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Your data matches 61 different statistics following compositions of up to 3 maps.
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Matching statistic: St000314
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1] => [1,2,3] => 3 = 1 + 2
[1,1,2] => [1,2,3] => 3 = 1 + 2
[1,2,1] => [1,2,3] => 3 = 1 + 2
[2,1,1] => [2,1,3] => 2 = 0 + 2
[1,1,3] => [1,2,3] => 3 = 1 + 2
[1,3,1] => [1,3,2] => 2 = 0 + 2
[1,2,2] => [1,2,3] => 3 = 1 + 2
[2,1,2] => [2,1,3] => 2 = 0 + 2
[1,2,3] => [1,2,3] => 3 = 1 + 2
[1,3,2] => [1,3,2] => 2 = 0 + 2
[2,1,3] => [2,1,3] => 2 = 0 + 2
[1,1,1,1] => [1,2,3,4] => 4 = 2 + 2
[1,1,1,2] => [1,2,3,4] => 4 = 2 + 2
[1,1,2,1] => [1,2,3,4] => 4 = 2 + 2
[1,2,1,1] => [1,2,3,4] => 4 = 2 + 2
[2,1,1,1] => [2,1,3,4] => 3 = 1 + 2
[1,1,1,3] => [1,2,3,4] => 4 = 2 + 2
[1,1,3,1] => [1,2,3,4] => 4 = 2 + 2
[1,3,1,1] => [1,3,2,4] => 3 = 1 + 2
[3,1,1,1] => [2,3,1,4] => 3 = 1 + 2
[1,1,1,4] => [1,2,3,4] => 4 = 2 + 2
[1,1,4,1] => [1,2,4,3] => 3 = 1 + 2
[1,4,1,1] => [1,3,4,2] => 3 = 1 + 2
[4,1,1,1] => [2,3,4,1] => 3 = 1 + 2
[1,1,2,2] => [1,2,3,4] => 4 = 2 + 2
[1,2,1,2] => [1,2,3,4] => 4 = 2 + 2
[1,2,2,1] => [1,2,3,4] => 4 = 2 + 2
[2,1,1,2] => [2,1,3,4] => 3 = 1 + 2
[2,1,2,1] => [2,1,3,4] => 3 = 1 + 2
[2,2,1,1] => [3,1,2,4] => 2 = 0 + 2
[1,1,2,3] => [1,2,3,4] => 4 = 2 + 2
[1,1,3,2] => [1,2,3,4] => 4 = 2 + 2
[1,2,1,3] => [1,2,3,4] => 4 = 2 + 2
[1,2,3,1] => [1,2,3,4] => 4 = 2 + 2
[1,3,1,2] => [1,3,2,4] => 3 = 1 + 2
[1,3,2,1] => [1,3,2,4] => 3 = 1 + 2
[2,1,1,3] => [2,1,3,4] => 3 = 1 + 2
[2,1,3,1] => [2,1,3,4] => 3 = 1 + 2
[2,3,1,1] => [3,1,2,4] => 2 = 0 + 2
[3,1,1,2] => [2,3,1,4] => 3 = 1 + 2
[3,1,2,1] => [2,3,1,4] => 3 = 1 + 2
[3,2,1,1] => [3,2,1,4] => 2 = 0 + 2
[1,1,2,4] => [1,2,3,4] => 4 = 2 + 2
[1,1,4,2] => [1,2,4,3] => 3 = 1 + 2
[1,2,1,4] => [1,2,3,4] => 4 = 2 + 2
[1,2,4,1] => [1,2,4,3] => 3 = 1 + 2
[1,4,1,2] => [1,3,4,2] => 3 = 1 + 2
[1,4,2,1] => [1,3,4,2] => 3 = 1 + 2
[2,1,1,4] => [2,1,3,4] => 3 = 1 + 2
[2,1,4,1] => [2,1,4,3] => 2 = 0 + 2
Description
The number of left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a '''left-to-right-maximum''' if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$.
