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Matching statistic: St000470
Mp00252: Permutations —restriction⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 1
[2,1] => [1] => [1] => 1
[1,2,3] => [1,2] => [1,2] => 1
[1,3,2] => [1,2] => [1,2] => 1
[2,1,3] => [2,1] => [1,2] => 1
[2,3,1] => [2,1] => [1,2] => 1
[3,1,2] => [1,2] => [1,2] => 1
[3,2,1] => [2,1] => [1,2] => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 2
[1,3,4,2] => [1,3,2] => [1,3,2] => 2
[1,4,2,3] => [1,2,3] => [1,2,3] => 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3,4] => [2,1,3] => [1,3,2] => 2
[2,1,4,3] => [2,1,3] => [1,3,2] => 2
[2,3,1,4] => [2,3,1] => [1,2,3] => 1
[2,3,4,1] => [2,3,1] => [1,2,3] => 1
[2,4,1,3] => [2,1,3] => [1,3,2] => 2
[2,4,3,1] => [2,3,1] => [1,2,3] => 1
[3,1,2,4] => [3,1,2] => [1,2,3] => 1
[3,1,4,2] => [3,1,2] => [1,2,3] => 1
[3,2,1,4] => [3,2,1] => [1,2,3] => 1
[3,2,4,1] => [3,2,1] => [1,2,3] => 1
[3,4,1,2] => [3,1,2] => [1,2,3] => 1
[3,4,2,1] => [3,2,1] => [1,2,3] => 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 2
[4,2,1,3] => [2,1,3] => [1,3,2] => 2
[4,2,3,1] => [2,3,1] => [1,2,3] => 1
[4,3,1,2] => [3,1,2] => [1,2,3] => 1
[4,3,2,1] => [3,2,1] => [1,2,3] => 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 2
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2,5] => [1,3,4,2] => [1,3,4,2] => 2
[1,3,4,5,2] => [1,3,4,2] => [1,3,4,2] => 2
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => 2
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2,5] => [1,4,3,2] => [1,4,2,3] => 2
[1,4,3,5,2] => [1,4,3,2] => [1,4,2,3] => 2
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,5,3,2] => [1,4,3,2] => [1,4,2,3] => 2
Description
The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
Matching statistic: St000884
Mp00252: Permutations —restriction⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 0 = 1 - 1
[2,1] => [1] => [1] => 0 = 1 - 1
[1,2,3] => [1,2] => [1,2] => 0 = 1 - 1
[1,3,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1,3] => [2,1] => [1,2] => 0 = 1 - 1
[2,3,1] => [2,1] => [1,2] => 0 = 1 - 1
[3,1,2] => [1,2] => [1,2] => 0 = 1 - 1
[3,2,1] => [2,1] => [1,2] => 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,3,4,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,3,4] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[2,3,4,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[2,4,1,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[3,1,2,4] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,1,4,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,2,1,4] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,2,4,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,4,1,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,4,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[4,2,1,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[4,3,1,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[4,3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,5,2] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
Description
The number of isolated descents of a permutation.
A descent $i$ is isolated if neither $i+1$ nor $i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].
Matching statistic: St001729
Mp00252: Permutations —restriction⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001729: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St001729: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 0 = 1 - 1
[2,1] => [1] => [1] => 0 = 1 - 1
[1,2,3] => [1,2] => [1,2] => 0 = 1 - 1
[1,3,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1,3] => [2,1] => [1,2] => 0 = 1 - 1
[2,3,1] => [2,1] => [1,2] => 0 = 1 - 1
[3,1,2] => [1,2] => [1,2] => 0 = 1 - 1
[3,2,1] => [2,1] => [1,2] => 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,3,4,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,3,4] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[2,3,4,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[2,4,1,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[3,1,2,4] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,1,4,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,2,1,4] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,2,4,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,4,1,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,4,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[4,2,1,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[4,3,1,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[4,3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,5,2] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
Description
The number of visible descents of a permutation.
A visible descent of a permutation $\pi$ is a position $i$ such that $\pi(i+1) \leq \min(i, \pi(i))$.
Matching statistic: St001928
Mp00252: Permutations —restriction⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 0 = 1 - 1
[2,1] => [1] => [1] => 0 = 1 - 1
[1,2,3] => [1,2] => [1,2] => 0 = 1 - 1
[1,3,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1,3] => [2,1] => [1,2] => 0 = 1 - 1
[2,3,1] => [2,1] => [1,2] => 0 = 1 - 1
[3,1,2] => [1,2] => [1,2] => 0 = 1 - 1
[3,2,1] => [2,1] => [1,2] => 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,3,4,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,3,4] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[2,3,4,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[2,4,1,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[3,1,2,4] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,1,4,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,2,1,4] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,2,4,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,4,1,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,4,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[4,2,1,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[4,3,1,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[4,3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,5,2] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
Description
The number of non-overlapping descents in a permutation.
In other words, any maximal descending subsequence $\pi_i,\pi_{i+1},\dots,\pi_k$ contributes $\lfloor\frac{k-i+1}{2}\rfloor$ to the total count.
