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Your data matches 149 different statistics following compositions of up to 3 maps.
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Matching statistic: St001394
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
St001394: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001394: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 0
[2,1,3] => [1,3,2] => 0
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,2,3] => 0
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,3,4,2] => 0
[1,4,2,3] => [1,4,2,3] => 1
[1,4,3,2] => [1,4,2,3] => 1
[2,1,3,4] => [1,3,4,2] => 0
[2,1,4,3] => [1,4,2,3] => 1
[2,3,1,4] => [1,4,2,3] => 1
[2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => 0
[2,4,3,1] => [1,2,4,3] => 0
[3,1,2,4] => [1,2,4,3] => 0
[3,1,4,2] => [1,4,2,3] => 1
[3,2,1,4] => [1,4,2,3] => 1
[3,2,4,1] => [1,2,4,3] => 0
[3,4,1,2] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => 0
[4,2,1,3] => [1,3,2,4] => 0
[4,2,3,1] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,4,5,3] => 0
[1,2,5,3,4] => [1,2,5,3,4] => 1
[1,2,5,4,3] => [1,2,5,3,4] => 1
[1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,3,4,2,5] => 0
[1,3,4,5,2] => [1,3,4,5,2] => 0
[1,3,5,2,4] => [1,3,5,2,4] => 1
[1,3,5,4,2] => [1,3,5,2,4] => 1
[1,4,2,3,5] => [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => 1
[1,4,3,2,5] => [1,4,2,5,3] => 1
[1,4,3,5,2] => [1,4,2,3,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => 1
Description
The genus of a permutation.
The genus $g(\pi)$ of a permutation $\pi\in\mathfrak S_n$ is defined via the relation
$$
n+1-2g(\pi) = z(\pi) + z(\pi^{-1} \zeta ),
$$
where $\zeta = (1,2,\dots,n)$ is the long cycle and $z(\cdot)$ is the number of cycles in the permutation.
Matching statistic: St001665
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001665: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001665: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,2,3] => 0
[2,1,3] => [1,3,2] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0
[1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,3,4,2] => [1,2,3,4] => 0
[1,4,2,3] => [1,4,2,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,4,2,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,3,4,2] => [1,2,3,4] => 0
[2,1,4,3] => [1,4,2,3] => [1,2,4,3] => 1
[2,3,1,4] => [1,4,2,3] => [1,2,4,3] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,2,3,4] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,3,4] => 0
[3,1,2,4] => [1,2,4,3] => [1,2,3,4] => 0
[3,1,4,2] => [1,4,2,3] => [1,2,4,3] => 1
[3,2,1,4] => [1,4,2,3] => [1,2,4,3] => 1
[3,2,4,1] => [1,2,4,3] => [1,2,3,4] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,2,3,4] => 0
[4,2,1,3] => [1,3,2,4] => [1,2,3,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,3,5,4] => 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,3,5,4] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,3,4,2,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,3,4,5,2] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,3,5,2,4] => [1,2,3,5,4] => 1
[1,3,5,4,2] => [1,3,5,2,4] => [1,2,3,5,4] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,2,4,3,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [1,2,4,5,3] => 1
[1,4,3,2,5] => [1,4,2,5,3] => [1,2,4,5,3] => 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,2,4,3,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,2,4,3,5] => 1
Description
The number of pure excedances of a permutation.
A pure excedance of a permutation $\pi$ is a position $i < \pi_i$ such that there is no $j < i$ with $i\leq \pi_j < \pi_i$.
Matching statistic: St000670
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000670: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000670: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,3,2] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,3,2] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[3,1,2,4] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,1,4,2] => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,2,1,4] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,2,4,1] => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,1,2] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[4,1,3,2] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[4,2,1,3] => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[4,2,3,1] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[4,3,1,2] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[4,3,2,1] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
Description
The reversal length of a permutation.
A reversal in a permutation $\pi = [\pi_1,\ldots,\pi_n]$ is a reversal of a subsequence of the form $\operatorname{reversal}_{i,j}(\pi) = [\pi_1,\ldots,\pi_{i-1},\pi_j,\pi_{j-1},\ldots,\pi_{i+1},\pi_i,\pi_{j+1},\ldots,\pi_n]$ for $1 \leq i < j \leq n$.
This statistic is then given by the minimal number of reversals needed to sort a permutation.
The reversal distance between two permutations plays an important role in studying DNA structures.