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
Matching statistic: St000053
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,2] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,1] => [2,1,3] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,2] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,1,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,1,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,3,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,1,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,1,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,1,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[4,1,1,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,2,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,2,1,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,2,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,3,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,1,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,3,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,1,2] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,2,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,1,3] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,3,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,3,1,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[3,1,1,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,2,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,2,1,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,2,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,4,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,1,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,1,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,2,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,1,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,4,1] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
Description
The number of valleys of the Dyck path.
Matching statistic: St000541
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1] => [1,2,3] => [3,2,1] => 2 = 1 + 1
[1,1,2] => [1,2,3] => [3,2,1] => 2 = 1 + 1
[1,2,1] => [1,2,3] => [3,2,1] => 2 = 1 + 1
[2,1,1] => [2,1,3] => [2,3,1] => 1 = 0 + 1
[1,1,3] => [1,2,3] => [3,2,1] => 2 = 1 + 1
[1,3,1] => [1,3,2] => [3,1,2] => 1 = 0 + 1
[1,2,2] => [1,2,3] => [3,2,1] => 2 = 1 + 1
[2,1,2] => [2,1,3] => [2,3,1] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [3,2,1] => 2 = 1 + 1
[1,3,2] => [1,3,2] => [3,1,2] => 1 = 0 + 1
[2,1,3] => [2,1,3] => [2,3,1] => 1 = 0 + 1
[1,1,1,1] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,1,1,2] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,1,2,1] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,2,1,1] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[2,1,1,1] => [2,1,3,4] => [3,4,2,1] => 2 = 1 + 1
[1,1,1,3] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,1,3,1] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,3,1,1] => [1,3,2,4] => [4,2,3,1] => 2 = 1 + 1
[3,1,1,1] => [2,3,1,4] => [3,2,4,1] => 2 = 1 + 1
[1,1,1,4] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,1,4,1] => [1,2,4,3] => [4,3,1,2] => 2 = 1 + 1
[1,4,1,1] => [1,3,4,2] => [4,2,1,3] => 2 = 1 + 1
[4,1,1,1] => [2,3,4,1] => [3,2,1,4] => 2 = 1 + 1
[1,1,2,2] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,2,1,2] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,2,2,1] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[2,1,1,2] => [2,1,3,4] => [3,4,2,1] => 2 = 1 + 1
[2,1,2,1] => [2,1,3,4] => [3,4,2,1] => 2 = 1 + 1
[2,2,1,1] => [3,1,2,4] => [2,4,3,1] => 1 = 0 + 1
[1,1,2,3] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,1,3,2] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,2,1,3] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,2,3,1] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,3,1,2] => [1,3,2,4] => [4,2,3,1] => 2 = 1 + 1
[1,3,2,1] => [1,3,2,4] => [4,2,3,1] => 2 = 1 + 1
[2,1,1,3] => [2,1,3,4] => [3,4,2,1] => 2 = 1 + 1
[2,1,3,1] => [2,1,3,4] => [3,4,2,1] => 2 = 1 + 1
[2,3,1,1] => [3,1,2,4] => [2,4,3,1] => 1 = 0 + 1
[3,1,1,2] => [2,3,1,4] => [3,2,4,1] => 2 = 1 + 1
[3,1,2,1] => [2,3,1,4] => [3,2,4,1] => 2 = 1 + 1
[3,2,1,1] => [3,2,1,4] => [2,3,4,1] => 1 = 0 + 1
[1,1,2,4] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,1,4,2] => [1,2,4,3] => [4,3,1,2] => 2 = 1 + 1
[1,2,1,4] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,2,4,1] => [1,2,4,3] => [4,3,1,2] => 2 = 1 + 1
[1,4,1,2] => [1,3,4,2] => [4,2,1,3] => 2 = 1 + 1
[1,4,2,1] => [1,3,4,2] => [4,2,1,3] => 2 = 1 + 1
[2,1,1,4] => [2,1,3,4] => [3,4,2,1] => 2 = 1 + 1
[2,1,4,1] => [2,1,4,3] => [3,4,1,2] => 1 = 0 + 1
Description
The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right.
