Identifier
-
Mp00031:
Dyck paths
—to 312-avoiding permutation⟶
Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St001728: Permutations ⟶ ℤ
Values
[1,0] => [1] => [1] => 0
[1,0,1,0] => [1,2] => [1,2] => 0
[1,1,0,0] => [2,1] => [2,1] => 0
[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0] => [1,3,2] => [1,3,2] => 0
[1,1,0,0,1,0] => [2,1,3] => [2,1,3] => 0
[1,1,0,1,0,0] => [2,3,1] => [3,2,1] => 1
[1,1,1,0,0,0] => [3,2,1] => [3,1,2] => 0
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,4,3] => 0
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3,2,4] => 0
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,4,3,2] => 1
[1,0,1,1,1,0,0,0] => [1,4,3,2] => [1,4,2,3] => 0
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => 0
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,4,3] => 0
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [3,2,1,4] => 1
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [4,2,3,1] => 1
[1,1,0,1,1,0,0,0] => [2,4,3,1] => [4,2,1,3] => 1
[1,1,1,0,0,0,1,0] => [3,2,1,4] => [3,1,2,4] => 0
[1,1,1,0,0,1,0,0] => [3,2,4,1] => [4,3,2,1] => 1
[1,1,1,0,1,0,0,0] => [3,4,2,1] => [4,1,3,2] => 0
[1,1,1,1,0,0,0,0] => [4,3,2,1] => [4,1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [1,2,5,3,4] => 0
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,4,3,2,5] => 1
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,5,3,4,2] => 1
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [1,5,3,2,4] => 1
[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [1,4,2,3,5] => 0
[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [1,5,4,3,2] => 1
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => [1,5,2,4,3] => 0
[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [1,5,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [2,1,4,3,5] => 0
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [2,1,5,4,3] => 1
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [2,1,5,3,4] => 0
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [3,2,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [3,2,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [4,2,3,1,5] => 1
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [5,2,3,1,4] => 1
[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [4,2,1,3,5] => 1
[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [5,2,4,3,1] => 1
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => [5,2,1,4,3] => 1
[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [5,2,1,3,4] => 1
[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [3,1,2,4,5] => 0
[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [3,1,2,5,4] => 0
[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [4,3,2,1,5] => 1
[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [5,3,2,4,1] => 1
[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [5,3,2,1,4] => 1
[1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => [4,1,3,2,5] => 0
[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => [5,4,3,2,1] => 2
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => [5,1,3,4,2] => 0
[1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => [5,1,3,2,4] => 0
[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [4,1,2,3,5] => 0
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [5,4,2,3,1] => 1
[1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => [5,1,4,3,2] => 0
[1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => [5,1,2,4,3] => 0
[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [5,1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => 0
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => [1,2,3,6,4,5] => 0
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 0
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => 0
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [1,2,5,4,3,6] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => [1,2,6,4,3,5] => 1
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => [1,2,5,3,4,6] => 0
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => [1,2,6,5,4,3] => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => [1,2,6,3,5,4] => 0
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => [1,2,6,3,4,5] => 0
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => 0
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => 0
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [1,3,2,6,5,4] => 1
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => [1,3,2,6,4,5] => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [1,4,3,2,5,6] => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [1,4,3,2,6,5] => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [1,5,3,4,2,6] => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [1,6,3,4,5,2] => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => [1,6,3,4,2,5] => 1
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => [1,5,3,2,4,6] => 1
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => [1,6,3,5,4,2] => 1
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => [1,6,3,2,5,4] => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => [1,6,3,2,4,5] => 1
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => [1,4,2,3,5,6] => 0
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => [1,4,2,3,6,5] => 0
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => [1,5,4,3,2,6] => 1
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => [1,6,4,3,5,2] => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => [1,6,4,3,2,5] => 1
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,3,2,6] => [1,5,2,4,3,6] => 0
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => [1,6,5,4,3,2] => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => [1,6,2,4,5,3] => 0
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => [1,6,2,4,3,5] => 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
first fundamental transformation
Description
Return the permutation whose cycles are the subsequences between successive left to right maxima.
Map
to 312-avoiding permutation
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