Matching statistic: St001269
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001269: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001269: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,3,2] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,3,2] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[3,1,2,4] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,1,4,2] => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,2,1,4] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,2,4,1] => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,1,2] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[4,1,3,2] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[4,2,1,3] => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[4,2,3,1] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[4,3,1,2] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[4,3,2,1] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
Description
The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation.
Matching statistic: St001470
Mp00223: Permutations —runsort⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St001470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St001470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [3,1,2] => 0
[2,1,3] => [1,3,2] => [1,3,2] => [3,1,2] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 1
[1,3,4,2] => [1,3,4,2] => [1,4,3,2] => [4,3,1,2] => 1
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [3,4,1,2] => 0
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => [3,4,1,2] => 0
[2,1,3,4] => [1,3,4,2] => [1,4,3,2] => [4,3,1,2] => 1
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => [3,4,1,2] => 0
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => [3,4,1,2] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 0
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 0
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => [3,4,1,2] => 0
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => [3,4,1,2] => 0
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 1
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => 1
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [4,5,1,2,3] => 0
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => [4,5,1,2,3] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [2,5,1,3,4] => 1
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => 1
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,3,4,2] => [3,5,4,1,2] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [1,5,3,2,4] => [4,3,5,1,2] => 1
[1,3,5,4,2] => [1,3,5,2,4] => [1,5,3,2,4] => [4,3,5,1,2] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [3,4,1,2,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [1,5,2,4,3] => [5,3,4,1,2] => 1
[1,4,3,2,5] => [1,4,2,5,3] => [1,5,2,4,3] => [5,3,4,1,2] => 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => [3,4,1,2,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,5,4,3,2] => [5,4,3,1,2] => 1
Description
The cyclic holeyness of a permutation.
For $S\subset [n]:=\{1,2,\dots,n\}$ let $\delta(S)$ be the number of elements $m\in S$ such that $(m\bmod n)+1\notin S$.
For a permutation $\pi$ of $[n]$ the cyclic holeyness of $\pi$ is $$\max_{S\subset [n]} (\delta(\pi(S))-\delta(S)).$$
Matching statistic: St001737
Mp00223: Permutations —runsort⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St001737: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St001737: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,3,2] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,3,1,4] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[3,1,2,4] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[3,1,4,2] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[3,2,1,4] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[3,2,4,1] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,2,1,3] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,3,5,4,2] => [1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,4,3,2,5] => [1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => 1
Description
The number of descents of type 2 in a permutation.
A position $i\in[1,n-1]$ is a descent of type 2 of a permutation $\pi$ of $n$ letters, if it is a descent and if $\pi(j) < \pi(i)$ for all $j < i$.
Matching statistic: St001928
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,3,2] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,3,2] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[3,1,2,4] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,1,4,2] => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,2,1,4] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,2,4,1] => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,1,2] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[4,1,3,2] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[4,2,1,3] => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[4,2,3,1] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[4,3,1,2] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[4,3,2,1] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
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: St000955
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000955: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000955: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[1,2] => [1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
Description
Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra.
Matching statistic: St000243
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000243: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000243: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ? = 0 + 1
[1,2] => [1,2] => [1,2] => 1 = 0 + 1
[2,1] => [1,2] => [1,2] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,3,1] => 1 = 0 + 1
[2,1,3] => [1,3,2] => [2,3,1] => 1 = 0 + 1
[2,3,1] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[3,1,2] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[3,2,1] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [2,3,4,1] => 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [2,3,1,4] => 2 = 1 + 1
[1,3,4,2] => [1,3,4,2] => [2,4,1,3] => 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [2,1,4,3] => 1 = 0 + 1
[1,4,3,2] => [1,4,2,3] => [2,1,4,3] => 1 = 0 + 1
[2,1,3,4] => [1,3,4,2] => [2,4,1,3] => 2 = 1 + 1
[2,1,4,3] => [1,4,2,3] => [2,1,4,3] => 1 = 0 + 1
[2,3,1,4] => [1,4,2,3] => [2,1,4,3] => 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => [2,3,1,4] => 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [2,3,4,1] => 1 = 0 + 1
[3,1,2,4] => [1,2,4,3] => [2,3,4,1] => 1 = 0 + 1
[3,1,4,2] => [1,4,2,3] => [2,1,4,3] => 1 = 0 + 1
[3,2,1,4] => [1,4,2,3] => [2,1,4,3] => 1 = 0 + 1
[3,2,4,1] => [1,2,4,3] => [2,3,4,1] => 1 = 0 + 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => [2,3,1,4] => 2 = 1 + 1
[4,2,1,3] => [1,3,2,4] => [2,3,1,4] => 2 = 1 + 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,3,5,1,4] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,3,1,5,4] => 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [2,3,1,5,4] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,3,1,4,5] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,4,2,5,1] => 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [2,4,1,3,5] => 2 = 1 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,1,3,4] => 2 = 1 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,1,4,5,3] => 1 = 0 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [2,1,4,5,3] => 1 = 0 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,1,4,3,5] => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,4,5,1,2] => 1 = 0 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [3,4,5,1,2] => 1 = 0 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [2,1,4,3,5] => 2 = 1 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [2,1,5,3,4] => 2 = 1 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [2,1,5,3,4] => 2 = 1 + 1
Description
The number of cyclic valleys and cyclic peaks of a permutation.
