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Your data matches 666 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000010: Integer partitions ⟶ 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] => [2]
=> 1
[1,1,1] => [1,1,1]
=> 3
[1,2] => [2,1]
=> 2
[2,1] => [2,1]
=> 2
[3] => [3]
=> 1
[1,1,1,1] => [1,1,1,1]
=> 4
[1,1,2] => [2,1,1]
=> 3
[1,3] => [3,1]
=> 2
[2,1,1] => [2,1,1]
=> 3
[2,2] => [2,2]
=> 2
[3,1] => [3,1]
=> 2
[4] => [4]
=> 1
[1,1,1,1,1] => [1,1,1,1,1]
=> 5
[1,1,1,2] => [2,1,1,1]
=> 4
[1,1,3] => [3,1,1]
=> 3
[1,4] => [4,1]
=> 2
[2,1,1,1] => [2,1,1,1]
=> 4
[2,1,2] => [2,2,1]
=> 3
[2,3] => [3,2]
=> 2
[3,1,1] => [3,1,1]
=> 3
[3,2] => [3,2]
=> 2
[4,1] => [4,1]
=> 2
[5] => [5]
=> 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 6
[1,1,1,1,2] => [2,1,1,1,1]
=> 5
[1,1,1,3] => [3,1,1,1]
=> 4
[1,1,4] => [4,1,1]
=> 3
[1,5] => [5,1]
=> 2
[2,1,1,1,1] => [2,1,1,1,1]
=> 5
[2,1,1,2] => [2,2,1,1]
=> 4
[2,1,3] => [3,2,1]
=> 3
[2,4] => [4,2]
=> 2
[3,1,1,1] => [3,1,1,1]
=> 4
[3,1,2] => [3,2,1]
=> 3
[3,3] => [3,3]
=> 2
[4,1,1] => [4,1,1]
=> 3
[4,2] => [4,2]
=> 2
[5,1] => [5,1]
=> 2
[6] => [6]
=> 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> 7
[1,1,1,1,1,2] => [2,1,1,1,1,1]
=> 6
[1,1,1,1,3] => [3,1,1,1,1]
=> 5
[1,1,1,4] => [4,1,1,1]
=> 4
[1,1,5] => [5,1,1]
=> 3
[1,6] => [6,1]
=> 2
[2,1,1,1,1,1] => [2,1,1,1,1,1]
=> 6
[2,1,1,1,2] => [2,2,1,1,1]
=> 5
[2,1,1,3] => [3,2,1,1]
=> 4
Description
The length of the partition.
Matching statistic: St000011
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1
[1,1] => [1,0,1,0]
=> 2
[2] => [1,1,0,0]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> 3
[1,2] => [1,0,1,1,0,0]
=> 2
[2,1] => [1,1,0,0,1,0]
=> 2
[3] => [1,1,1,0,0,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[1,3] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 4
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 3
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 4
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 3
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 4
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 3
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 6
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 5
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 3
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 2
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> 4
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000097
(load all 37 compositions to match this statistic)
(load all 37 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 1
[1,1] => ([(0,1)],2)
=> 2
[2] => ([],2)
=> 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2] => ([(1,2)],3)
=> 2
[2,1] => ([(0,2),(1,2)],3)
=> 2
[3] => ([],3)
=> 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,3] => ([(2,3)],4)
=> 2
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,2] => ([(1,3),(2,3)],4)
=> 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4] => ([],4)
=> 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,4] => ([(3,4)],5)
=> 2
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,3] => ([(2,4),(3,4)],5)
=> 2
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[5] => ([],5)
=> 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3
[1,5] => ([(4,5)],6)
=> 2
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,4] => ([(3,5),(4,5)],6)
=> 2
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[6] => ([],6)
=> 1
[1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 3
[1,6] => ([(5,6)],7)
=> 2
[2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
(load all 37 compositions to match this statistic)
(load all 37 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 1
[1,1] => ([(0,1)],2)
=> 2
[2] => ([],2)
=> 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2] => ([(1,2)],3)
=> 2
[2,1] => ([(0,2),(1,2)],3)
=> 2
[3] => ([],3)
=> 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,3] => ([(2,3)],4)
=> 2
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,2] => ([(1,3),(2,3)],4)
=> 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4] => ([],4)
=> 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,4] => ([(3,4)],5)
=> 2
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,3] => ([(2,4),(3,4)],5)
=> 2
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[5] => ([],5)
=> 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3
[1,5] => ([(4,5)],6)
=> 2
