searching the database
Your data matches 140 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000007
(load all 54 compositions to match this statistic)
(load all 54 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ 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
[1,2] => [1,2] => 1
[2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,2,3] => 1
[2,1,3] => [1,2,3] => 1
[2,3,1] => [1,2,3] => 1
[3,1,2] => [1,3,2] => 2
[3,2,1] => [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,3,4] => 1
[1,3,2,4] => [1,2,3,4] => 1
[1,3,4,2] => [1,2,3,4] => 1
[1,4,2,3] => [1,2,4,3] => 2
[1,4,3,2] => [1,2,4,3] => 2
[2,1,3,4] => [1,2,3,4] => 1
[2,1,4,3] => [1,2,3,4] => 1
[2,3,1,4] => [1,2,3,4] => 1
[2,3,4,1] => [1,2,3,4] => 1
[2,4,1,3] => [1,2,4,3] => 2
[2,4,3,1] => [1,2,4,3] => 2
[3,1,2,4] => [1,3,2,4] => 1
[3,1,4,2] => [1,3,4,2] => 2
[3,2,1,4] => [1,3,2,4] => 1
[3,2,4,1] => [1,3,4,2] => 2
[3,4,1,2] => [1,3,2,4] => 1
[3,4,2,1] => [1,3,2,4] => 1
[4,1,2,3] => [1,4,3,2] => 3
[4,1,3,2] => [1,4,2,3] => 2
[4,2,1,3] => [1,4,3,2] => 3
[4,2,3,1] => [1,4,2,3] => 2
[4,3,1,2] => [1,4,2,3] => 2
[4,3,2,1] => [1,4,2,3] => 2
[1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,4,5] => 1
[1,2,4,3,5] => [1,2,3,4,5] => 1
[1,2,4,5,3] => [1,2,3,4,5] => 1
[1,2,5,3,4] => [1,2,3,5,4] => 2
[1,2,5,4,3] => [1,2,3,5,4] => 2
[1,3,2,4,5] => [1,2,3,4,5] => 1
[1,3,2,5,4] => [1,2,3,4,5] => 1
[1,3,4,2,5] => [1,2,3,4,5] => 1
[1,3,4,5,2] => [1,2,3,4,5] => 1
[1,3,5,2,4] => [1,2,3,5,4] => 2
[1,3,5,4,2] => [1,2,3,5,4] => 2
[1,4,2,3,5] => [1,2,4,3,5] => 1
[1,4,2,5,3] => [1,2,4,5,3] => 2
[1,4,3,2,5] => [1,2,4,3,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => 2
[1,4,5,2,3] => [1,2,4,3,5] => 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 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ 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
[1,2] => [1,2] => ([(0,1)],2)
=> 1
[2,1] => [1,2] => ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[3,1,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[3,2,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,2,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[1,4,3,2] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,4,1,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[2,4,3,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,1,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[3,2,1,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,2,4,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[3,4,1,2] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,4,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 3
[4,1,3,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[4,2,1,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 3
[4,2,3,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[4,3,1,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[4,3,2,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,5,3,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,2,5,4,3] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,5,2,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,3,5,4,2] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,4,2,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,4,3,2,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,4,5,2,3] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
Description
The number of maximal elements of a poset.