Matching statistic: St000619
Mp00252: Permutations —restriction⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => ? = 1
[2,1] => [1] => [1] => ? = 1
[1,2,3] => [1,2] => [1,2] => 1
[1,3,2] => [1,2] => [1,2] => 1
[2,1,3] => [2,1] => [1,2] => 1
[2,3,1] => [2,1] => [1,2] => 1
[3,1,2] => [1,2] => [1,2] => 1
[3,2,1] => [2,1] => [1,2] => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 2
[1,3,4,2] => [1,3,2] => [1,3,2] => 2
[1,4,2,3] => [1,2,3] => [1,2,3] => 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3,4] => [2,1,3] => [1,3,2] => 2
[2,1,4,3] => [2,1,3] => [1,3,2] => 2
[2,3,1,4] => [2,3,1] => [1,2,3] => 1
[2,3,4,1] => [2,3,1] => [1,2,3] => 1
[2,4,1,3] => [2,1,3] => [1,3,2] => 2
[2,4,3,1] => [2,3,1] => [1,2,3] => 1
[3,1,2,4] => [3,1,2] => [1,2,3] => 1
[3,1,4,2] => [3,1,2] => [1,2,3] => 1
[3,2,1,4] => [3,2,1] => [1,2,3] => 1
[3,2,4,1] => [3,2,1] => [1,2,3] => 1
[3,4,1,2] => [3,1,2] => [1,2,3] => 1
[3,4,2,1] => [3,2,1] => [1,2,3] => 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 2
[4,2,1,3] => [2,1,3] => [1,3,2] => 2
[4,2,3,1] => [2,3,1] => [1,2,3] => 1
[4,3,1,2] => [3,1,2] => [1,2,3] => 1
[4,3,2,1] => [3,2,1] => [1,2,3] => 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 2
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2,5] => [1,3,4,2] => [1,3,4,2] => 2
[1,3,4,5,2] => [1,3,4,2] => [1,3,4,2] => 2
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => 2
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2,5] => [1,4,3,2] => [1,4,2,3] => 2
[1,4,3,5,2] => [1,4,3,2] => [1,4,2,3] => 2
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,5,3,2] => [1,4,3,2] => [1,4,2,3] => 2
[1,5,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,5,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
Description
The number of cyclic descents of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is given by the number of indices $1 \leq i \leq n$ such that $\pi(i) > \pi(i+1)$ where we set $\pi(n+1) = \pi(1)$.
Matching statistic: St000354
Mp00252: Permutations —restriction⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => [1] => ? = 1 - 1
[2,1] => [1] => [1] => [1] => ? = 1 - 1
[1,2,3] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[1,3,2] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1,3] => [2,1] => [1,2] => [1,2] => 0 = 1 - 1
[2,3,1] => [2,1] => [1,2] => [1,2] => 0 = 1 - 1
[3,1,2] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[3,2,1] => [2,1] => [1,2] => [1,2] => 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,3,4,2] => [1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,4,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,3,4] => [2,1,3] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,3,4,1] => [2,3,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,4,1,3] => [2,1,3] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,1,2,4] => [3,1,2] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,1,4,2] => [3,1,2] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,2,1,4] => [3,2,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,2,4,1] => [3,2,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,4,1,2] => [3,1,2] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,4,2,1] => [3,2,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[4,1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[4,2,1,3] => [2,1,3] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[4,3,1,2] => [3,1,2] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[4,3,2,1] => [3,2,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [1,3,4,2] => [1,4,2,3] => 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,2] => [1,3,4,2] => [1,4,2,3] => 1 = 2 - 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [1,4,2,3] => [1,3,4,2] => 1 = 2 - 1
[1,4,3,5,2] => [1,4,3,2] => [1,4,2,3] => [1,3,4,2] => 1 = 2 - 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [1,4,2,3] => [1,3,4,2] => 1 = 2 - 1
[1,5,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,5,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[8,7,6,2,1,3,4,5] => [7,6,2,1,3,4,5] => [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 2 - 1
[6,7,8,2,1,3,4,5] => [6,7,2,1,3,4,5] => [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 2 - 1
[8,7,5,2,1,3,4,6] => [7,5,2,1,3,4,6] => [1,3,4,6,2,5,7] => [1,5,2,3,6,4,7] => ? = 2 - 1
[7,8,5,2,1,3,4,6] => [7,5,2,1,3,4,6] => [1,3,4,6,2,5,7] => [1,5,2,3,6,4,7] => ? = 2 - 1
[7,8,4,2,1,3,5,6] => [7,4,2,1,3,5,6] => [1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => ? = 2 - 1
[8,7,3,2,1,4,5,6] => [7,3,2,1,4,5,6] => [1,4,5,6,2,3,7] => [1,5,6,2,3,4,7] => ? = 2 - 1