For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.
Matching statistic: St001169
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001169: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001169: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,2] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,1] => [2,1,3] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,2] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,1,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,1,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,3,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,1,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,1,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,1,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[4,1,1,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,2,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,2,1,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,2,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,3,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,1,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,3,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,1,2] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,2,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,1,3] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,3,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,3,1,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[3,1,1,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,2,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,2,1,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,2,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,4,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,1,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,1,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,2,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,1,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,4,1] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
Description
Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St000007
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1] => [1,2,3] => [3,2,1] => 3 = 1 + 2
[1,1,2] => [1,2,3] => [3,2,1] => 3 = 1 + 2
[1,2,1] => [1,2,3] => [3,2,1] => 3 = 1 + 2
[2,1,1] => [2,1,3] => [3,1,2] => 2 = 0 + 2
[1,1,3] => [1,2,3] => [3,2,1] => 3 = 1 + 2
[1,3,1] => [1,3,2] => [2,3,1] => 2 = 0 + 2
[1,2,2] => [1,2,3] => [3,2,1] => 3 = 1 + 2
[2,1,2] => [2,1,3] => [3,1,2] => 2 = 0 + 2
[1,2,3] => [1,2,3] => [3,2,1] => 3 = 1 + 2
[1,3,2] => [1,3,2] => [2,3,1] => 2 = 0 + 2
[2,1,3] => [2,1,3] => [3,1,2] => 2 = 0 + 2
[1,1,1,1] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,1,1,2] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,1,2,1] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,2,1,1] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[2,1,1,1] => [2,1,3,4] => [4,3,1,2] => 3 = 1 + 2
[1,1,1,3] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,1,3,1] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,3,1,1] => [1,3,2,4] => [4,2,3,1] => 3 = 1 + 2
[3,1,1,1] => [2,3,1,4] => [4,1,3,2] => 3 = 1 + 2
[1,1,1,4] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,1,4,1] => [1,2,4,3] => [3,4,2,1] => 3 = 1 + 2
[1,4,1,1] => [1,3,4,2] => [2,4,3,1] => 3 = 1 + 2
[4,1,1,1] => [2,3,4,1] => [1,4,3,2] => 3 = 1 + 2
[1,1,2,2] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,2,1,2] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,2,2,1] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[2,1,1,2] => [2,1,3,4] => [4,3,1,2] => 3 = 1 + 2
[2,1,2,1] => [2,1,3,4] => [4,3,1,2] => 3 = 1 + 2
[2,2,1,1] => [3,1,2,4] => [4,2,1,3] => 2 = 0 + 2
[1,1,2,3] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,1,3,2] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,2,1,3] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,2,3,1] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,3,1,2] => [1,3,2,4] => [4,2,3,1] => 3 = 1 + 2
[1,3,2,1] => [1,3,2,4] => [4,2,3,1] => 3 = 1 + 2
[2,1,1,3] => [2,1,3,4] => [4,3,1,2] => 3 = 1 + 2
[2,1,3,1] => [2,1,3,4] => [4,3,1,2] => 3 = 1 + 2
[2,3,1,1] => [3,1,2,4] => [4,2,1,3] => 2 = 0 + 2
[3,1,1,2] => [2,3,1,4] => [4,1,3,2] => 3 = 1 + 2
[3,1,2,1] => [2,3,1,4] => [4,1,3,2] => 3 = 1 + 2
[3,2,1,1] => [3,2,1,4] => [4,1,2,3] => 2 = 0 + 2
