searching the database
Your data matches 30 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: St000221
(load all 1278 compositions to match this statistic)
(load all 1278 compositions to match this statistic)
St000221: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 2
[2,1] => 0
[1,2,3] => 3
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 4
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 2
[2,1,4,3] => 0
[2,3,1,4] => 1
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 0
[3,1,2,4] => 1
[3,1,4,2] => 0
[3,2,1,4] => 1
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 5
[1,2,3,5,4] => 3
[1,2,4,3,5] => 3
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 2
[1,3,2,4,5] => 3
[1,3,2,5,4] => 1
[1,3,4,2,5] => 2
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 2
[1,4,2,5,3] => 1
[1,4,3,2,5] => 2
[1,4,3,5,2] => 1
[1,4,5,2,3] => 1
Description
The number of strong fixed points of a permutation.
$i$ is called a strong fixed point of $\pi$ if
1. $j < i$ implies $\pi_j < \pi_i$, and
2. $j > i$ implies $\pi_j > \pi_i$
This can be described as an occurrence of the mesh pattern ([1], {(0,1),(1,0)}), i.e., the upper left and the lower right quadrants are shaded, see [3].
The generating function for the joint-distribution (RLmin, LRmax, strong fixed points) has a continued fraction expression as given in [4, Lemma 3.2], for LRmax see [[St000314]].
Matching statistic: St000315
(load all 152 compositions to match this statistic)
(load all 152 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
St000315: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000315: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 1
[1,2] => ([],2)
=> 2
[2,1] => ([(0,1)],2)
=> 0
[1,2,3] => ([],3)
=> 3
[1,3,2] => ([(1,2)],3)
=> 1
[2,1,3] => ([(1,2)],3)
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> 0
[3,1,2] => ([(0,2),(1,2)],3)
=> 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
[1,2,3,4] => ([],4)
=> 4
[1,2,4,3] => ([(2,3)],4)
=> 2
[1,3,2,4] => ([(2,3)],4)
=> 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => ([(2,3)],4)
=> 2
[2,1,4,3] => ([(0,3),(1,2)],4)
=> 0
[2,3,1,4] => ([(1,3),(2,3)],4)
=> 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 0
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[3,1,2,4] => ([(1,3),(2,3)],4)
=> 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,2,3,4,5] => ([],5)
=> 5
[1,2,3,5,4] => ([(3,4)],5)
=> 3
[1,2,4,3,5] => ([(3,4)],5)
=> 3
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => ([(3,4)],5)
=> 3
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 1
Description
The number of isolated vertices of a graph.
Matching statistic: St000022
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
St000022: 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] => 2
[2,1] => [2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 3
[1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => [2,3,1] => 0
[3,1,2] => [3,2,1] => [2,3,1] => 0
[3,2,1] => [3,2,1] => [2,3,1] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 1
[1,4,2,3] => [1,4,3,2] => [1,3,4,2] => 1
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 1
[2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 0
[2,4,1,3] => [3,4,1,2] => [4,3,2,1] => 0
[2,4,3,1] => [4,3,2,1] => [3,4,1,2] => 0
[3,1,2,4] => [3,2,1,4] => [2,3,1,4] => 1
[3,1,4,2] => [4,2,3,1] => [2,3,4,1] => 0
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 1
[3,2,4,1] => [4,2,3,1] => [2,3,4,1] => 0
[3,4,1,2] => [4,3,2,1] => [3,4,1,2] => 0
[3,4,2,1] => [4,3,2,1] => [3,4,1,2] => 0
[4,1,2,3] => [4,2,3,1] => [2,3,4,1] => 0
[4,1,3,2] => [4,2,3,1] => [2,3,4,1] => 0
[4,2,1,3] => [4,3,2,1] => [3,4,1,2] => 0
[4,2,3,1] => [4,3,2,1] => [3,4,1,2] => 0
[4,3,1,2] => [4,3,2,1] => [3,4,1,2] => 0
[4,3,2,1] => [4,3,2,1] => [3,4,1,2] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 3
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 3
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 3
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,4,3,2] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,4,5,2,3] => 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,3,4,2,5] => 2
[1,4,2,5,3] => [1,5,3,4,2] => [1,3,4,5,2] => 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 2
[1,4,3,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,4,5,2,3] => 1
Description
The number of fixed points of a permutation.
Matching statistic: St000117
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000117: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000117: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 0
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
Description
The number of centered tunnels of a Dyck path.
A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b==n then the tunnel is called centered.