[7,8,3,2,1,4,5,6] => [7,3,2,1,4,5,6] => [1,4,5,6,2,3,7] => [1,5,6,2,3,4,7] => ? = 2 - 1
[7,8,2,1,3,4,5,6] => [7,2,1,3,4,5,6] => [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 2 - 1
[8,5,6,2,1,3,4,7] => [5,6,2,1,3,4,7] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => ? = 2 - 1
[8,6,4,2,1,3,5,7] => [6,4,2,1,3,5,7] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? = 2 - 1
[8,3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => [1,4,5,6,7,2,3] => [1,6,7,2,3,4,5] => ? = 2 - 1
[8,2,3,1,4,5,6,7] => [2,3,1,4,5,6,7] => [1,4,5,6,7,2,3] => [1,6,7,2,3,4,5] => ? = 2 - 1
[8,2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [1,3,4,5,6,7,2] => [1,7,2,3,4,5,6] => ? = 2 - 1
[7,2,1,3,4,5,6,8] => [7,2,1,3,4,5,6] => [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 2 - 1
[6,2,3,1,4,5,7,8] => [6,2,3,1,4,5,7] => [1,4,5,7,2,3,6] => [1,5,6,2,3,7,4] => ? = 2 - 1
[3,2,1,4,5,6,7,8] => [3,2,1,4,5,6,7] => [1,4,5,6,7,2,3] => [1,6,7,2,3,4,5] => ? = 2 - 1
[2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7] => [1,3,4,5,6,7,2] => [1,7,2,3,4,5,6] => ? = 2 - 1
[1,3,5,7,8,6,4,2] => [1,3,5,7,6,4,2] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? = 2 - 1
[1,3,5,6,7,8,4,2] => [1,3,5,6,7,4,2] => [1,3,5,6,7,2,4] => [1,6,2,7,3,4,5] => ? = 2 - 1
[1,3,4,8,7,6,5,2] => [1,3,4,7,6,5,2] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => ? = 2 - 1
[1,3,4,5,7,8,6,2] => [1,3,4,5,7,6,2] => [1,3,4,5,7,2,6] => [1,6,2,3,4,7,5] => ? = 2 - 1
[1,3,4,5,6,8,7,2] => [1,3,4,5,6,7,2] => [1,3,4,5,6,7,2] => [1,7,2,3,4,5,6] => ? = 2 - 1
[1,3,4,5,6,7,8,2] => [1,3,4,5,6,7,2] => [1,3,4,5,6,7,2] => [1,7,2,3,4,5,6] => ? = 2 - 1
[2,1,4,5,8,7,6,3] => [2,1,4,5,7,6,3] => [1,4,5,7,2,3,6] => [1,5,6,2,3,7,4] => ? = 2 - 1
[2,1,4,5,7,8,6,3] => [2,1,4,5,7,6,3] => [1,4,5,7,2,3,6] => [1,5,6,2,3,7,4] => ? = 2 - 1
[2,1,4,5,6,7,8,3] => [2,1,4,5,6,7,3] => [1,4,5,6,7,2,3] => [1,6,7,2,3,4,5] => ? = 2 - 1
[2,1,3,4,8,7,6,5] => [2,1,3,4,7,6,5] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => ? = 2 - 1
[2,1,3,4,7,8,6,5] => [2,1,3,4,7,6,5] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => ? = 2 - 1
[2,1,3,4,7,6,8,5] => [2,1,3,4,7,6,5] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => ? = 2 - 1
[2,1,3,4,6,7,8,5] => [2,1,3,4,6,7,5] => [1,3,4,6,7,2,5] => [1,6,2,3,7,4,5] => ? = 2 - 1
[3,2,1,4,5,8,7,6] => [3,2,1,4,5,7,6] => [1,4,5,7,2,3,6] => [1,5,6,2,3,7,4] => ? = 2 - 1
[2,3,1,4,5,7,8,6] => [2,3,1,4,5,7,6] => [1,4,5,7,2,3,6] => [1,5,6,2,3,7,4] => ? = 2 - 1
[2,1,4,5,6,3,8,7] => [2,1,4,5,6,3,7] => [1,4,5,6,2,3,7] => [1,5,6,2,3,4,7] => ? = 2 - 1
[2,1,3,4,5,6,8,7] => [2,1,3,4,5,6,7] => [1,3,4,5,6,7,2] => [1,7,2,3,4,5,6] => ? = 2 - 1
[1,3,4,6,7,5,2,8] => [1,3,4,6,7,5,2] => [1,3,4,6,7,2,5] => [1,6,2,3,7,4,5] => ? = 2 - 1
[1,3,4,5,7,6,2,8] => [1,3,4,5,7,6,2] => [1,3,4,5,7,2,6] => [1,6,2,3,4,7,5] => ? = 2 - 1
[1,3,8,5,6,7,4,2] => [1,3,5,6,7,4,2] => [1,3,5,6,7,2,4] => [1,6,2,7,3,4,5] => ? = 2 - 1
[2,1,3,8,5,7,6,4] => [2,1,3,5,7,6,4] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? = 2 - 1
[2,1,4,5,8,6,7,3] => [2,1,4,5,6,7,3] => [1,4,5,6,7,2,3] => [1,6,7,2,3,4,5] => ? = 2 - 1
[2,4,6,1,3,5,8,7] => [2,4,6,1,3,5,7] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? = 2 - 1
[2,4,6,1,3,8,5,7] => [2,4,6,1,3,5,7] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? = 2 - 1
[2,4,6,1,8,3,5,7] => [2,4,6,1,3,5,7] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? = 2 - 1
[2,4,6,8,1,3,5,7] => [2,4,6,1,3,5,7] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? = 2 - 1
[2,4,1,3,6,7,8,5] => [2,4,1,3,6,7,5] => [1,3,6,7,2,4,5] => [1,5,2,6,7,3,4] => ? = 2 - 1
[2,4,7,1,3,5,8,6] => [2,4,7,1,3,5,6] => [1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => ? = 2 - 1
[2,4,7,1,3,8,5,6] => [2,4,7,1,3,5,6] => [1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => ? = 2 - 1
[2,4,7,8,1,3,5,6] => [2,4,7,1,3,5,6] => [1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => ? = 2 - 1
[2,1,4,5,7,3,8,6] => [2,1,4,5,7,3,6] => [1,4,5,7,2,3,6] => [1,5,6,2,3,7,4] => ? = 2 - 1
Description
The number of recoils of a permutation.
A '''recoil''', or '''inverse descent''' of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$.
In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
Matching statistic: St000035
Mp00252: Permutations —restriction⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 75%
Mp00223: Permutations —runsort⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 75%
Values
[1,2] => [1] => [1] => 0 = 1 - 1
[2,1] => [1] => [1] => 0 = 1 - 1
[1,2,3] => [1,2] => [1,2] => 0 = 1 - 1
