Identifier
-
Mp00031:
Dyck paths
—to 312-avoiding permutation⟶
Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000646: Permutations ⟶ ℤ
Values
[1,0,1,0] => [1,2] => [1,2] => [2,1] => 0
[1,1,0,0] => [2,1] => [2,1] => [1,2] => 0
[1,0,1,0,1,0] => [1,2,3] => [1,3,2] => [2,3,1] => 0
[1,0,1,1,0,0] => [1,3,2] => [1,3,2] => [2,3,1] => 0
[1,1,0,0,1,0] => [2,1,3] => [2,1,3] => [3,1,2] => 0
[1,1,0,1,0,0] => [2,3,1] => [2,3,1] => [1,3,2] => 1
[1,1,1,0,0,0] => [3,2,1] => [3,2,1] => [1,2,3] => 0
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,4,3,2] => [2,3,4,1] => 0
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,4,3,2] => [2,3,4,1] => 0
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,4,3,2] => [2,3,4,1] => 0
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,4,3,2] => [2,3,4,1] => 0
[1,0,1,1,1,0,0,0] => [1,4,3,2] => [1,4,3,2] => [2,3,4,1] => 0
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 0
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 1
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [2,4,3,1] => [1,3,4,2] => 1
[1,1,0,1,1,0,0,0] => [2,4,3,1] => [2,4,3,1] => [1,3,4,2] => 1
[1,1,1,0,0,0,1,0] => [3,2,1,4] => [3,2,1,4] => [4,1,2,3] => 0
[1,1,1,0,0,1,0,0] => [3,2,4,1] => [3,2,4,1] => [1,4,2,3] => 1
[1,1,1,0,1,0,0,0] => [3,4,2,1] => [3,4,2,1] => [1,2,4,3] => 1
[1,1,1,1,0,0,0,0] => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [2,1,5,4,3] => [3,4,5,1,2] => 0
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [2,1,5,4,3] => [3,4,5,1,2] => 0
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [2,1,5,4,3] => [3,4,5,1,2] => 0
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [2,1,5,4,3] => [3,4,5,1,2] => 0
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [2,1,5,4,3] => [3,4,5,1,2] => 0
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [2,5,1,4,3] => [3,4,1,5,2] => 1
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [2,5,1,4,3] => [3,4,1,5,2] => 1
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [2,5,4,1,3] => [3,1,4,5,2] => 1
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [2,5,4,3,1] => [1,3,4,5,2] => 1
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [2,5,4,3,1] => [1,3,4,5,2] => 1
[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [2,5,4,1,3] => [3,1,4,5,2] => 1
[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [2,5,4,3,1] => [1,3,4,5,2] => 1
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => [2,5,4,3,1] => [1,3,4,5,2] => 1
[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [2,5,4,3,1] => [1,3,4,5,2] => 1
[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [3,2,1,5,4] => [4,5,1,2,3] => 0
[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [3,2,1,5,4] => [4,5,1,2,3] => 0
[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [3,2,5,1,4] => [4,1,5,2,3] => 1
[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [3,2,5,4,1] => [1,4,5,2,3] => 1
[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [3,2,5,4,1] => [1,4,5,2,3] => 1
[1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => [3,5,2,1,4] => [4,1,2,5,3] => 1
[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => [3,5,2,4,1] => [1,4,2,5,3] => 2
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => [3,5,4,2,1] => [1,2,4,5,3] => 1
[1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => [3,5,4,2,1] => [1,2,4,5,3] => 1
[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [4,3,2,1,5] => [5,1,2,3,4] => 0
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [4,3,2,5,1] => [1,5,2,3,4] => 1
[1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => [4,3,5,2,1] => [1,2,5,3,4] => 1
[1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => [4,5,3,2,1] => [1,2,3,5,4] => 1
[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,3,2,6] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 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] => [2,3,4,5,6,1] => 0
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
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Description
The number of big ascents of a permutation.
For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i+1)−\pi(i) > 1$.
For the number of small ascents, see St000441The number of successions of a permutation..
For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i+1)−\pi(i) > 1$.
For the number of small ascents, see St000441The number of successions of a permutation..
Map
Simion-Schmidt map
Description
The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
Map
to 312-avoiding permutation
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