[1,1,2,4] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,1,4,2] => [1,2,4,3] => [3,4,2,1] => 3 = 1 + 2
[1,2,1,4] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,2,4,1] => [1,2,4,3] => [3,4,2,1] => 3 = 1 + 2
[1,4,1,2] => [1,3,4,2] => [2,4,3,1] => 3 = 1 + 2
[1,4,2,1] => [1,3,4,2] => [2,4,3,1] => 3 = 1 + 2
[2,1,1,4] => [2,1,3,4] => [4,3,1,2] => 3 = 1 + 2
[2,1,4,1] => [2,1,4,3] => [3,4,1,2] => 2 = 0 + 2
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000015
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1] => [1,2,3] => [1,0,1,0,1,0]
=> 3 = 1 + 2
[1,1,2] => [1,2,3] => [1,0,1,0,1,0]
=> 3 = 1 + 2
[1,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 3 = 1 + 2
[2,1,1] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 0 + 2
[1,1,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3 = 1 + 2
[1,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
[1,2,2] => [1,2,3] => [1,0,1,0,1,0]
=> 3 = 1 + 2
[2,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 0 + 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3 = 1 + 2
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,2,1,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[2,1,1,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[1,1,1,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,3,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,3,1,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[3,1,1,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[1,1,1,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[1,4,1,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
[4,1,1,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,2,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,2,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,2,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[2,1,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[2,1,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[2,2,1,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,2,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,3,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,2,1,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,2,3,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,3,1,2] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[1,3,2,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[2,1,1,3] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[2,1,3,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[2,3,1,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[3,1,1,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[3,1,2,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[3,2,1,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,2,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,4,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[1,2,1,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,2,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[1,4,1,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
[1,4,2,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
[2,1,1,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[2,1,4,1] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
Description
The number of peaks of a Dyck path.
Matching statistic: St000542
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1] => [1,2,3] => [3,2,1] => 3 = 1 + 2
[1,1,2] => [1,2,3] => [3,2,1] => 3 = 1 + 2
[1,2,1] => [1,2,3] => [3,2,1] => 3 = 1 + 2
[2,1,1] => [2,1,3] => [2,3,1] => 2 = 0 + 2
[1,1,3] => [1,2,3] => [3,2,1] => 3 = 1 + 2
[1,3,1] => [1,3,2] => [3,1,2] => 2 = 0 + 2
[1,2,2] => [1,2,3] => [3,2,1] => 3 = 1 + 2
[2,1,2] => [2,1,3] => [2,3,1] => 2 = 0 + 2
[1,2,3] => [1,2,3] => [3,2,1] => 3 = 1 + 2
[1,3,2] => [1,3,2] => [3,1,2] => 2 = 0 + 2
[2,1,3] => [2,1,3] => [2,3,1] => 2 = 0 + 2
[1,1,1,1] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,1,1,2] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,1,2,1] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,2,1,1] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[2,1,1,1] => [2,1,3,4] => [3,4,2,1] => 3 = 1 + 2
[1,1,1,3] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,1,3,1] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,3,1,1] => [1,3,2,4] => [4,2,3,1] => 3 = 1 + 2
[3,1,1,1] => [2,3,1,4] => [3,2,4,1] => 3 = 1 + 2
[1,1,1,4] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,1,4,1] => [1,2,4,3] => [4,3,1,2] => 3 = 1 + 2