[1,3,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1,3] => [2,1] => [1,2] => 0 = 1 - 1
[2,3,1] => [2,1] => [1,2] => 0 = 1 - 1
[3,1,2] => [1,2] => [1,2] => 0 = 1 - 1
[3,2,1] => [2,1] => [1,2] => 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,3,4,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,3,4] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[2,3,4,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[2,4,1,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[3,1,2,4] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,1,4,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,2,1,4] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,2,4,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,4,1,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,4,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[4,2,1,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[4,3,1,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[4,3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,5,2] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[8,7,5,4,3,6,2,1] => [7,5,4,3,6,2,1] => [1,2,3,6,4,5,7] => ? = 2 - 1
[5,4,6,7,3,8,2,1] => [5,4,6,7,3,2,1] => [1,2,3,4,6,7,5] => ? = 2 - 1
[7,6,4,3,5,8,2,1] => [7,6,4,3,5,2,1] => [1,2,3,5,4,6,7] => ? = 2 - 1
[4,3,5,6,7,8,2,1] => [4,3,5,6,7,2,1] => [1,2,3,5,6,7,4] => ? = 2 - 1
[8,7,6,5,3,2,4,1] => [7,6,5,3,2,4,1] => [1,2,4,3,5,6,7] => ? = 2 - 1
[8,6,5,7,3,2,4,1] => [6,5,7,3,2,4,1] => [1,2,4,3,5,7,6] => ? = 3 - 1
[5,6,7,8,3,2,4,1] => [5,6,7,3,2,4,1] => [1,2,4,3,5,6,7] => ? = 2 - 1
[7,8,3,2,4,5,6,1] => [7,3,2,4,5,6,1] => [1,2,4,5,6,3,7] => ? = 2 - 1
[6,5,4,3,7,2,8,1] => [6,5,4,3,7,2,1] => [1,2,3,7,4,5,6] => ? = 2 - 1
[5,4,6,3,7,2,8,1] => [5,4,6,3,7,2,1] => [1,2,3,7,4,6,5] => ? = 3 - 1
[6,5,3,4,7,2,8,1] => [6,5,3,4,7,2,1] => [1,2,3,4,7,5,6] => ? = 2 - 1
[7,6,5,3,2,4,8,1] => [7,6,5,3,2,4,1] => [1,2,4,3,5,6,7] => ? = 2 - 1
[5,4,3,2,6,7,8,1] => [5,4,3,2,6,7,1] => [1,2,6,7,3,4,5] => ? = 2 - 1
[3,4,5,2,6,7,8,1] => [3,4,5,2,6,7,1] => [1,2,6,7,3,4,5] => ? = 2 - 1
[5,2,3,4,6,7,8,1] => [5,2,3,4,6,7,1] => [1,2,3,4,6,7,5] => ? = 2 - 1
[4,3,2,5,6,7,8,1] => [4,3,2,5,6,7,1] => [1,2,5,6,7,3,4] => ? = 2 - 1
[3,4,2,5,6,7,8,1] => [3,4,2,5,6,7,1] => [1,2,5,6,7,3,4] => ? = 2 - 1
[4,2,3,5,6,7,8,1] => [4,2,3,5,6,7,1] => [1,2,3,5,6,7,4] => ? = 2 - 1
[3,2,4,5,6,7,8,1] => [3,2,4,5,6,7,1] => [1,2,4,5,6,7,3] => ? = 2 - 1
[7,8,5,4,3,6,1,2] => [7,5,4,3,6,1,2] => [1,2,3,6,4,5,7] => ? = 2 - 1
[7,8,4,5,3,6,1,2] => [7,4,5,3,6,1,2] => [1,2,3,6,4,5,7] => ? = 2 - 1
[7,8,4,3,5,6,1,2] => [7,4,3,5,6,1,2] => [1,2,3,5,6,4,7] => ? = 2 - 1
[8,4,3,5,6,7,1,2] => [4,3,5,6,7,1,2] => [1,2,3,5,6,7,4] => ? = 2 - 1
[5,6,4,7,3,8,1,2] => [5,6,4,7,3,1,2] => [1,2,3,4,7,5,6] => ? = 2 - 1
[5,4,6,7,3,8,1,2] => [5,4,6,7,3,1,2] => [1,2,3,4,6,7,5] => ? = 2 - 1
[5,4,6,3,7,8,1,2] => [5,4,6,3,7,1,2] => [1,2,3,7,4,6,5] => ? = 3 - 1
[5,6,3,4,7,8,1,2] => [5,6,3,4,7,1,2] => [1,2,3,4,7,5,6] => ? = 2 - 1
[6,4,3,5,7,8,1,2] => [6,4,3,5,7,1,2] => [1,2,3,5,7,4,6] => ? = 2 - 1
[5,4,3,6,7,8,1,2] => [5,4,3,6,7,1,2] => [1,2,3,6,7,4,5] => ? = 2 - 1
[4,5,3,6,7,8,1,2] => [4,5,3,6,7,1,2] => [1,2,3,6,7,4,5] => ? = 2 - 1
[4,3,5,6,7,8,1,2] => [4,3,5,6,7,1,2] => [1,2,3,5,6,7,4] => ? = 2 - 1
[6,5,4,7,8,1,2,3] => [6,5,4,7,1,2,3] => [1,2,3,4,7,5,6] => ? = 2 - 1
[8,7,6,5,3,2,1,4] => [7,6,5,3,2,1,4] => [1,4,2,3,5,6,7] => ? = 2 - 1
[7,8,5,6,3,2,1,4] => [7,5,6,3,2,1,4] => [1,4,2,3,5,6,7] => ? = 2 - 1
[7,6,5,8,3,2,1,4] => [7,6,5,3,2,1,4] => [1,4,2,3,5,6,7] => ? = 2 - 1
[5,6,7,8,3,2,1,4] => [5,6,7,3,2,1,4] => [1,4,2,3,5,6,7] => ? = 2 - 1
[8,7,6,5,2,3,1,4] => [7,6,5,2,3,1,4] => [1,4,2,3,5,6,7] => ? = 2 - 1
[7,8,5,6,2,3,1,4] => [7,5,6,2,3,1,4] => [1,4,2,3,5,6,7] => ? = 2 - 1
[6,7,5,8,2,3,1,4] => [6,7,5,2,3,1,4] => [1,4,2,3,5,6,7] => ? = 2 - 1
[7,5,6,8,2,3,1,4] => [7,5,6,2,3,1,4] => [1,4,2,3,5,6,7] => ? = 2 - 1
[5,6,7,8,2,3,1,4] => [5,6,7,2,3,1,4] => [1,4,2,3,5,6,7] => ? = 2 - 1
[8,7,6,5,3,1,2,4] => [7,6,5,3,1,2,4] => [1,2,4,3,5,6,7] => ? = 2 - 1
[7,8,5,6,3,1,2,4] => [7,5,6,3,1,2,4] => [1,2,4,3,5,6,7] => ? = 2 - 1
[6,7,5,8,3,1,2,4] => [6,7,5,3,1,2,4] => [1,2,4,3,5,6,7] => ? = 2 - 1
[7,5,6,8,3,1,2,4] => [7,5,6,3,1,2,4] => [1,2,4,3,5,6,7] => ? = 2 - 1
[5,6,7,8,3,1,2,4] => [5,6,7,3,1,2,4] => [1,2,4,3,5,6,7] => ? = 2 - 1
[8,7,6,5,2,1,3,4] => [7,6,5,2,1,3,4] => [1,3,4,2,5,6,7] => ? = 2 - 1
[7,8,6,5,2,1,3,4] => [7,6,5,2,1,3,4] => [1,3,4,2,5,6,7] => ? = 2 - 1
[6,7,8,5,2,1,3,4] => [6,7,5,2,1,3,4] => [1,3,4,2,5,6,7] => ? = 2 - 1
[7,8,5,6,2,1,3,4] => [7,5,6,2,1,3,4] => [1,3,4,2,5,6,7] => ? = 2 - 1
Description
The number of left outer peaks of a permutation.