Matching statistic: St000475
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> 1
[1,2] => ([],2)
=> [1,1]
=> 2
[2,1] => ([(0,1)],2)
=> [2]
=> 0
[1,2,3] => ([],3)
=> [1,1,1]
=> 3
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> 1
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> 0
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 4
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> 2
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> 2
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> 0
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 0
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 0
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> 5
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 3
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 3
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 3
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 1
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000895
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [[1]]
=> 1
[1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> 2
[2,1] => [1,1,0,0]
=> [[0,1],[1,0]]
=> 0
[1,2,3] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3
[1,3,2] => [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 0
[3,1,2] => [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> 0
[3,2,1] => [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> 0
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> 0
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 3
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 3
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
Description
The number of ones on the main diagonal of an alternating sign matrix.
Matching statistic: St001008
(load all 36 compositions to match this statistic)
(load all 36 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001008: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001008: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
Description
Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000164
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
St000164: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
St000164: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [(1,2)]
=> [(1,2)]
=> 1
[1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> [(1,2),(3,4)]
=> 2
[2,1] => [1,1,0,0]
=> [(1,4),(2,3)]
=> [(1,3),(2,4)]
=> 0
[1,2,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [(1,2),(3,4),(5,6)]
=> 3
[1,3,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [(1,2),(3,5),(4,6)]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [(1,3),(2,4),(5,6)]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [(1,3),(2,5),(4,6)]
=> 0
[3,1,2] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [(1,4),(2,5),(3,6)]
=> 0
[3,2,1] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [(1,4),(2,5),(3,6)]
=> 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [(1,2),(3,4),(5,7),(6,8)]
=> 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [(1,2),(3,5),(4,6),(7,8)]
=> 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [(1,2),(3,5),(4,7),(6,8)]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [(1,2),(3,6),(4,7),(5,8)]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [(1,2),(3,6),(4,7),(5,8)]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [(1,3),(2,4),(5,6),(7,8)]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [(1,3),(2,4),(5,7),(6,8)]
=> 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [(1,3),(2,5),(4,6),(7,8)]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [(1,3),(2,5),(4,7),(6,8)]
=> 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [(1,3),(2,6),(4,7),(5,8)]
=> 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [(1,3),(2,6),(4,7),(5,8)]
=> 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [(1,4),(2,5),(3,6),(7,8)]
=> 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [(1,4),(2,5),(3,7),(6,8)]
=> 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [(1,4),(2,5),(3,6),(7,8)]
=> 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [(1,4),(2,5),(3,7),(6,8)]
=> 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [(1,4),(2,6),(3,7),(5,8)]
=> 0
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [(1,4),(2,6),(3,7),(5,8)]
=> 0
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [(1,5),(2,6),(3,7),(4,8)]
=> 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [(1,5),(2,6),(3,7),(4,8)]
=> 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [(1,5),(2,6),(3,7),(4,8)]
=> 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [(1,5),(2,6),(3,7),(4,8)]
=> 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [(1,5),(2,6),(3,7),(4,8)]
=> 0
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [(1,5),(2,6),(3,7),(4,8)]
=> 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [(1,2),(3,4),(5,6),(7,9),(8,10)]
=> 3
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [(1,2),(3,4),(5,7),(6,8),(9,10)]
=> 3
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [(1,2),(3,4),(5,7),(6,9),(8,10)]
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [(1,2),(3,4),(5,8),(6,9),(7,10)]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [(1,2),(3,4),(5,8),(6,9),(7,10)]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [(1,2),(3,5),(4,6),(7,8),(9,10)]
=> 3
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [(1,2),(3,5),(4,6),(7,9),(8,10)]
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [(1,2),(3,5),(4,7),(6,8),(9,10)]
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [(1,2),(3,5),(4,7),(6,9),(8,10)]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [(1,2),(3,5),(4,8),(6,9),(7,10)]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [(1,2),(3,5),(4,8),(6,9),(7,10)]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [(1,2),(3,6),(4,7),(5,8),(9,10)]
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [(1,2),(3,6),(4,7),(5,9),(8,10)]
=> 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [(1,2),(3,6),(4,7),(5,8),(9,10)]
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [(1,2),(3,6),(4,7),(5,9),(8,10)]
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [(1,2),(3,6),(4,8),(5,9),(7,10)]
=> 1
Description
The number of short pairs.
A short pair is a matching pair of the form $(i,i+1)$.
