Identifier
-
Mp00028:
Dyck paths
—reverse⟶
Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001728: Permutations ⟶ ℤ
Values
[1,0] => [1,0] => [1] => [1] => 0
[1,0,1,0] => [1,0,1,0] => [2,1] => [2,1] => 0
[1,1,0,0] => [1,1,0,0] => [1,2] => [1,2] => 0
[1,0,1,0,1,0] => [1,0,1,0,1,0] => [2,3,1] => [3,2,1] => 1
[1,0,1,1,0,0] => [1,1,0,0,1,0] => [1,3,2] => [1,3,2] => 0
[1,1,0,0,1,0] => [1,0,1,1,0,0] => [2,1,3] => [2,1,3] => 0
[1,1,0,1,0,0] => [1,1,0,1,0,0] => [3,1,2] => [3,1,2] => 0
[1,1,1,0,0,0] => [1,1,1,0,0,0] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [2,3,4,1] => [4,2,3,1] => 1
[1,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,0] => [1,3,4,2] => [1,4,3,2] => 1
[1,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => [2,1,4,3] => [2,1,4,3] => 0
[1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0] => [3,1,4,2] => [4,1,3,2] => 0
[1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0] => [1,2,4,3] => [1,2,4,3] => 0
[1,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,0] => [2,3,1,4] => [3,2,1,4] => 1
[1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => [1,3,2,4] => [1,3,2,4] => 0
[1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0] => [2,4,1,3] => [4,2,1,3] => 1
[1,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0] => [3,4,1,2] => [4,3,2,1] => 1
[1,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => [1,4,2,3] => [1,4,2,3] => 0
[1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => 0
[1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0] => [3,1,2,4] => [3,1,2,4] => 0
[1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0] => [4,1,2,3] => [4,1,2,3] => 0
[1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,0,1,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => [1,5,3,4,2] => 1
[1,0,1,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => [2,1,5,4,3] => 1
[1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => [3,1,4,5,2] => [5,1,3,4,2] => 0
[1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,0,1,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => [3,2,1,5,4] => 1
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0,1,0] => [2,4,1,5,3] => [5,2,1,4,3] => 1
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [3,4,1,5,2] => [5,3,2,4,1] => 1
[1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => [1,5,2,4,3] => 0
[1,0,1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [3,1,2,5,4] => [3,1,2,5,4] => 0
[1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0,1,0] => [4,1,2,5,3] => [5,1,2,4,3] => 0
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,1,0,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => [4,2,3,1,5] => 1
[1,1,0,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => [1,4,3,2,5] => 1
[1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => [2,1,4,3,5] => 0
[1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => [3,1,4,2,5] => [4,1,3,2,5] => 0
[1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4] => [5,2,3,1,4] => 1
[1,1,0,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => [1,3,5,2,4] => [1,5,3,2,4] => 1
[1,1,0,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3] => [5,2,4,3,1] => 1
[1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => [5,4,3,2,1] => 2
[1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => [1,5,4,3,2] => 1
[1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => [2,1,5,3,4] => [2,1,5,3,4] => 0
[1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4] => [5,1,3,2,4] => 0
[1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => [5,1,4,3,2] => 0
[1,1,0,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => [3,2,1,4,5] => 1
[1,1,1,0,0,0,1,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => [2,4,1,3,5] => [4,2,1,3,5] => 1
[1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => [4,3,2,1,5] => 1
[1,1,1,0,0,1,1,0,0,0] => [1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4] => [5,2,1,3,4] => 1
[1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4] => [5,3,2,1,4] => 1
[1,1,1,0,1,0,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => [5,4,2,3,1] => 1
[1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[1,1,1,1,0,0,0,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => [3,1,2,4,5] => 0
[1,1,1,1,0,0,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => [4,1,2,3,5] => 0
[1,1,1,1,0,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => [5,1,2,3,4] => 0
[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] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => [6,2,3,4,5,1] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => [1,3,4,5,6,2] => [1,6,3,4,5,2] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3] => [2,1,6,4,5,3] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => [3,1,4,5,6,2] => [6,1,3,4,5,2] => 0
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => [2,3,1,5,6,4] => [3,2,1,6,5,4] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => [1,3,2,5,6,4] => [1,3,2,6,5,4] => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0,1,0,1,0] => [2,4,1,5,6,3] => [6,2,1,4,5,3] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => [3,4,1,5,6,2] => [6,3,2,4,5,1] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,0,1,0,1,0] => [1,4,2,5,6,3] => [1,6,2,4,5,3] => 0
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,3,5,6,4] => [2,1,3,6,5,4] => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0,1,0] => [3,1,2,5,6,4] => [3,1,2,6,5,4] => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0,1,0,1,0] => [4,1,2,5,6,3] => [6,1,2,4,5,3] => 0
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,1,6,5] => [4,2,3,1,6,5] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0,1,0] => [1,3,4,2,6,5] => [1,4,3,2,6,5] => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0,1,0] => [3,1,4,2,6,5] => [4,1,3,2,6,5] => 0
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,1,0,0,1,0] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,1,0,0,1,0] => [2,3,5,1,6,4] => [6,2,3,1,5,4] => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,1,0,0,1,0] => [1,3,5,2,6,4] => [1,6,3,2,5,4] => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,1,0,0,1,0] => [2,4,5,1,6,3] => [6,2,4,3,5,1] => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => [3,4,5,1,6,2] => [6,4,3,2,5,1] => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0,1,0] => [1,4,5,2,6,3] => [1,6,4,3,5,2] => 1
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,5,3,6,4] => [2,1,6,3,5,4] => 0
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0,1,0] => [3,1,5,2,6,4] => [6,1,3,2,5,4] => 0
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,1,0,0,1,0] => [4,1,5,2,6,3] => [6,1,4,3,5,2] => 0
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,1,0,0,1,0] => [1,2,5,3,6,4] => [1,2,6,3,5,4] => 0
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => [2,3,1,4,6,5] => [3,2,1,4,6,5] => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => 0
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,6,5] => [4,2,1,3,6,5] => 1
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,0,1,1,0,0,0,1,0] => [3,4,1,2,6,5] => [4,3,2,1,6,5] => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,1,0,0,1,1,0,0,0,1,0] => [1,4,2,3,6,5] => [1,4,2,3,6,5] => 0
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,1,0,0,0,1,0] => [2,5,1,3,6,4] => [6,2,1,3,5,4] => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0,1,0] => [3,5,1,2,6,4] => [6,3,2,1,5,4] => 1
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0,1,0] => [4,5,1,2,6,3] => [6,4,2,3,5,1] => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0,1,0] => [1,5,2,3,6,4] => [1,6,2,3,5,4] => 0
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Description
The number of invisible descents of a permutation.
A visible descent of a permutation $\pi$ is a position $i$ such that $\pi(i+1) \leq \min(i, \pi(i))$. Thus, an invisible descent satisfies $\pi(i) > \pi(i+1) > i$.
A visible descent of a permutation $\pi$ is a position $i$ such that $\pi(i+1) \leq \min(i, \pi(i))$. Thus, an invisible descent satisfies $\pi(i) > \pi(i+1) > i$.
Map
reverse
Description
The reversal of a Dyck path.
This is the Dyck path obtained by reading the path backwards.
This is the Dyck path obtained by reading the path backwards.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
Map
Corteel
Description
Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.
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