A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$.
In other words, it is a peak in the word $[0,w_1,..., w_n]$.
This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.
Matching statistic: St000245
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => [1] => 0 = 1 - 1
[2,1] => [1] => [1] => [1] => 0 = 1 - 1
[1,2,3] => [1,2] => [1,2] => [2,1] => 0 = 1 - 1
[1,3,2] => [1,2] => [1,2] => [2,1] => 0 = 1 - 1
[2,1,3] => [2,1] => [1,2] => [2,1] => 0 = 1 - 1
[2,3,1] => [2,1] => [1,2] => [2,1] => 0 = 1 - 1
[3,1,2] => [1,2] => [1,2] => [2,1] => 0 = 1 - 1
[3,2,1] => [2,1] => [1,2] => [2,1] => 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[1,3,4,2] => [1,3,2] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[1,4,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[2,1,3,4] => [2,1,3] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[2,3,4,1] => [2,3,1] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[2,4,1,3] => [2,1,3] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[3,1,2,4] => [3,1,2] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[3,1,4,2] => [3,1,2] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[3,2,1,4] => [3,2,1] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[3,2,4,1] => [3,2,1] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[3,4,1,2] => [3,1,2] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[3,4,2,1] => [3,2,1] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[4,1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[4,2,1,3] => [2,1,3] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[4,3,1,2] => [3,1,2] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[4,3,2,1] => [3,2,1] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [1,3,4,2] => [4,2,1,3] => 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,2] => [1,3,4,2] => [4,2,1,3] => 1 = 2 - 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => [4,2,1,3] => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [1,4,2,3] => [4,1,3,2] => 1 = 2 - 1
[1,4,3,5,2] => [1,4,3,2] => [1,4,2,3] => [4,1,3,2] => 1 = 2 - 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [1,4,2,3] => [4,1,3,2] => 1 = 2 - 1
[7,5,4,6,8,3,2,1] => [7,5,4,6,3,2,1] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 2 - 1
[8,7,5,4,3,6,2,1] => [7,5,4,3,6,2,1] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 2 - 1
[5,4,6,7,3,8,2,1] => [5,4,6,7,3,2,1] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 2 - 1
[7,6,4,3,5,8,2,1] => [7,6,4,3,5,2,1] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 2 - 1
[4,3,5,6,7,8,2,1] => [4,3,5,6,7,2,1] => [1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => ? = 2 - 1
[8,7,6,5,3,2,4,1] => [7,6,5,3,2,4,1] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 2 - 1
[8,6,5,7,3,2,4,1] => [6,5,7,3,2,4,1] => [1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => ? = 3 - 1
[5,6,7,8,3,2,4,1] => [5,6,7,3,2,4,1] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 2 - 1
[7,8,3,2,4,5,6,1] => [7,3,2,4,5,6,1] => [1,2,4,5,6,3,7] => [7,6,4,3,2,5,1] => ? = 2 - 1
[6,5,4,3,7,2,8,1] => [6,5,4,3,7,2,1] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 2 - 1
[5,4,6,3,7,2,8,1] => [5,4,6,3,7,2,1] => [1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => ? = 3 - 1
[6,5,3,4,7,2,8,1] => [6,5,3,4,7,2,1] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 2 - 1
[7,6,5,3,2,4,8,1] => [7,6,5,3,2,4,1] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 2 - 1
[5,4,3,2,6,7,8,1] => [5,4,3,2,6,7,1] => [1,2,6,7,3,4,5] => [7,6,2,1,5,4,3] => ? = 2 - 1
[3,4,5,2,6,7,8,1] => [3,4,5,2,6,7,1] => [1,2,6,7,3,4,5] => [7,6,2,1,5,4,3] => ? = 2 - 1
[5,2,3,4,6,7,8,1] => [5,2,3,4,6,7,1] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 2 - 1
[4,3,2,5,6,7,8,1] => [4,3,2,5,6,7,1] => [1,2,5,6,7,3,4] => [7,6,3,2,1,5,4] => ? = 2 - 1
[3,4,2,5,6,7,8,1] => [3,4,2,5,6,7,1] => [1,2,5,6,7,3,4] => [7,6,3,2,1,5,4] => ? = 2 - 1
[4,2,3,5,6,7,8,1] => [4,2,3,5,6,7,1] => [1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => ? = 2 - 1
[3,2,4,5,6,7,8,1] => [3,2,4,5,6,7,1] => [1,2,4,5,6,7,3] => [7,6,4,3,2,1,5] => ? = 2 - 1
[7,8,5,4,3,6,1,2] => [7,5,4,3,6,1,2] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 2 - 1
[7,8,4,5,3,6,1,2] => [7,4,5,3,6,1,2] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 2 - 1
[7,8,5,3,4,6,1,2] => [7,5,3,4,6,1,2] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 2 - 1
[7,8,4,3,5,6,1,2] => [7,4,3,5,6,1,2] => [1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => ? = 2 - 1
[8,4,3,5,6,7,1,2] => [4,3,5,6,7,1,2] => [1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => ? = 2 - 1
[5,6,4,7,3,8,1,2] => [5,6,4,7,3,1,2] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 2 - 1
[5,4,6,7,3,8,1,2] => [5,4,6,7,3,1,2] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 2 - 1
[5,4,6,3,7,8,1,2] => [5,4,6,3,7,1,2] => [1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => ? = 3 - 1
