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Your data matches 16 different statistics following compositions of up to 3 maps.
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Matching statistic: St000007
(load all 34 compositions to match this statistic)
(load all 34 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,1] => [1,2] => 1
[1,2] => [1,2] => 1
[2,1] => [2,1] => 2
[1,1,1] => [1,2,3] => 1
[1,1,2] => [1,2,3] => 1
[1,2,1] => [1,2,3] => 1
[2,1,1] => [2,1,3] => 1
[1,1,3] => [1,2,3] => 1
[1,3,1] => [1,3,2] => 2
[3,1,1] => [2,3,1] => 2
[1,2,2] => [1,2,3] => 1
[2,1,2] => [2,1,3] => 1
[2,2,1] => [3,1,2] => 2
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => 2
[3,1,2] => [2,3,1] => 2
[3,2,1] => [3,2,1] => 3
[1,1,1,1] => [1,2,3,4] => 1
[1,1,1,2] => [1,2,3,4] => 1
[1,1,2,1] => [1,2,3,4] => 1
[1,2,1,1] => [1,2,3,4] => 1
[2,1,1,1] => [2,1,3,4] => 1
[1,1,1,3] => [1,2,3,4] => 1
[1,1,3,1] => [1,2,3,4] => 1
[1,3,1,1] => [1,3,2,4] => 1
[3,1,1,1] => [2,3,1,4] => 1
[1,1,1,4] => [1,2,3,4] => 1
[1,1,4,1] => [1,2,4,3] => 2
[1,4,1,1] => [1,3,4,2] => 2
[4,1,1,1] => [2,3,4,1] => 2
[1,1,2,2] => [1,2,3,4] => 1
[1,2,1,2] => [1,2,3,4] => 1
[1,2,2,1] => [1,2,3,4] => 1
[2,1,1,2] => [2,1,3,4] => 1
[2,1,2,1] => [2,1,3,4] => 1
[2,2,1,1] => [3,1,2,4] => 1
[1,1,2,3] => [1,2,3,4] => 1
[1,1,3,2] => [1,2,3,4] => 1
[1,2,1,3] => [1,2,3,4] => 1
[1,2,3,1] => [1,2,3,4] => 1
[1,3,1,2] => [1,3,2,4] => 1
[1,3,2,1] => [1,3,2,4] => 1
[2,1,1,3] => [2,1,3,4] => 1
[2,1,3,1] => [2,1,3,4] => 1
[2,3,1,1] => [3,1,2,4] => 1
[3,1,1,2] => [2,3,1,4] => 1
[3,1,2,1] => [2,3,1,4] => 1
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: St000069
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 1
[1,1] => [1,2] => ([(0,1)],2)
=> 1
[1,2] => [1,2] => ([(0,1)],2)
=> 1
[2,1] => [2,1] => ([],2)
=> 2
[1,1,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[2,1,1] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[1,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[3,1,1] => [2,3,1] => ([(1,2)],3)
=> 2
[1,2,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[2,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[2,2,1] => [3,1,2] => ([(1,2)],3)
=> 2
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[2,3,1] => [3,1,2] => ([(1,2)],3)
=> 2
[3,1,2] => [2,3,1] => ([(1,2)],3)
=> 2
[3,2,1] => [3,2,1] => ([],3)
=> 3
[1,1,1,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,1,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,1,1] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[1,1,1,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,1,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,1,1,1] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[1,1,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,4,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[1,4,1,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[4,1,1,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[1,1,2,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,1,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[2,1,2,1] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[2,2,1,1] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[1,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,3,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,1,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,1,2] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,1,3] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[2,1,3,1] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[2,3,1,1] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[3,1,1,2] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[3,1,2,1] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
Description
The number of maximal elements of a poset.