[1,4,1,1] => [1,3,4,2] => [4,2,1,3] => 3 = 1 + 2
[4,1,1,1] => [2,3,4,1] => [3,2,1,4] => 3 = 1 + 2
[1,1,2,2] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,2,1,2] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,2,2,1] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[2,1,1,2] => [2,1,3,4] => [3,4,2,1] => 3 = 1 + 2
[2,1,2,1] => [2,1,3,4] => [3,4,2,1] => 3 = 1 + 2
[2,2,1,1] => [3,1,2,4] => [2,4,3,1] => 2 = 0 + 2
[1,1,2,3] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,1,3,2] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,2,1,3] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,2,3,1] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,3,1,2] => [1,3,2,4] => [4,2,3,1] => 3 = 1 + 2
[1,3,2,1] => [1,3,2,4] => [4,2,3,1] => 3 = 1 + 2
[2,1,1,3] => [2,1,3,4] => [3,4,2,1] => 3 = 1 + 2
[2,1,3,1] => [2,1,3,4] => [3,4,2,1] => 3 = 1 + 2
[2,3,1,1] => [3,1,2,4] => [2,4,3,1] => 2 = 0 + 2
[3,1,1,2] => [2,3,1,4] => [3,2,4,1] => 3 = 1 + 2
[3,1,2,1] => [2,3,1,4] => [3,2,4,1] => 3 = 1 + 2
[3,2,1,1] => [3,2,1,4] => [2,3,4,1] => 2 = 0 + 2
[1,1,2,4] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,1,4,2] => [1,2,4,3] => [4,3,1,2] => 3 = 1 + 2
[1,2,1,4] => [1,2,3,4] => [4,3,2,1] => 4 = 2 + 2
[1,2,4,1] => [1,2,4,3] => [4,3,1,2] => 3 = 1 + 2
[1,4,1,2] => [1,3,4,2] => [4,2,1,3] => 3 = 1 + 2
[1,4,2,1] => [1,3,4,2] => [4,2,1,3] => 3 = 1 + 2
[2,1,1,4] => [2,1,3,4] => [3,4,2,1] => 3 = 1 + 2
[2,1,4,1] => [2,1,4,3] => [3,4,1,2] => 2 = 0 + 2
Description
The number of left-to-right-minima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
Matching statistic: St000991
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1] => [1,2,3] => [1,2,3] => 3 = 1 + 2
[1,1,2] => [1,2,3] => [1,2,3] => 3 = 1 + 2
[1,2,1] => [1,2,3] => [1,2,3] => 3 = 1 + 2
[2,1,1] => [2,1,3] => [2,1,3] => 2 = 0 + 2
[1,1,3] => [1,2,3] => [1,2,3] => 3 = 1 + 2
[1,3,1] => [1,3,2] => [1,3,2] => 2 = 0 + 2
[1,2,2] => [1,2,3] => [1,2,3] => 3 = 1 + 2
[2,1,2] => [2,1,3] => [2,1,3] => 2 = 0 + 2
[1,2,3] => [1,2,3] => [1,2,3] => 3 = 1 + 2
[1,3,2] => [1,3,2] => [1,3,2] => 2 = 0 + 2
[2,1,3] => [2,1,3] => [2,1,3] => 2 = 0 + 2
[1,1,1,1] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[1,1,1,2] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[1,1,2,1] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[1,2,1,1] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[2,1,1,1] => [2,1,3,4] => [2,1,3,4] => 3 = 1 + 2
[1,1,1,3] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[1,1,3,1] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[1,3,1,1] => [1,3,2,4] => [1,3,2,4] => 3 = 1 + 2
[3,1,1,1] => [2,3,1,4] => [3,1,2,4] => 3 = 1 + 2
[1,1,1,4] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[1,1,4,1] => [1,2,4,3] => [1,2,4,3] => 3 = 1 + 2
[1,4,1,1] => [1,3,4,2] => [1,4,2,3] => 3 = 1 + 2
[4,1,1,1] => [2,3,4,1] => [4,1,2,3] => 3 = 1 + 2
[1,1,2,2] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[1,2,1,2] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[1,2,2,1] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[2,1,1,2] => [2,1,3,4] => [2,1,3,4] => 3 = 1 + 2
[2,1,2,1] => [2,1,3,4] => [2,1,3,4] => 3 = 1 + 2
[2,2,1,1] => [3,1,2,4] => [2,3,1,4] => 2 = 0 + 2
[1,1,2,3] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[1,1,3,2] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[1,2,1,3] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[1,2,3,1] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[1,3,1,2] => [1,3,2,4] => [1,3,2,4] => 3 = 1 + 2
[1,3,2,1] => [1,3,2,4] => [1,3,2,4] => 3 = 1 + 2
[2,1,1,3] => [2,1,3,4] => [2,1,3,4] => 3 = 1 + 2
[2,1,3,1] => [2,1,3,4] => [2,1,3,4] => 3 = 1 + 2
[2,3,1,1] => [3,1,2,4] => [2,3,1,4] => 2 = 0 + 2
[3,1,1,2] => [2,3,1,4] => [3,1,2,4] => 3 = 1 + 2
[3,1,2,1] => [2,3,1,4] => [3,1,2,4] => 3 = 1 + 2
[3,2,1,1] => [3,2,1,4] => [3,2,1,4] => 2 = 0 + 2
[1,1,2,4] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[1,1,4,2] => [1,2,4,3] => [1,2,4,3] => 3 = 1 + 2
[1,2,1,4] => [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[1,2,4,1] => [1,2,4,3] => [1,2,4,3] => 3 = 1 + 2
[1,4,1,2] => [1,3,4,2] => [1,4,2,3] => 3 = 1 + 2
[1,4,2,1] => [1,3,4,2] => [1,4,2,3] => 3 = 1 + 2
[2,1,1,4] => [2,1,3,4] => [2,1,3,4] => 3 = 1 + 2
[2,1,4,1] => [2,1,4,3] => [2,1,4,3] => 2 = 0 + 2
Description
The number of right-to-left minima of a permutation.
