i, \pi(i))$.
A visible descent of a permutation $\pi$ is a position $i$ such that $\pi(i+1) \leq \min(i, \pi(i))$.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
first fundamental transformation
Description
Return the permutation whose cycles are the subsequences between successive left to right maxima.
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.
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