Matching statistic: St000314
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,1] => [1,2] => [2,1] => 1
[1,2] => [1,2] => [2,1] => 1
[2,1] => [2,1] => [1,2] => 2
[1,1,1] => [1,2,3] => [3,2,1] => 1
[1,1,2] => [1,2,3] => [3,2,1] => 1
[1,2,1] => [1,2,3] => [3,2,1] => 1
[2,1,1] => [2,1,3] => [3,1,2] => 1
[1,1,3] => [1,2,3] => [3,2,1] => 1
[1,3,1] => [1,3,2] => [2,3,1] => 2
[3,1,1] => [2,3,1] => [1,3,2] => 2
[1,2,2] => [1,2,3] => [3,2,1] => 1
[2,1,2] => [2,1,3] => [3,1,2] => 1
[2,2,1] => [3,1,2] => [2,1,3] => 2
[1,2,3] => [1,2,3] => [3,2,1] => 1
[1,3,2] => [1,3,2] => [2,3,1] => 2
[2,1,3] => [2,1,3] => [3,1,2] => 1
[2,3,1] => [3,1,2] => [2,1,3] => 2
[3,1,2] => [2,3,1] => [1,3,2] => 2
[3,2,1] => [3,2,1] => [1,2,3] => 3
[1,1,1,1] => [1,2,3,4] => [4,3,2,1] => 1
[1,1,1,2] => [1,2,3,4] => [4,3,2,1] => 1
[1,1,2,1] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,1,1] => [1,2,3,4] => [4,3,2,1] => 1
[2,1,1,1] => [2,1,3,4] => [4,3,1,2] => 1
[1,1,1,3] => [1,2,3,4] => [4,3,2,1] => 1
[1,1,3,1] => [1,2,3,4] => [4,3,2,1] => 1
[1,3,1,1] => [1,3,2,4] => [4,2,3,1] => 1
[3,1,1,1] => [2,3,1,4] => [4,1,3,2] => 1
[1,1,1,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,1,4,1] => [1,2,4,3] => [3,4,2,1] => 2
[1,4,1,1] => [1,3,4,2] => [2,4,3,1] => 2
[4,1,1,1] => [2,3,4,1] => [1,4,3,2] => 2
[1,1,2,2] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,1,2] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,2,1] => [1,2,3,4] => [4,3,2,1] => 1
[2,1,1,2] => [2,1,3,4] => [4,3,1,2] => 1
[2,1,2,1] => [2,1,3,4] => [4,3,1,2] => 1
[2,2,1,1] => [3,1,2,4] => [4,2,1,3] => 1
[1,1,2,3] => [1,2,3,4] => [4,3,2,1] => 1
[1,1,3,2] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,1,3] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,3,1] => [1,2,3,4] => [4,3,2,1] => 1
[1,3,1,2] => [1,3,2,4] => [4,2,3,1] => 1
[1,3,2,1] => [1,3,2,4] => [4,2,3,1] => 1
[2,1,1,3] => [2,1,3,4] => [4,3,1,2] => 1
[2,1,3,1] => [2,1,3,4] => [4,3,1,2] => 1
[2,3,1,1] => [3,1,2,4] => [4,2,1,3] => 1
[3,1,1,2] => [2,3,1,4] => [4,1,3,2] => 1
[3,1,2,1] => [2,3,1,4] => [4,1,3,2] => 1
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: St000991
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,1] => [1,2] => [2,1] => 1
[1,2] => [1,2] => [2,1] => 1
[2,1] => [2,1] => [1,2] => 2
[1,1,1] => [1,2,3] => [3,2,1] => 1
[1,1,2] => [1,2,3] => [3,2,1] => 1
[1,2,1] => [1,2,3] => [3,2,1] => 1
[2,1,1] => [2,1,3] => [2,3,1] => 1
[1,1,3] => [1,2,3] => [3,2,1] => 1
[1,3,1] => [1,3,2] => [3,1,2] => 2
[3,1,1] => [2,3,1] => [2,1,3] => 2
[1,2,2] => [1,2,3] => [3,2,1] => 1
[2,1,2] => [2,1,3] => [2,3,1] => 1
[2,2,1] => [3,1,2] => [1,3,2] => 2
[1,2,3] => [1,2,3] => [3,2,1] => 1
[1,3,2] => [1,3,2] => [3,1,2] => 2
[2,1,3] => [2,1,3] => [2,3,1] => 1
[2,3,1] => [3,1,2] => [1,3,2] => 2
[3,1,2] => [2,3,1] => [2,1,3] => 2
[3,2,1] => [3,2,1] => [1,2,3] => 3
[1,1,1,1] => [1,2,3,4] => [4,3,2,1] => 1
[1,1,1,2] => [1,2,3,4] => [4,3,2,1] => 1
[1,1,2,1] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,1,1] => [1,2,3,4] => [4,3,2,1] => 1
[2,1,1,1] => [2,1,3,4] => [3,4,2,1] => 1
[1,1,1,3] => [1,2,3,4] => [4,3,2,1] => 1
[1,1,3,1] => [1,2,3,4] => [4,3,2,1] => 1
[1,3,1,1] => [1,3,2,4] => [4,2,3,1] => 1
[3,1,1,1] => [2,3,1,4] => [3,2,4,1] => 1
[1,1,1,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,1,4,1] => [1,2,4,3] => [4,3,1,2] => 2
[1,4,1,1] => [1,3,4,2] => [4,2,1,3] => 2
[4,1,1,1] => [2,3,4,1] => [3,2,1,4] => 2
[1,1,2,2] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,1,2] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,2,1] => [1,2,3,4] => [4,3,2,1] => 1
[2,1,1,2] => [2,1,3,4] => [3,4,2,1] => 1
[2,1,2,1] => [2,1,3,4] => [3,4,2,1] => 1
