Identifier
-
Mp00129:
Dyck paths
—to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶
Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000365: Permutations ⟶ ℤ
Values
[1,0] => [1] => [1] => [1] => 0
[1,0,1,0] => [2,1] => [2,1] => [1,2] => 0
[1,1,0,0] => [1,2] => [1,2] => [2,1] => 0
[1,0,1,0,1,0] => [2,3,1] => [3,2,1] => [1,2,3] => 1
[1,0,1,1,0,0] => [2,1,3] => [2,1,3] => [2,3,1] => 0
[1,1,0,0,1,0] => [1,3,2] => [1,3,2] => [3,1,2] => 0
[1,1,0,1,0,0] => [3,1,2] => [2,3,1] => [2,1,3] => 0
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,0,1,0,1,0,1,0] => [2,3,4,1] => [4,2,3,1] => [1,3,2,4] => 0
[1,0,1,0,1,1,0,0] => [2,3,1,4] => [3,2,1,4] => [2,3,4,1] => 1
[1,0,1,1,0,0,1,0] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0
[1,0,1,1,0,1,0,0] => [2,4,1,3] => [3,2,4,1] => [2,3,1,4] => 0
[1,0,1,1,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 0
[1,1,0,0,1,0,1,0] => [1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 1
[1,1,0,0,1,1,0,0] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[1,1,0,1,0,0,1,0] => [3,1,4,2] => [3,4,1,2] => [2,1,4,3] => 0
[1,1,0,1,0,1,0,0] => [3,4,1,2] => [2,4,3,1] => [3,1,2,4] => 1
[1,1,0,1,1,0,0,0] => [3,1,2,4] => [2,3,1,4] => [3,2,4,1] => 0
[1,1,1,0,0,0,1,0] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 0
[1,1,1,0,0,1,0,0] => [1,4,2,3] => [1,3,4,2] => [4,2,1,3] => 0
[1,1,1,0,1,0,0,0] => [4,1,2,3] => [2,3,4,1] => [3,2,1,4] => 0
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => [5,2,3,4,1] => [1,4,3,2,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => [4,2,3,1,5] => [2,4,3,5,1] => 0
[1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => [3,2,1,5,4] => [3,4,5,1,2] => 1
[1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4] => [4,2,3,5,1] => [2,4,3,1,5] => 0
[1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => [3,2,1,4,5] => [3,4,5,2,1] => 1
[1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => [2,1,5,4,3] => [4,5,1,2,3] => 1
[1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => [2,1,4,3,5] => [4,5,2,3,1] => 0
[1,0,1,1,0,1,0,0,1,0] => [2,4,1,5,3] => [4,2,5,1,3] => [2,4,1,5,3] => 0
[1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3] => [3,2,5,4,1] => [3,4,1,2,5] => 1
[1,0,1,1,0,1,1,0,0,0] => [2,4,1,3,5] => [3,2,4,1,5] => [3,4,2,5,1] => 0
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 0
[1,0,1,1,1,0,0,1,0,0] => [2,1,5,3,4] => [2,1,4,5,3] => [4,5,2,1,3] => 0
[1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4] => [3,2,4,5,1] => [3,4,2,1,5] => 0
[1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => [2,1,3,4,5] => [4,5,3,2,1] => 0
[1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => [1,5,3,4,2] => [5,1,3,2,4] => 0
[1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => 1
[1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 0
[1,1,0,0,1,1,0,1,0,0] => [1,3,5,2,4] => [1,4,3,5,2] => [5,2,3,1,4] => 0
[1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 0
[1,1,0,1,0,0,1,0,1,0] => [3,1,4,5,2] => [3,5,1,4,2] => [3,1,5,2,4] => 0
[1,1,0,1,0,0,1,1,0,0] => [3,1,4,2,5] => [3,4,1,2,5] => [3,2,5,4,1] => 0
[1,1,0,1,0,1,0,0,1,0] => [3,4,1,5,2] => [4,5,3,1,2] => [2,1,3,5,4] => 1
[1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => [2,5,3,4,1] => [4,1,3,2,5] => 0
[1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => [2,4,3,1,5] => [4,2,3,5,1] => 1
[1,1,0,1,1,0,0,0,1,0] => [3,1,2,5,4] => [2,3,1,5,4] => [4,3,5,1,2] => 0
[1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4] => [3,4,1,5,2] => [3,2,5,1,4] => 0
[1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4] => [2,4,3,5,1] => [4,2,3,1,5] => 0
[1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => [2,3,1,4,5] => [4,3,5,2,1] => 0
[1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => 1
[1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 0
[1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => [1,4,5,2,3] => [5,2,1,4,3] => 0
[1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => [1,3,5,4,2] => [5,3,1,2,4] => 1
[1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => [1,3,4,2,5] => [5,3,2,4,1] => 0
[1,1,1,0,1,0,0,0,1,0] => [4,1,2,5,3] => [2,4,5,1,3] => [4,2,1,5,3] => 0
[1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => [4,3,5,1,2] => [2,3,1,5,4] => 0
[1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => [2,3,5,4,1] => [4,3,1,2,5] => 1
[1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => [2,3,4,1,5] => [4,3,2,5,1] => 0
