Identifier
-
Mp00102:
Dyck paths
—rise composition⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000649: Permutations ⟶ ℤ
Values
[1,0] => [1] => [1,0] => [2,1] => 0
[1,0,1,0] => [1,1] => [1,0,1,0] => [3,1,2] => 0
[1,1,0,0] => [2] => [1,1,0,0] => [2,3,1] => 0
[1,0,1,0,1,0] => [1,1,1] => [1,0,1,0,1,0] => [4,1,2,3] => 1
[1,0,1,1,0,0] => [1,2] => [1,0,1,1,0,0] => [3,1,4,2] => 0
[1,1,0,0,1,0] => [2,1] => [1,1,0,0,1,0] => [2,4,1,3] => 0
[1,1,0,1,0,0] => [2,1] => [1,1,0,0,1,0] => [2,4,1,3] => 0
[1,1,1,0,0,0] => [3] => [1,1,1,0,0,0] => [2,3,4,1] => 0
[1,0,1,0,1,0,1,0] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [1,1,2] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 1
[1,0,1,1,0,0,1,0] => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[1,0,1,1,0,1,0,0] => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[1,0,1,1,1,0,0,0] => [1,3] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 0
[1,1,0,0,1,0,1,0] => [2,1,1] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[1,1,0,0,1,1,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 0
[1,1,0,1,0,0,1,0] => [2,1,1] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[1,1,0,1,0,1,0,0] => [2,1,1] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[1,1,0,1,1,0,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 0
[1,1,1,0,0,0,1,0] => [3,1] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 0
[1,1,1,0,0,1,0,0] => [3,1] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 0
[1,1,1,0,1,0,0,0] => [3,1] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 0
[1,1,1,1,0,0,0,0] => [4] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 0
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 0
[1,0,1,0,1,1,0,0,1,0] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 1
[1,0,1,0,1,1,0,1,0,0] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 1
[1,0,1,0,1,1,1,0,0,0] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 1
[1,0,1,1,0,0,1,0,1,0] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 0
[1,0,1,1,0,1,0,0,1,0] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 1
[1,0,1,1,0,1,0,1,0,0] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 1
[1,0,1,1,0,1,1,0,0,0] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 0
[1,0,1,1,1,0,0,0,1,0] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 0
[1,0,1,1,1,0,0,1,0,0] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 0
[1,0,1,1,1,0,1,0,0,0] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 0
[1,0,1,1,1,1,0,0,0,0] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 0
[1,1,0,0,1,0,1,0,1,0] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 1
[1,1,0,0,1,1,0,0,1,0] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 0
[1,1,0,0,1,1,0,1,0,0] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 0
[1,1,0,0,1,1,1,0,0,0] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 0
[1,1,0,1,0,0,1,0,1,0] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 0
[1,1,0,1,0,0,1,1,0,0] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 1
[1,1,0,1,0,1,0,0,1,0] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 0
[1,1,0,1,0,1,0,1,0,0] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 0
[1,1,0,1,0,1,1,0,0,0] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 1
[1,1,0,1,1,0,0,0,1,0] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 0
[1,1,0,1,1,0,0,1,0,0] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 0
[1,1,0,1,1,0,1,0,0,0] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 0
[1,1,0,1,1,1,0,0,0,0] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 0
[1,1,1,0,0,0,1,0,1,0] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 1
[1,1,1,0,0,0,1,1,0,0] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 0
[1,1,1,0,0,1,0,0,1,0] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 1
[1,1,1,0,0,1,0,1,0,0] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 1
[1,1,1,0,0,1,1,0,0,0] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 0
[1,1,1,0,1,0,0,0,1,0] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 1
[1,1,1,0,1,0,0,1,0,0] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 1
[1,1,1,0,1,0,1,0,0,0] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 1
[1,1,1,0,1,1,0,0,0,0] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 0
[1,1,1,1,0,0,0,0,1,0] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 0
[1,1,1,1,0,0,0,1,0,0] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 0
[1,1,1,1,0,0,1,0,0,0] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 0
[1,1,1,1,0,1,0,0,0,0] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 0
[1,1,1,1,1,0,0,0,0,0] => [5] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 0
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Description
The number of 3-excedences of a permutation.
This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+3$.
This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+3$.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
rise composition
Description
Send a Dyck path to the composition of sizes of its rises.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
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