[2,2,1,1] => [3,1,2,4] => [2,4,3,1] => 1
[1,1,2,3] => [1,2,3,4] => [4,3,2,1] => 1
[1,1,3,2] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,1,3] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,3,1] => [1,2,3,4] => [4,3,2,1] => 1
[1,3,1,2] => [1,3,2,4] => [4,2,3,1] => 1
[1,3,2,1] => [1,3,2,4] => [4,2,3,1] => 1
[2,1,1,3] => [2,1,3,4] => [3,4,2,1] => 1
[2,1,3,1] => [2,1,3,4] => [3,4,2,1] => 1
[2,3,1,1] => [3,1,2,4] => [2,4,3,1] => 1
[3,1,1,2] => [2,3,1,4] => [3,2,4,1] => 1
[3,1,2,1] => [2,3,1,4] => [3,2,4,1] => 1
Description
The number of right-to-left minima of a permutation.
For the number of left-to-right maxima, see [[St000314]].
Matching statistic: St000015
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ 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%
Mp00064: Permutations —reverse⟶ 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%
Values
[1] => [1] => [1] => [1,0]
=> 1
[1,1] => [1,2] => [2,1] => [1,1,0,0]
=> 1
[1,2] => [1,2] => [2,1] => [1,1,0,0]
=> 1
[2,1] => [2,1] => [1,2] => [1,0,1,0]
=> 2
[1,1,1] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[1,1,2] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[1,2,1] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[2,1,1] => [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[1,1,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[1,3,1] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[3,1,1] => [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[1,2,2] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[2,1,2] => [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[2,2,1] => [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[1,2,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[1,3,2] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[2,1,3] => [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[2,3,1] => [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[3,1,2] => [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[3,2,1] => [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[1,1,1,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,2,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,1,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,1,1] => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,3,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,1,1] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[3,1,1,1] => [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,4,1] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[1,4,1,1] => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[4,1,1,1] => [2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[1,1,2,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,1,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,2,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,1,2] => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,2,1] => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[2,2,1,1] => [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,2,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,3,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,1,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,1,2] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,2,1] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,1,3] => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,3,1] => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[2,3,1,1] => [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1
[3,1,1,2] => [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1
[3,1,2,1] => [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1
Description
The number of peaks of a Dyck path.