For the number of left-to-right maxima, see [[St000314]].
Matching statistic: St001068
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1] => [1,2,3] => [1,0,1,0,1,0]
=> 3 = 1 + 2
[1,1,2] => [1,2,3] => [1,0,1,0,1,0]
=> 3 = 1 + 2
[1,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 3 = 1 + 2
[2,1,1] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 0 + 2
[1,1,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3 = 1 + 2
[1,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
[1,2,2] => [1,2,3] => [1,0,1,0,1,0]
=> 3 = 1 + 2
[2,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 0 + 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3 = 1 + 2
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,2,1,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[2,1,1,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[1,1,1,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,3,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,3,1,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[3,1,1,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[1,1,1,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[1,4,1,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
[4,1,1,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,2,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,2,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,2,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[2,1,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[2,1,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[2,2,1,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,2,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,3,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,2,1,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,2,3,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,3,1,2] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[1,3,2,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[2,1,1,3] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[2,1,3,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[2,3,1,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[3,1,1,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[3,1,2,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[3,2,1,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,2,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,4,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[1,2,1,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,2,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[1,4,1,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
[1,4,2,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
[2,1,1,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[2,1,4,1] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St001124
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1] => [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,1,2] => [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 1
[2,1,1] => [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 0
[1,1,3] => [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 0
[1,2,2] => [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 1
[2,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 0
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 0
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 0
[1,1,1,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,1,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,1,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,2,1,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[2,1,1,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[1,1,1,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,1,3,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,3,1,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[3,1,1,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[1,1,1,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,1,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[1,4,1,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[4,1,1,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,1,2,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,2,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,2,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[2,1,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[2,1,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[2,2,1,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[1,1,2,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,1,3,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,2,1,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,2,3,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,3,1,2] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[1,3,2,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[2,1,1,3] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[2,1,3,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[2,3,1,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[3,1,1,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[3,1,2,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[3,2,1,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[1,1,2,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,1,4,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[1,2,1,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,2,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[1,4,1,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[1,4,2,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[2,1,1,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[2,1,4,1] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
The following 51 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000021The number of descents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000245The number of ascents of a permutation. St000292The number of ascents of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000340The number of non-final maximal constant sub-paths of length greater than one. St000354The number of recoils of a permutation. St000546The number of global descents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000061The number of nodes on the left branch of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000069The number of maximal elements of a poset. St000084The number of subtrees. St000105The number of blocks in the set partition. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000239The number of small weak excedances. St000291The number of descents of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000325The width of the tree associated to a permutation. St000390The number of runs of ones in a binary word. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000925The number of topologically connected components of a set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St000031The number of cycles in the cycle decomposition of a permutation. St000068The number of minimal elements in a poset. St000159The number of distinct parts of the integer partition. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000702The number of weak deficiencies of a permutation. St000942The number of critical left to right maxima of the parking functions. St001712The number of natural descents of a standard Young tableau. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
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