Matching statistic: St000241
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000241: 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] => 2
[2,1] => [2,1] => [2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [2,3,1] => 3
[1,3,2] => [1,3,2] => [1,3,2] => [3,2,1] => 1
[2,1,3] => [2,1,3] => [2,1,3] => [1,3,2] => 1
[2,3,1] => [3,2,1] => [2,3,1] => [1,2,3] => 0
[3,1,2] => [3,2,1] => [2,3,1] => [1,2,3] => 0
[3,2,1] => [3,2,1] => [2,3,1] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 4
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [2,4,3,1] => 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [3,2,4,1] => 2
[1,3,4,2] => [1,4,3,2] => [1,3,4,2] => [4,2,3,1] => 1
[1,4,2,3] => [1,4,3,2] => [1,3,4,2] => [4,2,3,1] => 1
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => [4,2,3,1] => 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 2
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [1,4,3,2] => 0
[2,3,1,4] => [3,2,1,4] => [2,3,1,4] => [1,2,4,3] => 1
[2,3,4,1] => [4,2,3,1] => [2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [3,4,1,2] => [4,3,2,1] => [3,2,1,4] => 0
[2,4,3,1] => [4,3,2,1] => [3,4,1,2] => [4,1,2,3] => 0
[3,1,2,4] => [3,2,1,4] => [2,3,1,4] => [1,2,4,3] => 1
[3,1,4,2] => [4,2,3,1] => [2,3,4,1] => [1,2,3,4] => 0
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => [1,2,4,3] => 1
[3,2,4,1] => [4,2,3,1] => [2,3,4,1] => [1,2,3,4] => 0
[3,4,1,2] => [4,3,2,1] => [3,4,1,2] => [4,1,2,3] => 0
[3,4,2,1] => [4,3,2,1] => [3,4,1,2] => [4,1,2,3] => 0
[4,1,2,3] => [4,2,3,1] => [2,3,4,1] => [1,2,3,4] => 0
[4,1,3,2] => [4,2,3,1] => [2,3,4,1] => [1,2,3,4] => 0
[4,2,1,3] => [4,3,2,1] => [3,4,1,2] => [4,1,2,3] => 0
[4,2,3,1] => [4,3,2,1] => [3,4,1,2] => [4,1,2,3] => 0
[4,3,1,2] => [4,3,2,1] => [3,4,1,2] => [4,1,2,3] => 0
[4,3,2,1] => [4,3,2,1] => [3,4,1,2] => [4,1,2,3] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 5
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => 3
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => 3
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => [2,5,3,4,1] => 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => [2,5,3,4,1] => 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => [2,5,3,4,1] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => 3
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,1] => 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [4,2,3,5,1] => 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => [5,2,3,4,1] => 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,4,3,2] => [5,4,3,2,1] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,4,5,2,3] => [4,5,2,3,1] => 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,3,4,2,5] => [4,2,3,5,1] => 2
[1,4,2,5,3] => [1,5,3,4,2] => [1,3,4,5,2] => [5,2,3,4,1] => 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [4,2,3,5,1] => 2
[1,4,3,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => [5,2,3,4,1] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,4,5,2,3] => [4,5,2,3,1] => 1
Description
The number of cyclical small excedances.
A cyclical small excedance is an index $i$ such that $\pi_i = i+1$ considered cyclically.
Matching statistic: St000445
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => [1] => [1,0]
=> 1
[1,2] => 1 => [1,1] => [1,0,1,0]
=> 2
[2,1] => 0 => [2] => [1,1,0,0]
=> 0
[1,2,3] => 11 => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,3,2] => 10 => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => 01 => [2,1] => [1,1,0,0,1,0]
=> 1
[2,3,1] => 00 => [3] => [1,1,1,0,0,0]
=> 0
[3,1,2] => 00 => [3] => [1,1,1,0,0,0]
=> 0
[3,2,1] => 00 => [3] => [1,1,1,0,0,0]
=> 0
[1,2,3,4] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,4,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,3,2,4] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,3,4,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,3,4] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[2,1,4,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[2,3,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,3,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,4,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,4,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
[3,1,2,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,4,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
[3,2,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,2,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
[3,4,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
[3,4,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
[4,1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
[4,1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
[4,2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
[4,2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
[4,3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
[4,3,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,3,5,4] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,2,4,3,5] => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,2,4,5,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,2,5,3,4] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,2,5,4,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,3,2,4,5] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,3,2,5,4] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,3,4,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,4,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,2,3,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,5,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,3,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,3,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,5,2,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
Description
The number of rises of length 1 of a Dyck path.
The following 20 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001691The number of kings in a graph. St000011The number of touch points (or returns) of a Dyck path. St000674The number of hills of a Dyck path. St000247The number of singleton blocks of a set partition. St001826The maximal number of leaves on a vertex of a graph. St001672The restrained domination number of a graph. St001479The number of bridges of a graph. St001342The number of vertices in the center of a graph. St001368The number of vertices of maximal degree in a graph. St000214The number of adjacencies of a permutation. St000237The number of small exceedances. St000456The monochromatic index of a connected graph. St000894The trace of an alternating sign matrix. St000239The number of small weak excedances. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001903The number of fixed points of a parking function. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset.
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!