Matching statistic: St000542
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
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] => 1
[1,1] => [1,2] => [2,1] => [1,2] => 1
[1,2] => [1,2] => [2,1] => [1,2] => 1
[2,1] => [2,1] => [1,2] => [2,1] => 2
[1,1,1] => [1,2,3] => [3,2,1] => [1,2,3] => 1
[1,1,2] => [1,2,3] => [3,2,1] => [1,2,3] => 1
[1,2,1] => [1,2,3] => [3,2,1] => [1,2,3] => 1
[2,1,1] => [2,1,3] => [3,1,2] => [1,3,2] => 1
[1,1,3] => [1,2,3] => [3,2,1] => [1,2,3] => 1
[1,3,1] => [1,3,2] => [2,3,1] => [2,1,3] => 2
[3,1,1] => [2,3,1] => [1,3,2] => [3,1,2] => 2
[1,2,2] => [1,2,3] => [3,2,1] => [1,2,3] => 1
[2,1,2] => [2,1,3] => [3,1,2] => [1,3,2] => 1
[2,2,1] => [3,1,2] => [2,1,3] => [2,3,1] => 2
[1,2,3] => [1,2,3] => [3,2,1] => [1,2,3] => 1
[1,3,2] => [1,3,2] => [2,3,1] => [2,1,3] => 2
[2,1,3] => [2,1,3] => [3,1,2] => [1,3,2] => 1
[2,3,1] => [3,1,2] => [2,1,3] => [2,3,1] => 2
[3,1,2] => [2,3,1] => [1,3,2] => [3,1,2] => 2
[3,2,1] => [3,2,1] => [1,2,3] => [3,2,1] => 3
[1,1,1,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1
[1,1,1,2] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1
[1,1,2,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1
[1,2,1,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1
[2,1,1,1] => [2,1,3,4] => [4,3,1,2] => [1,2,4,3] => 1
[1,1,1,3] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1
[1,1,3,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1
[1,3,1,1] => [1,3,2,4] => [4,2,3,1] => [1,3,2,4] => 1
[3,1,1,1] => [2,3,1,4] => [4,1,3,2] => [1,4,2,3] => 1
[1,1,1,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1
[1,1,4,1] => [1,2,4,3] => [3,4,2,1] => [2,1,3,4] => 2
[1,4,1,1] => [1,3,4,2] => [2,4,3,1] => [3,1,2,4] => 2
[4,1,1,1] => [2,3,4,1] => [1,4,3,2] => [4,1,2,3] => 2
[1,1,2,2] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1
[1,2,1,2] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1
[1,2,2,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1
[2,1,1,2] => [2,1,3,4] => [4,3,1,2] => [1,2,4,3] => 1
[2,1,2,1] => [2,1,3,4] => [4,3,1,2] => [1,2,4,3] => 1
[2,2,1,1] => [3,1,2,4] => [4,2,1,3] => [1,3,4,2] => 1
[1,1,2,3] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1
[1,1,3,2] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1
[1,2,1,3] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1
[1,2,3,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1
[1,3,1,2] => [1,3,2,4] => [4,2,3,1] => [1,3,2,4] => 1
[1,3,2,1] => [1,3,2,4] => [4,2,3,1] => [1,3,2,4] => 1
[2,1,1,3] => [2,1,3,4] => [4,3,1,2] => [1,2,4,3] => 1
[2,1,3,1] => [2,1,3,4] => [4,3,1,2] => [1,2,4,3] => 1
[2,3,1,1] => [3,1,2,4] => [4,2,1,3] => [1,3,4,2] => 1
[3,1,1,2] => [2,3,1,4] => [4,1,3,2] => [1,4,2,3] => 1
[3,1,2,1] => [2,3,1,4] => [4,1,3,2] => [1,4,2,3] => 1
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: St000740
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1
[1,1] => [1,2] => [1,2] => [2,1] => 1
[1,2] => [1,2] => [1,2] => [2,1] => 1
[2,1] => [2,1] => [2,1] => [1,2] => 2
[1,1,1] => [1,2,3] => [1,2,3] => [3,2,1] => 1
[1,1,2] => [1,2,3] => [1,2,3] => [3,2,1] => 1
[1,2,1] => [1,2,3] => [1,2,3] => [3,2,1] => 1
[2,1,1] => [2,1,3] => [2,1,3] => [2,3,1] => 1
[1,1,3] => [1,2,3] => [1,2,3] => [3,2,1] => 1
[1,3,1] => [1,3,2] => [1,3,2] => [3,1,2] => 2
[3,1,1] => [2,3,1] => [1,3,2] => [3,1,2] => 2
[1,2,2] => [1,2,3] => [1,2,3] => [3,2,1] => 1
[2,1,2] => [2,1,3] => [2,1,3] => [2,3,1] => 1
[2,2,1] => [3,1,2] => [3,1,2] => [1,3,2] => 2
[1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 1
[1,3,2] => [1,3,2] => [1,3,2] => [3,1,2] => 2
[2,1,3] => [2,1,3] => [2,1,3] => [2,3,1] => 1
[2,3,1] => [3,1,2] => [3,1,2] => [1,3,2] => 2
[3,1,2] => [2,3,1] => [1,3,2] => [3,1,2] => 2
[3,2,1] => [3,2,1] => [3,2,1] => [1,2,3] => 3
[1,1,1,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,1,1,2] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,1,2,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,1,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[2,1,1,1] => [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 1
[1,1,1,3] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,1,3,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,3,1,1] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[3,1,1,1] => [2,3,1,4] => [1,3,2,4] => [4,2,3,1] => 1
[1,1,1,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,1,4,1] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 2
[1,4,1,1] => [1,3,4,2] => [1,2,4,3] => [4,3,1,2] => 2
[4,1,1,1] => [2,3,4,1] => [1,2,4,3] => [4,3,1,2] => 2
[1,1,2,2] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,1,2] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,2,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[2,1,1,2] => [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 1
[2,1,2,1] => [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 1
[2,2,1,1] => [3,1,2,4] => [3,1,2,4] => [2,4,3,1] => 1
[1,1,2,3] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,1,3,2] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,1,3] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,3,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,3,1,2] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[1,3,2,1] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[2,1,1,3] => [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 1
[2,1,3,1] => [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 1
[2,3,1,1] => [3,1,2,4] => [3,1,2,4] => [2,4,3,1] => 1
[3,1,1,2] => [2,3,1,4] => [1,3,2,4] => [4,2,3,1] => 1
[3,1,2,1] => [2,3,1,4] => [1,3,2,4] => [4,2,3,1] => 1
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St001068
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ 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%
Mp00064: Permutations —reverse⟶ 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%