[1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 0
[1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => [1,2,4,5,3] => [5,4,2,1,3] => 0
[1,1,1,1,0,0,1,0,0,0] => [1,5,2,3,4] => [1,3,4,5,2] => [5,3,2,1,4] => 0
[1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => [2,3,4,5,1] => [4,3,2,1,5] => 0
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 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] => [1,5,4,3,2,6] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,1,6] => [5,2,3,4,1,6] => [2,5,4,3,6,1] => 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] => [3,5,4,6,1,2] => 0
[1,0,1,0,1,0,1,1,0,1,0,0] => [2,3,4,6,1,5] => [5,2,3,4,6,1] => [2,5,4,3,1,6] => 0
[1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,1,5,6] => [4,2,3,1,5,6] => [3,5,4,6,2,1] => 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] => [4,5,6,1,2,3] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [2,3,1,5,4,6] => [3,2,1,5,4,6] => [4,5,6,2,3,1] => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [2,3,5,1,6,4] => [5,2,3,6,1,4] => [2,5,4,1,6,3] => 0
[1,0,1,0,1,1,0,1,0,1,0,0] => [2,3,5,6,1,4] => [4,2,3,6,5,1] => [3,5,4,1,2,6] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => [4,2,3,5,1,6] => [3,5,4,2,6,1] => 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] => [4,5,6,3,1,2] => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [2,3,1,6,4,5] => [3,2,1,5,6,4] => [4,5,6,2,1,3] => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [2,3,6,1,4,5] => [4,2,3,5,6,1] => [3,5,4,2,1,6] => 0
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,1,4,5,6] => [3,2,1,4,5,6] => [4,5,6,3,2,1] => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3] => [2,1,6,4,5,3] => [5,6,1,3,2,4] => 0
[1,0,1,1,0,0,1,0,1,1,0,0] => [2,1,4,5,3,6] => [2,1,5,4,3,6] => [5,6,2,3,4,1] => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => [5,6,3,4,1,2] => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [2,1,4,6,3,5] => [2,1,5,4,6,3] => [5,6,2,3,1,4] => 0
[1,0,1,1,0,0,1,1,1,0,0,0] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => [5,6,3,4,2,1] => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [2,4,1,5,6,3] => [4,2,6,1,5,3] => [3,5,1,6,2,4] => 0
[1,0,1,1,0,1,0,0,1,1,0,0] => [2,4,1,5,3,6] => [4,2,5,1,3,6] => [3,5,2,6,4,1] => 0
[1,0,1,1,0,1,0,1,0,0,1,0] => [2,4,5,1,6,3] => [5,2,6,4,1,3] => [2,5,1,3,6,4] => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [2,4,5,6,1,3] => [3,2,6,4,5,1] => [4,5,1,3,2,6] => 0
[1,0,1,1,0,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => [3,2,5,4,1,6] => [4,5,2,3,6,1] => 1
[1,0,1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,6,5] => [3,2,4,1,6,5] => [4,5,3,6,1,2] => 0
[1,0,1,1,0,1,1,0,0,1,0,0] => [2,4,1,6,3,5] => [4,2,5,1,6,3] => [3,5,2,6,1,4] => 0
[1,0,1,1,0,1,1,0,1,0,0,0] => [2,4,6,1,3,5] => [3,2,5,4,6,1] => [4,5,2,3,1,6] => 0
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,4,1,3,5,6] => [3,2,4,1,5,6] => [4,5,3,6,2,1] => 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] => [5,6,4,1,2,3] => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => [5,6,4,2,3,1] => 0
[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,5,3,6,4] => [2,1,5,6,3,4] => [5,6,2,1,4,3] => 0
[1,0,1,1,1,0,0,1,0,1,0,0] => [2,1,5,6,3,4] => [2,1,4,6,5,3] => [5,6,3,1,2,4] => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [2,1,5,3,4,6] => [2,1,4,5,3,6] => [5,6,3,2,4,1] => 0
[1,0,1,1,1,0,1,0,0,0,1,0] => [2,5,1,3,6,4] => [3,2,5,6,1,4] => [4,5,2,1,6,3] => 0
[1,0,1,1,1,0,1,0,0,1,0,0] => [2,5,1,6,3,4] => [5,2,4,6,1,3] => [2,5,3,1,6,4] => 0
[1,0,1,1,1,0,1,0,1,0,0,0] => [2,5,6,1,3,4] => [3,2,4,6,5,1] => [4,5,3,1,2,6] => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,5,1,3,4,6] => [3,2,4,5,1,6] => [4,5,3,2,6,1] => 0
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Description
The number of double ascents of a permutation.
A double ascent of a permutation $\pi$ is a position $i$ such that $\pi(i) < \pi(i+1) < \pi(i+2)$.
A double ascent of a permutation $\pi$ is a position $i$ such that $\pi(i) < \pi(i+1) < \pi(i+2)$.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
Map
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
Map
first fundamental transformation
Description
Return the permutation whose cycles are the subsequences between successive left to right maxima.
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