Matching statistic: St000314
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ 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
[1,2] => [1,2] => [2,1] => 1
[2,1] => [1,2] => [2,1] => 1
[1,2,3] => [1,2,3] => [3,2,1] => 1
[1,3,2] => [1,2,3] => [3,2,1] => 1
[2,1,3] => [1,2,3] => [3,2,1] => 1
[2,3,1] => [1,2,3] => [3,2,1] => 1
[3,1,2] => [1,3,2] => [2,3,1] => 2
[3,2,1] => [1,3,2] => [2,3,1] => 2
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 1
[1,4,2,3] => [1,2,4,3] => [3,4,2,1] => 2
[1,4,3,2] => [1,2,4,3] => [3,4,2,1] => 2
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 1
[2,4,1,3] => [1,2,4,3] => [3,4,2,1] => 2
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => 2
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[3,1,4,2] => [1,3,4,2] => [2,4,3,1] => 2
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 1
[3,2,4,1] => [1,3,4,2] => [2,4,3,1] => 2
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => 1
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => 1
[4,1,2,3] => [1,4,3,2] => [2,3,4,1] => 3
[4,1,3,2] => [1,4,2,3] => [3,2,4,1] => 2
[4,2,1,3] => [1,4,3,2] => [2,3,4,1] => 3
[4,2,3,1] => [1,4,2,3] => [3,2,4,1] => 2
[4,3,1,2] => [1,4,2,3] => [3,2,4,1] => 2
[4,3,2,1] => [1,4,2,3] => [3,2,4,1] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,5,3,4] => [1,2,3,5,4] => [4,5,3,2,1] => 2
[1,2,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => 2
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,3,5,2,4] => [1,2,3,5,4] => [4,5,3,2,1] => 2
[1,3,5,4,2] => [1,2,3,5,4] => [4,5,3,2,1] => 2
[1,4,2,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [3,5,4,2,1] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [5,3,4,2,1] => 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: St000542
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00325: Permutations —ones to leading⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => 1
[2,1] => [1,2] => [1,2] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 1
[1,3,2] => [1,2,3] => [1,2,3] => 1
[2,1,3] => [1,2,3] => [1,2,3] => 1
[2,3,1] => [1,2,3] => [1,2,3] => 1
[3,1,2] => [1,3,2] => [2,3,1] => 2
[3,2,1] => [1,3,2] => [2,3,1] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 1
[1,4,2,3] => [1,2,4,3] => [2,3,4,1] => 2
[1,4,3,2] => [1,2,4,3] => [2,3,4,1] => 2
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 1
[2,4,1,3] => [1,2,4,3] => [2,3,4,1] => 2
[2,4,3,1] => [1,2,4,3] => [2,3,4,1] => 2
[3,1,2,4] => [1,3,2,4] => [1,2,4,3] => 1
[3,1,4,2] => [1,3,4,2] => [2,3,1,4] => 2
[3,2,1,4] => [1,3,2,4] => [1,2,4,3] => 1
[3,2,4,1] => [1,3,4,2] => [2,3,1,4] => 2
[3,4,1,2] => [1,3,2,4] => [1,2,4,3] => 1
[3,4,2,1] => [1,3,2,4] => [1,2,4,3] => 1
[4,1,2,3] => [1,4,3,2] => [3,4,2,1] => 3
[4,1,3,2] => [1,4,2,3] => [3,4,1,2] => 2
[4,2,1,3] => [1,4,3,2] => [3,4,2,1] => 3
[4,2,3,1] => [1,4,2,3] => [3,4,1,2] => 2
[4,3,1,2] => [1,4,2,3] => [3,4,1,2] => 2
[4,3,2,1] => [1,4,2,3] => [3,4,1,2] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,5,1] => 2
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,5,1] => 2
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,5,1] => 2
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,5,1] => 2
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,3,5,4] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,4,1,5] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,3,5,4] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,4,1,5] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,3,5,4] => 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: St000991
(load all 34 compositions to match this statistic)
(load all 34 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ 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
[1,2] => [1,2] => [2,1] => 1
[2,1] => [1,2] => [2,1] => 1
[1,2,3] => [1,2,3] => [3,2,1] => 1
[1,3,2] => [1,2,3] => [3,2,1] => 1
[2,1,3] => [1,2,3] => [3,2,1] => 1
[2,3,1] => [1,2,3] => [3,2,1] => 1
[3,1,2] => [1,3,2] => [3,1,2] => 2
[3,2,1] => [1,3,2] => [3,1,2] => 2
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 1
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => 2
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => 2
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 1
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => 2
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => 2
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => 2
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 1
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => 2
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => 1
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => 1
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => 3
[4,1,3,2] => [1,4,2,3] => [4,1,3,2] => 2
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => 3
[4,2,3,1] => [1,4,2,3] => [4,1,3,2] => 2
[4,3,1,2] => [1,4,2,3] => [4,1,3,2] => 2
[4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,4,3,1,2] => 2
[1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => 2
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,4,3,1,2] => 2
[1,3,5,4,2] => [1,2,3,5,4] => [5,4,3,1,2] => 2
[1,4,2,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => 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
Mp00090: Permutations —cycle-as-one-line notation⟶ 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
[1,2] => [1,2] => [2,1] => [1,1,0,0]
=> 1
[2,1] => [1,2] => [2,1] => [1,1,0,0]
=> 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[1,3,2] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[2,1,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[2,3,1] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[3,1,2] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[3,2,1] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,4,2,3] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[1,4,3,2] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[2,4,1,3] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[3,1,4,2] => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[3,2,4,1] => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[4,1,2,3] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[4,1,3,2] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[4,2,1,3] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[4,2,3,1] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[4,3,1,2] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[4,3,2,1] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,5,3,4] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,2,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,5,2,4] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,3,5,4,2] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,4,2,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,4,5,2,3] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
Description
The number of peaks of a Dyck path.