[5,6,3,4,7,8,1,2] => [5,6,3,4,7,1,2] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 2 - 1
[6,4,3,5,7,8,1,2] => [6,4,3,5,7,1,2] => [1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => ? = 2 - 1
[5,4,3,6,7,8,1,2] => [5,4,3,6,7,1,2] => [1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => ? = 2 - 1
[4,5,3,6,7,8,1,2] => [4,5,3,6,7,1,2] => [1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => ? = 2 - 1
[4,3,5,6,7,8,1,2] => [4,3,5,6,7,1,2] => [1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => ? = 2 - 1
[8,7,6,5,4,2,1,3] => [7,6,5,4,2,1,3] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => ? = 2 - 1
[7,6,8,5,4,2,1,3] => [7,6,5,4,2,1,3] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => ? = 2 - 1
[5,6,7,8,4,2,1,3] => [5,6,7,4,2,1,3] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => ? = 2 - 1
[4,5,6,7,8,2,1,3] => [4,5,6,7,2,1,3] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => ? = 2 - 1
[6,5,4,7,8,1,2,3] => [6,5,4,7,1,2,3] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 2 - 1
[8,7,6,5,3,2,1,4] => [7,6,5,3,2,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 2 - 1
[7,8,5,6,3,2,1,4] => [7,5,6,3,2,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 2 - 1
[7,6,5,8,3,2,1,4] => [7,6,5,3,2,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 2 - 1
[5,6,7,8,3,2,1,4] => [5,6,7,3,2,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 2 - 1
[8,7,6,5,2,3,1,4] => [7,6,5,2,3,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 2 - 1
[7,8,5,6,2,3,1,4] => [7,5,6,2,3,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 2 - 1
[6,7,5,8,2,3,1,4] => [6,7,5,2,3,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 2 - 1
[7,5,6,8,2,3,1,4] => [7,5,6,2,3,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 2 - 1
[5,6,7,8,2,3,1,4] => [5,6,7,2,3,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 2 - 1
[8,7,6,5,3,1,2,4] => [7,6,5,3,1,2,4] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 2 - 1
[7,8,5,6,3,1,2,4] => [7,5,6,3,1,2,4] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 2 - 1
[6,7,5,8,3,1,2,4] => [6,7,5,3,1,2,4] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 2 - 1
Description
The number of ascents of a permutation.
Matching statistic: St000834
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => [1] => 0 = 1 - 1
[2,1] => [1] => [1] => [1] => 0 = 1 - 1
[1,2,3] => [1,2] => [1,2] => [2,1] => 0 = 1 - 1
[1,3,2] => [1,2] => [1,2] => [2,1] => 0 = 1 - 1
[2,1,3] => [2,1] => [1,2] => [2,1] => 0 = 1 - 1
[2,3,1] => [2,1] => [1,2] => [2,1] => 0 = 1 - 1
[3,1,2] => [1,2] => [1,2] => [2,1] => 0 = 1 - 1
[3,2,1] => [2,1] => [1,2] => [2,1] => 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[1,3,4,2] => [1,3,2] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[1,4,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[2,1,3,4] => [2,1,3] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[2,3,4,1] => [2,3,1] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[2,4,1,3] => [2,1,3] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[3,1,2,4] => [3,1,2] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[3,1,4,2] => [3,1,2] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[3,2,1,4] => [3,2,1] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[3,2,4,1] => [3,2,1] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[3,4,1,2] => [3,1,2] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[3,4,2,1] => [3,2,1] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[4,1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[4,2,1,3] => [2,1,3] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[4,3,1,2] => [3,1,2] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[4,3,2,1] => [3,2,1] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [1,3,4,2] => [4,2,1,3] => 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,2] => [1,3,4,2] => [4,2,1,3] => 1 = 2 - 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => [4,2,1,3] => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [1,4,2,3] => [4,1,3,2] => 1 = 2 - 1
[1,4,3,5,2] => [1,4,3,2] => [1,4,2,3] => [4,1,3,2] => 1 = 2 - 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [1,4,2,3] => [4,1,3,2] => 1 = 2 - 1
[7,5,4,6,8,3,2,1] => [7,5,4,6,3,2,1] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 2 - 1
[8,7,5,4,3,6,2,1] => [7,5,4,3,6,2,1] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 2 - 1
[5,4,6,7,3,8,2,1] => [5,4,6,7,3,2,1] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 2 - 1
[7,6,4,3,5,8,2,1] => [7,6,4,3,5,2,1] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 2 - 1
[4,3,5,6,7,8,2,1] => [4,3,5,6,7,2,1] => [1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => ? = 2 - 1
[8,7,6,5,3,2,4,1] => [7,6,5,3,2,4,1] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 2 - 1
[8,6,5,7,3,2,4,1] => [6,5,7,3,2,4,1] => [1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => ? = 3 - 1