Values
[1] => [1] => [1] => [1,0]
=> 1
[1,1] => [1,2] => [2,1] => [1,1,0,0]
=> 1
[1,2] => [1,2] => [2,1] => [1,1,0,0]
=> 1
[2,1] => [2,1] => [1,2] => [1,0,1,0]
=> 2
[1,1,1] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[1,1,2] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[1,2,1] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[2,1,1] => [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[1,1,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[1,3,1] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[3,1,1] => [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[1,2,2] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[2,1,2] => [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[2,2,1] => [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[1,2,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[1,3,2] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[2,1,3] => [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[2,3,1] => [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[3,1,2] => [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[3,2,1] => [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[1,1,1,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,2,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,1,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,1,1] => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,3,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,1,1] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[3,1,1,1] => [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,4,1] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[1,4,1,1] => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[4,1,1,1] => [2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[1,1,2,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,1,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,2,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,1,2] => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,2,1] => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[2,2,1,1] => [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,2,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,3,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,1,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,1,2] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,2,1] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,1,3] => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,3,1] => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[2,3,1,1] => [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1
[3,1,1,2] => [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1
[3,1,2,1] => [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St000053
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ 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%
Mp00064: Permutations —reverse⟶ 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%
Values
[1] => [1] => [1] => [1,0]
=> 0 = 1 - 1
[1,1] => [1,2] => [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[1,2] => [1,2] => [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[2,1] => [2,1] => [1,2] => [1,0,1,0]
=> 1 = 2 - 1
[1,1,1] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,2] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,2,1] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[2,1,1] => [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,3,1] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 1 = 2 - 1
[3,1,1] => [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,2,2] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[2,1,2] => [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[2,2,1] => [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,3,2] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 1 = 2 - 1
[2,1,3] => [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[2,3,1] => [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[3,1,2] => [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,1] => [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,1,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,2,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,1,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[2,1,1,1] => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,3,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,3,1,1] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[3,1,1,1] => [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,4,1] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,4,1,1] => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[4,1,1,1] => [2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,2,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,1,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,2,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[2,1,1,2] => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[2,1,2,1] => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[2,2,1,1] => [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,2,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,3,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,1,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,3,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,3,1,2] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,3,2,1] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[2,1,1,3] => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[2,1,3,1] => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[2,3,1,1] => [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[3,1,1,2] => [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[3,1,2,1] => [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
Description
The number of valleys of the Dyck path.