Matching statistic: St000031
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [2,1] => [2,1] => 1
[2,1] => [1,2] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [3,2,1] => [3,1,2] => 1
[1,3,2] => [1,2,3] => [3,2,1] => [3,1,2] => 1
[2,1,3] => [1,2,3] => [3,2,1] => [3,1,2] => 1
[2,3,1] => [1,2,3] => [3,2,1] => [3,1,2] => 1
[3,1,2] => [1,3,2] => [2,3,1] => [3,2,1] => 2
[3,2,1] => [1,3,2] => [2,3,1] => [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [4,1,2,3] => 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [4,1,2,3] => 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [4,1,2,3] => 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [4,1,2,3] => 1
[1,4,2,3] => [1,2,4,3] => [3,4,2,1] => [4,1,3,2] => 2
[1,4,3,2] => [1,2,4,3] => [3,4,2,1] => [4,1,3,2] => 2
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [4,1,2,3] => 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [4,1,2,3] => 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [4,1,2,3] => 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [4,1,2,3] => 1
[2,4,1,3] => [1,2,4,3] => [3,4,2,1] => [4,1,3,2] => 2
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => [4,1,3,2] => 2
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [4,3,1,2] => 1
[3,1,4,2] => [1,3,4,2] => [2,4,3,1] => [4,2,1,3] => 2
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [4,3,1,2] => 1
[3,2,4,1] => [1,3,4,2] => [2,4,3,1] => [4,2,1,3] => 2
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [4,3,1,2] => 1
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [4,3,1,2] => 1
[4,1,2,3] => [1,4,3,2] => [2,3,4,1] => [4,2,3,1] => 3
[4,1,3,2] => [1,4,2,3] => [3,2,4,1] => [4,3,2,1] => 2
[4,2,1,3] => [1,4,3,2] => [2,3,4,1] => [4,2,3,1] => 3
[4,2,3,1] => [1,4,2,3] => [3,2,4,1] => [4,3,2,1] => 2
[4,3,1,2] => [1,4,2,3] => [3,2,4,1] => [4,3,2,1] => 2
[4,3,2,1] => [1,4,2,3] => [3,2,4,1] => [4,3,2,1] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => 1
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => 1
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => 1
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => 1
[1,2,5,3,4] => [1,2,3,5,4] => [4,5,3,2,1] => [5,1,2,4,3] => 2
[1,2,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => [5,1,2,4,3] => 2
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => 1
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => 1
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => 1
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => 1
[1,3,5,2,4] => [1,2,3,5,4] => [4,5,3,2,1] => [5,1,2,4,3] => 2
[1,3,5,4,2] => [1,2,3,5,4] => [4,5,3,2,1] => [5,1,2,4,3] => 2
[1,4,2,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [5,1,4,2,3] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [5,1,3,2,4] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => [5,1,4,2,3] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [3,5,4,2,1] => [5,1,3,2,4] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [5,3,4,2,1] => [5,1,4,2,3] => 1
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St000740
Mp00090: Permutations —cycle-as-one-line notation⟶ 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
[1,2] => [1,2] => [1,2] => [2,1] => 1
[2,1] => [1,2] => [1,2] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 1
[1,3,2] => [1,2,3] => [1,2,3] => [3,2,1] => 1
[2,1,3] => [1,2,3] => [1,2,3] => [3,2,1] => 1
[2,3,1] => [1,2,3] => [1,2,3] => [3,2,1] => 1
[3,1,2] => [1,3,2] => [1,3,2] => [3,1,2] => 2
[3,2,1] => [1,3,2] => [1,3,2] => [3,1,2] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 2
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 2
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 2
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 2
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[3,1,4,2] => [1,3,4,2] => [1,2,4,3] => [4,3,1,2] => 2
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[3,2,4,1] => [1,3,4,2] => [1,2,4,3] => [4,3,1,2] => 2
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 3
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 2
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 3
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 2
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 2
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 2
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 2
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 2
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 2
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,3,5,4] => [5,4,3,1,2] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,3,5,4] => [5,4,3,1,2] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St001068
Mp00090: Permutations —cycle-as-one-line notation⟶ 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
[1,2] => [1,2] => [2,1] => [1,1,0,0]
=> 1
[2,1] => [1,2] => [2,1] => [1,1,0,0]
=> 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[1,3,2] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[2,1,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[2,3,1] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[3,1,2] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[3,2,1] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,4,2,3] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[1,4,3,2] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[2,4,1,3] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[3,1,4,2] => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[3,2,4,1] => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[4,1,2,3] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[4,1,3,2] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[4,2,1,3] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[4,2,3,1] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[4,3,1,2] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[4,3,2,1] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,5,3,4] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,2,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,5,2,4] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,3,5,4,2] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,4,2,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,4,5,2,3] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St001499