[5,6,7,8,3,2,4,1] => [5,6,7,3,2,4,1] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 2 - 1
[7,8,3,2,4,5,6,1] => [7,3,2,4,5,6,1] => [1,2,4,5,6,3,7] => [7,6,4,3,2,5,1] => ? = 2 - 1
[6,5,4,3,7,2,8,1] => [6,5,4,3,7,2,1] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 2 - 1
[5,4,6,3,7,2,8,1] => [5,4,6,3,7,2,1] => [1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => ? = 3 - 1
[6,5,3,4,7,2,8,1] => [6,5,3,4,7,2,1] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 2 - 1
[7,6,5,3,2,4,8,1] => [7,6,5,3,2,4,1] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 2 - 1
[5,4,3,2,6,7,8,1] => [5,4,3,2,6,7,1] => [1,2,6,7,3,4,5] => [7,6,2,1,5,4,3] => ? = 2 - 1
[3,4,5,2,6,7,8,1] => [3,4,5,2,6,7,1] => [1,2,6,7,3,4,5] => [7,6,2,1,5,4,3] => ? = 2 - 1
[5,2,3,4,6,7,8,1] => [5,2,3,4,6,7,1] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 2 - 1
[4,3,2,5,6,7,8,1] => [4,3,2,5,6,7,1] => [1,2,5,6,7,3,4] => [7,6,3,2,1,5,4] => ? = 2 - 1
[3,4,2,5,6,7,8,1] => [3,4,2,5,6,7,1] => [1,2,5,6,7,3,4] => [7,6,3,2,1,5,4] => ? = 2 - 1
[4,2,3,5,6,7,8,1] => [4,2,3,5,6,7,1] => [1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => ? = 2 - 1
[3,2,4,5,6,7,8,1] => [3,2,4,5,6,7,1] => [1,2,4,5,6,7,3] => [7,6,4,3,2,1,5] => ? = 2 - 1
[7,8,5,4,3,6,1,2] => [7,5,4,3,6,1,2] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 2 - 1
[7,8,4,5,3,6,1,2] => [7,4,5,3,6,1,2] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 2 - 1
[7,8,5,3,4,6,1,2] => [7,5,3,4,6,1,2] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 2 - 1
[7,8,4,3,5,6,1,2] => [7,4,3,5,6,1,2] => [1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => ? = 2 - 1
[8,4,3,5,6,7,1,2] => [4,3,5,6,7,1,2] => [1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => ? = 2 - 1
[5,6,4,7,3,8,1,2] => [5,6,4,7,3,1,2] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 2 - 1
[5,4,6,7,3,8,1,2] => [5,4,6,7,3,1,2] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 2 - 1
[5,4,6,3,7,8,1,2] => [5,4,6,3,7,1,2] => [1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => ? = 3 - 1
[5,6,3,4,7,8,1,2] => [5,6,3,4,7,1,2] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 2 - 1
[6,4,3,5,7,8,1,2] => [6,4,3,5,7,1,2] => [1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => ? = 2 - 1
[5,4,3,6,7,8,1,2] => [5,4,3,6,7,1,2] => [1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => ? = 2 - 1
[4,5,3,6,7,8,1,2] => [4,5,3,6,7,1,2] => [1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => ? = 2 - 1
[4,3,5,6,7,8,1,2] => [4,3,5,6,7,1,2] => [1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => ? = 2 - 1
[8,7,6,5,4,2,1,3] => [7,6,5,4,2,1,3] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => ? = 2 - 1
[7,6,8,5,4,2,1,3] => [7,6,5,4,2,1,3] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => ? = 2 - 1
[5,6,7,8,4,2,1,3] => [5,6,7,4,2,1,3] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => ? = 2 - 1
[4,5,6,7,8,2,1,3] => [4,5,6,7,2,1,3] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => ? = 2 - 1
[6,5,4,7,8,1,2,3] => [6,5,4,7,1,2,3] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 2 - 1
[8,7,6,5,3,2,1,4] => [7,6,5,3,2,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 2 - 1
[7,8,5,6,3,2,1,4] => [7,5,6,3,2,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 2 - 1
[7,6,5,8,3,2,1,4] => [7,6,5,3,2,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 2 - 1
[5,6,7,8,3,2,1,4] => [5,6,7,3,2,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 2 - 1
[8,7,6,5,2,3,1,4] => [7,6,5,2,3,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 2 - 1
[7,8,5,6,2,3,1,4] => [7,5,6,2,3,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 2 - 1
[6,7,5,8,2,3,1,4] => [6,7,5,2,3,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 2 - 1
[7,5,6,8,2,3,1,4] => [7,5,6,2,3,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 2 - 1
[5,6,7,8,2,3,1,4] => [5,6,7,2,3,1,4] => [1,4,2,3,5,6,7] => [7,4,6,5,3,2,1] => ? = 2 - 1
[8,7,6,5,3,1,2,4] => [7,6,5,3,1,2,4] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 2 - 1
[7,8,5,6,3,1,2,4] => [7,5,6,3,1,2,4] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 2 - 1
[6,7,5,8,3,1,2,4] => [6,7,5,3,1,2,4] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 2 - 1
Description
The number of right outer peaks of a permutation.
A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St000099
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 75%
Mp00223: Permutations —runsort⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 75%
Values
[1,2] => [1] => [1] => 1
[2,1] => [1] => [1] => 1
[1,2,3] => [1,2] => [1,2] => 1
[1,3,2] => [1,2] => [1,2] => 1
[2,1,3] => [2,1] => [1,2] => 1
[2,3,1] => [2,1] => [1,2] => 1
[3,1,2] => [1,2] => [1,2] => 1
[3,2,1] => [2,1] => [1,2] => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 2
[1,3,4,2] => [1,3,2] => [1,3,2] => 2
[1,4,2,3] => [1,2,3] => [1,2,3] => 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3,4] => [2,1,3] => [1,3,2] => 2
[2,1,4,3] => [2,1,3] => [1,3,2] => 2