Matching statistic: St000133
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000133: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000133: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0 = 1 - 1
[1,1] => [1,2] => [1,2] => [2,1] => 0 = 1 - 1
[1,2] => [1,2] => [1,2] => [2,1] => 0 = 1 - 1
[2,1] => [2,1] => [2,1] => [1,2] => 1 = 2 - 1
[1,1,1] => [1,2,3] => [1,2,3] => [2,3,1] => 0 = 1 - 1
[1,1,2] => [1,2,3] => [1,2,3] => [2,3,1] => 0 = 1 - 1
[1,2,1] => [1,2,3] => [1,2,3] => [2,3,1] => 0 = 1 - 1
[2,1,1] => [2,1,3] => [2,1,3] => [3,2,1] => 0 = 1 - 1
[1,1,3] => [1,2,3] => [1,2,3] => [2,3,1] => 0 = 1 - 1
[1,3,1] => [1,3,2] => [1,3,2] => [2,1,3] => 1 = 2 - 1
[3,1,1] => [2,3,1] => [1,3,2] => [2,1,3] => 1 = 2 - 1
[1,2,2] => [1,2,3] => [1,2,3] => [2,3,1] => 0 = 1 - 1
[2,1,2] => [2,1,3] => [2,1,3] => [3,2,1] => 0 = 1 - 1
[2,2,1] => [3,1,2] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[1,2,3] => [1,2,3] => [1,2,3] => [2,3,1] => 0 = 1 - 1
[1,3,2] => [1,3,2] => [1,3,2] => [2,1,3] => 1 = 2 - 1
[2,1,3] => [2,1,3] => [2,1,3] => [3,2,1] => 0 = 1 - 1
[2,3,1] => [3,1,2] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[3,1,2] => [2,3,1] => [1,3,2] => [2,1,3] => 1 = 2 - 1
[3,2,1] => [3,2,1] => [3,2,1] => [1,3,2] => 2 = 3 - 1
[1,1,1,1] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,1,1,2] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,1,2,1] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,2,1,1] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[2,1,1,1] => [2,1,3,4] => [2,1,3,4] => [3,2,4,1] => 0 = 1 - 1
[1,1,1,3] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,1,3,1] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,3,1,1] => [1,3,2,4] => [1,3,2,4] => [2,4,3,1] => 0 = 1 - 1
[3,1,1,1] => [2,3,1,4] => [1,3,2,4] => [2,4,3,1] => 0 = 1 - 1
[1,1,1,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,1,4,1] => [1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 1 = 2 - 1
[1,4,1,1] => [1,3,4,2] => [1,2,4,3] => [2,3,1,4] => 1 = 2 - 1
[4,1,1,1] => [2,3,4,1] => [1,2,4,3] => [2,3,1,4] => 1 = 2 - 1
[1,1,2,2] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,2,1,2] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,2,2,1] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[2,1,1,2] => [2,1,3,4] => [2,1,3,4] => [3,2,4,1] => 0 = 1 - 1
[2,1,2,1] => [2,1,3,4] => [2,1,3,4] => [3,2,4,1] => 0 = 1 - 1
[2,2,1,1] => [3,1,2,4] => [3,1,2,4] => [3,4,2,1] => 0 = 1 - 1
[1,1,2,3] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,1,3,2] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,2,1,3] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,2,3,1] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,3,1,2] => [1,3,2,4] => [1,3,2,4] => [2,4,3,1] => 0 = 1 - 1
[1,3,2,1] => [1,3,2,4] => [1,3,2,4] => [2,4,3,1] => 0 = 1 - 1
[2,1,1,3] => [2,1,3,4] => [2,1,3,4] => [3,2,4,1] => 0 = 1 - 1
[2,1,3,1] => [2,1,3,4] => [2,1,3,4] => [3,2,4,1] => 0 = 1 - 1
[2,3,1,1] => [3,1,2,4] => [3,1,2,4] => [3,4,2,1] => 0 = 1 - 1
[3,1,1,2] => [2,3,1,4] => [1,3,2,4] => [2,4,3,1] => 0 = 1 - 1
[3,1,2,1] => [2,3,1,4] => [1,3,2,4] => [2,4,3,1] => 0 = 1 - 1
Description
The "bounce" of a permutation.
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000031The number of cycles in the cycle decomposition of a permutation. St000068The number of minimal elements in a poset. St001330The hat guessing number of a graph. St000942The number of critical left to right maxima of the parking functions.
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