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,2] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[2,1] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 4
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 3
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 3
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 3
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 3
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 3
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
Description
The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra.
We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.
The following 130 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000052The number of valleys of a Dyck path not on the x-axis. St000053The number of valleys of the Dyck path. St000133The "bounce" of a permutation. St000204The number of internal nodes of a binary tree. St000245The number of ascents 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. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. 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. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000062The length of the longest increasing subsequence of the permutation. St000619The number of cyclic descents of a permutation. St000061The number of nodes on the left branch of a binary tree. St000155The number of exceedances (also excedences) of a permutation. St000711The number of big exceedences of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000068The number of minimal elements in a poset. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000331The number of upper interactions of a Dyck path. St001432The order dimension of the partition. St000390The number of runs of ones in a binary word. St000783The side length of the largest staircase partition fitting into a partition. St000668The least common multiple of the parts of the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001128The exponens consonantiae of a partition. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000260The radius of a connected graph. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000993The multiplicity of the largest part of an integer partition. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001118The acyclic chromatic index of a graph. St000460The hook length of the last cell along the main diagonal of an integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000770The major index of an integer partition when read from bottom to top. St001060The distinguishing index of a graph. 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 $eA$. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001933The largest multiplicity of a part in an integer partition. St000454The largest eigenvalue of a graph if it is integral. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000456The monochromatic index of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001875The number of simple modules with projective dimension at most 1. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000284The Plancherel distribution on integer partitions. St000681The Grundy value of Chomp on Ferrers diagrams. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001964The interval resolution global dimension of a poset. St000307The number of rowmotion orbits of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000632The jump number of the poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000640The rank of the largest boolean interval in a poset. St001820The size of the image of the pop stack sorting operator. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001960The number of descents of a permutation minus one if its first entry is not one. St000259The diameter of a connected graph. St000264The girth of a graph, which is not a tree. St000741The Colin de Verdière graph invariant. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000934The 2-degree of an integer partition. St001587Half of the largest even part of an integer partition. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001624The breadth of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000451The length of the longest pattern of the form k 1 2. St000534The number of 2-rises of a permutation. St000842The breadth of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number 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. St001330The hat guessing number of a graph. St001870The number of positive entries followed by a negative entry in a signed permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001890The maximum magnitude of the Möbius function of a poset. St000942The number of critical left to right maxima of the parking functions. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001862The number of crossings of a signed permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!