This is given by the number of indices $i$ such that $\pi_{i-1} > \pi_i < \pi_{i+1}$ with indices considered cyclically. Equivalently, this is the number of indices $i$ such that $\pi_{i-1} < \pi_i > \pi_{i+1}$ with indices considered cyclically.
Matching statistic: St000779
Mp00223: Permutations —runsort⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000779: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000779: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,3] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,3,4,2] => [1,4,3,2] => [1,4,2,3] => 0
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 1
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 1
[2,1,3,4] => [1,3,4,2] => [1,4,3,2] => [1,4,2,3] => 0
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 1
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 1
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 1
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,3,4] => 0
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => [1,4,2,3,5] => 0
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,4,3,2] => [1,5,2,3,4] => 0
[1,3,5,2,4] => [1,3,5,2,4] => [1,5,3,2,4] => [1,4,2,5,3] => 1
[1,3,5,4,2] => [1,3,5,2,4] => [1,5,3,2,4] => [1,4,2,5,3] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [1,5,4,2,3] => [1,3,5,2,4] => 1
[1,4,3,2,5] => [1,4,2,5,3] => [1,5,4,2,3] => [1,3,5,2,4] => 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,5,2,4,3] => [1,4,5,3,2] => 1
[1,4,5,3,2] => [1,4,5,2,3] => [1,5,2,4,3] => [1,4,5,3,2] => 1
Description
The tier of a permutation.
This is the number of elements $i$ such that $[i+1,k,i]$ is an occurrence of the pattern $[2,3,1]$. For example, $[3,5,6,1,2,4]$ has tier $2$, with witnesses $[3,5,2]$ (or $[3,6,2]$) and $[5,6,4]$.
According to [1], this is the number of passes minus one needed to sort the permutation using a single stack. The generating function for this statistic appears as [[OEIS:A122890]] and [[OEIS:A158830]] in the form of triangles read by rows, see [sec. 4, 1].
The following 139 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001280The number of parts of an integer partition that are at least two. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001498The normalised height of a Nakayama algebra with magnitude 1. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000934The 2-degree of an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001597The Frobenius rank of a skew partition. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000681The Grundy value of Chomp on Ferrers diagrams. St000284The Plancherel distribution on integer partitions. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000455The second largest eigenvalue of a graph if it is integral. St001964The interval resolution global dimension of a poset. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000225Difference between largest and smallest parts in a partition. St000031The number of cycles in the cycle decomposition of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001960The number of descents of a permutation minus one if its first entry is not one. St001569The maximal modular displacement of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000260The radius of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000936The number of even values of the symmetric group character corresponding to the partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001570The minimal number of edges to add to make a graph Hamiltonian. St001877Number of indecomposable injective modules with projective dimension 2. St001846The number of elements which do not have a complement in the lattice. St001820The size of the image of the pop stack sorting operator. St001862The number of crossings of a signed permutation. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000068The number of minimal elements in a poset. St000454The largest eigenvalue of a graph if it is integral. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001845The number of join irreducibles minus the rank of a lattice. St001864The number of excedances of a signed permutation. St001875The number of simple modules with projective dimension at most 1. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St001866The nesting alignments of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001490The number of connected components of a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001867The number of alignments of type EN of a signed permutation. St000632The jump number of the poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001779The order of promotion on the set of linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001821The sorting index of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St001768The number of reduced words of a signed permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000095The number of triangles of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000322The skewness of a graph. St000449The number of pairs of vertices of a graph with distance 4. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001578The minimal number of edges to add or remove to make a graph a line graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001871The number of triconnected components of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001765The number of connected components of the friends and strangers graph.
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