[2,3,1,4] => [2,3,1] => [1,2,3] => 1
[2,3,4,1] => [2,3,1] => [1,2,3] => 1
[2,4,1,3] => [2,1,3] => [1,3,2] => 2
[2,4,3,1] => [2,3,1] => [1,2,3] => 1
[3,1,2,4] => [3,1,2] => [1,2,3] => 1
[3,1,4,2] => [3,1,2] => [1,2,3] => 1
[3,2,1,4] => [3,2,1] => [1,2,3] => 1
[3,2,4,1] => [3,2,1] => [1,2,3] => 1
[3,4,1,2] => [3,1,2] => [1,2,3] => 1
[3,4,2,1] => [3,2,1] => [1,2,3] => 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 2
[4,2,1,3] => [2,1,3] => [1,3,2] => 2
[4,2,3,1] => [2,3,1] => [1,2,3] => 1
[4,3,1,2] => [3,1,2] => [1,2,3] => 1
[4,3,2,1] => [3,2,1] => [1,2,3] => 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 2
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2,5] => [1,3,4,2] => [1,3,4,2] => 2
[1,3,4,5,2] => [1,3,4,2] => [1,3,4,2] => 2
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => 2
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2,5] => [1,4,3,2] => [1,4,2,3] => 2
[1,4,3,5,2] => [1,4,3,2] => [1,4,2,3] => 2
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,5,3,2] => [1,4,3,2] => [1,4,2,3] => 2
[8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ? = 1
[7,8,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ? = 1
[7,6,8,5,4,3,2,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ? = 1
[6,7,8,5,4,3,2,1] => [6,7,5,4,3,2,1] => [1,2,3,4,5,6,7] => ? = 1
[7,8,5,6,4,3,2,1] => [7,5,6,4,3,2,1] => [1,2,3,4,5,6,7] => ? = 1
[7,6,5,8,4,3,2,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ? = 1
[7,5,6,8,4,3,2,1] => [7,5,6,4,3,2,1] => [1,2,3,4,5,6,7] => ? = 1
[6,5,7,8,4,3,2,1] => [6,5,7,4,3,2,1] => [1,2,3,4,5,7,6] => ? = 2
[5,6,7,8,4,3,2,1] => [5,6,7,4,3,2,1] => [1,2,3,4,5,6,7] => ? = 1
[8,7,6,4,5,3,2,1] => [7,6,4,5,3,2,1] => [1,2,3,4,5,6,7] => ? = 1
[7,8,6,4,5,3,2,1] => [7,6,4,5,3,2,1] => [1,2,3,4,5,6,7] => ? = 1
[8,6,7,4,5,3,2,1] => [6,7,4,5,3,2,1] => [1,2,3,4,5,6,7] => ? = 1
[7,6,5,4,8,3,2,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ? = 1
[6,5,7,4,8,3,2,1] => [6,5,7,4,3,2,1] => [1,2,3,4,5,7,6] => ? = 2
[7,6,4,5,8,3,2,1] => [7,6,4,5,3,2,1] => [1,2,3,4,5,6,7] => ? = 1
[7,5,4,6,8,3,2,1] => [7,5,4,6,3,2,1] => [1,2,3,4,6,5,7] => ? = 2
[7,4,5,6,8,3,2,1] => [7,4,5,6,3,2,1] => [1,2,3,4,5,6,7] => ? = 1
[4,5,6,7,8,3,2,1] => [4,5,6,7,3,2,1] => [1,2,3,4,5,6,7] => ? = 1
[8,7,6,5,3,4,2,1] => [7,6,5,3,4,2,1] => [1,2,3,4,5,6,7] => ? = 1
[7,8,6,5,3,4,2,1] => [7,6,5,3,4,2,1] => [1,2,3,4,5,6,7] => ? = 1
[8,6,7,5,3,4,2,1] => [6,7,5,3,4,2,1] => [1,2,3,4,5,6,7] => ? = 1
[8,7,5,6,3,4,2,1] => [7,5,6,3,4,2,1] => [1,2,3,4,5,6,7] => ? = 1
[5,6,7,8,3,4,2,1] => [5,6,7,3,4,2,1] => [1,2,3,4,5,6,7] => ? = 1
[6,7,8,3,4,5,2,1] => [6,7,3,4,5,2,1] => [1,2,3,4,5,6,7] => ? = 1
[8,7,5,4,3,6,2,1] => [7,5,4,3,6,2,1] => [1,2,3,6,4,5,7] => ? = 2
[7,6,5,4,3,8,2,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ? = 1
[6,4,5,7,3,8,2,1] => [6,4,5,7,3,2,1] => [1,2,3,4,5,7,6] => ? = 2
[5,4,6,7,3,8,2,1] => [5,4,6,7,3,2,1] => [1,2,3,4,6,7,5] => ? = 2
[7,6,5,3,4,8,2,1] => [7,6,5,3,4,2,1] => [1,2,3,4,5,6,7] => ? = 1
[7,6,4,3,5,8,2,1] => [7,6,4,3,5,2,1] => [1,2,3,5,4,6,7] => ? = 2
[7,6,3,4,5,8,2,1] => [7,6,3,4,5,2,1] => [1,2,3,4,5,6,7] => ? = 1
[6,3,4,5,7,8,2,1] => [6,3,4,5,7,2,1] => [1,2,3,4,5,7,6] => ? = 2
[4,3,5,6,7,8,2,1] => [4,3,5,6,7,2,1] => [1,2,3,5,6,7,4] => ? = 2
[8,7,6,5,4,2,3,1] => [7,6,5,4,2,3,1] => [1,2,3,4,5,6,7] => ? = 1
[7,8,6,5,4,2,3,1] => [7,6,5,4,2,3,1] => [1,2,3,4,5,6,7] => ? = 1
[8,6,7,5,4,2,3,1] => [6,7,5,4,2,3,1] => [1,2,3,4,5,6,7] => ? = 1
[8,7,5,6,4,2,3,1] => [7,5,6,4,2,3,1] => [1,2,3,4,5,6,7] => ? = 1
[5,6,7,8,4,2,3,1] => [5,6,7,4,2,3,1] => [1,2,3,4,5,6,7] => ? = 1
[8,7,6,4,5,2,3,1] => [7,6,4,5,2,3,1] => [1,2,3,4,5,6,7] => ? = 1
[8,6,7,4,5,2,3,1] => [6,7,4,5,2,3,1] => [1,2,3,4,5,6,7] => ? = 1
[4,5,6,7,8,2,3,1] => [4,5,6,7,2,3,1] => [1,2,3,4,5,6,7] => ? = 1
[8,7,6,5,3,2,4,1] => [7,6,5,3,2,4,1] => [1,2,4,3,5,6,7] => ? = 2
[8,6,5,7,3,2,4,1] => [6,5,7,3,2,4,1] => [1,2,4,3,5,7,6] => ? = 3
[5,6,7,8,3,2,4,1] => [5,6,7,3,2,4,1] => [1,2,4,3,5,6,7] => ? = 2
[8,7,6,5,2,3,4,1] => [7,6,5,2,3,4,1] => [1,2,3,4,5,6,7] => ? = 1
[6,7,8,5,2,3,4,1] => [6,7,5,2,3,4,1] => [1,2,3,4,5,6,7] => ? = 1
[8,5,6,7,2,3,4,1] => [5,6,7,2,3,4,1] => [1,2,3,4,5,6,7] => ? = 1
[5,6,7,8,2,3,4,1] => [5,6,7,2,3,4,1] => [1,2,3,4,5,6,7] => ? = 1
[7,8,3,2,4,5,6,1] => [7,3,2,4,5,6,1] => [1,2,4,5,6,3,7] => ? = 2
[7,6,5,4,3,2,8,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ? = 1
Description
The number of valleys of a permutation, including the boundary.
The number of valleys excluding the boundary is [[St000353]].
The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000325The width of the tree associated to a permutation. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000092The number of outer peaks of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000353The number of inner valleys of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset.
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