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,4] => ([(3,5),(4,5)],6)
=> 2
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[6] => ([],6)
=> 1
[1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 3
[1,6] => ([(5,6)],7)
=> 2
[2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St000288
(load all 104 compositions to match this statistic)
(load all 104 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 1
[1,1] => 11 => 2
[2] => 10 => 1
[1,1,1] => 111 => 3
[1,2] => 110 => 2
[2,1] => 101 => 2
[3] => 100 => 1
[1,1,1,1] => 1111 => 4
[1,1,2] => 1110 => 3
[1,3] => 1100 => 2
[2,1,1] => 1011 => 3
[2,2] => 1010 => 2
[3,1] => 1001 => 2
[4] => 1000 => 1
[1,1,1,1,1] => 11111 => 5
[1,1,1,2] => 11110 => 4
[1,1,3] => 11100 => 3
[1,4] => 11000 => 2
[2,1,1,1] => 10111 => 4
[2,1,2] => 10110 => 3
[2,3] => 10100 => 2
[3,1,1] => 10011 => 3
[3,2] => 10010 => 2
[4,1] => 10001 => 2
[5] => 10000 => 1
[1,1,1,1,1,1] => 111111 => 6
[1,1,1,1,2] => 111110 => 5
[1,1,1,3] => 111100 => 4
[1,1,4] => 111000 => 3
[1,5] => 110000 => 2
[2,1,1,1,1] => 101111 => 5
[2,1,1,2] => 101110 => 4
[2,1,3] => 101100 => 3
[2,4] => 101000 => 2
[3,1,1,1] => 100111 => 4
[3,1,2] => 100110 => 3
[3,3] => 100100 => 2
[4,1,1] => 100011 => 3
[4,2] => 100010 => 2
[5,1] => 100001 => 2
[6] => 100000 => 1
[1,1,1,1,1,1,1] => 1111111 => 7
[1,1,1,1,1,2] => 1111110 => 6
[1,1,1,1,3] => 1111100 => 5
[1,1,1,4] => 1111000 => 4
[1,1,5] => 1110000 => 3
[1,6] => 1100000 => 2
[2,1,1,1,1,1] => 1011111 => 6
[2,1,1,1,2] => 1011110 => 5
[2,1,1,3] => 1011100 => 4
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000381
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,1] => [2] => 2
[2] => [1,1] => 1
[1,1,1] => [3] => 3
[1,2] => [1,2] => 2
[2,1] => [2,1] => 2
[3] => [1,1,1] => 1
[1,1,1,1] => [4] => 4
[1,1,2] => [1,3] => 3
[1,3] => [1,1,2] => 2
[2,1,1] => [3,1] => 3
[2,2] => [1,2,1] => 2
[3,1] => [2,1,1] => 2
[4] => [1,1,1,1] => 1
[1,1,1,1,1] => [5] => 5
[1,1,1,2] => [1,4] => 4
[1,1,3] => [1,1,3] => 3
[1,4] => [1,1,1,2] => 2
[2,1,1,1] => [4,1] => 4
[2,1,2] => [1,3,1] => 3
[2,3] => [1,1,2,1] => 2
[3,1,1] => [3,1,1] => 3
[3,2] => [1,2,1,1] => 2
[4,1] => [2,1,1,1] => 2
[5] => [1,1,1,1,1] => 1
[1,1,1,1,1,1] => [6] => 6
[1,1,1,1,2] => [1,5] => 5
[1,1,1,3] => [1,1,4] => 4
[1,1,4] => [1,1,1,3] => 3
[1,5] => [1,1,1,1,2] => 2
[2,1,1,1,1] => [5,1] => 5
[2,1,1,2] => [1,4,1] => 4
[2,1,3] => [1,1,3,1] => 3
[2,4] => [1,1,1,2,1] => 2
[3,1,1,1] => [4,1,1] => 4
[3,1,2] => [1,3,1,1] => 3
[3,3] => [1,1,2,1,1] => 2
[4,1,1] => [3,1,1,1] => 3
[4,2] => [1,2,1,1,1] => 2
[5,1] => [2,1,1,1,1] => 2
[6] => [1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1] => [7] => 7
[1,1,1,1,1,2] => [1,6] => 6
[1,1,1,1,3] => [1,1,5] => 5
[1,1,1,4] => [1,1,1,4] => 4
[1,1,5] => [1,1,1,1,3] => 3
[1,6] => [1,1,1,1,1,2] => 2
[2,1,1,1,1,1] => [6,1] => 6
[2,1,1,1,2] => [1,5,1] => 5
[2,1,1,3] => [1,1,4,1] => 4
Description
The largest part of an integer composition.
Matching statistic: St000684
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000684: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000684: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1
[1,1] => [1,0,1,0]
=> 2
[2] => [1,1,0,0]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> 3
[1,2] => [1,0,1,1,0,0]
=> 2
[2,1] => [1,1,0,0,1,0]
=> 2
[3] => [1,1,1,0,0,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[1,3] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 4
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 3
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 4
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 3
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 4
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 3
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 6
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 5
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 3
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 2
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> 4
Description
The global dimension of the LNakayama algebra associated to a Dyck path.
An n-LNakayama algebra is a quiver algebra with a directed line as a connected quiver with $n$ points for $n \geq 2$. Number those points from the left to the right by $0,1,\ldots,n-1$.
The algebra is then uniquely determined by the dimension $c_i$ of the projective indecomposable modules at point $i$. Such algebras are then uniquely determined by lists of the form $[c_0,c_1,...,c_{n-1}]$ with the conditions: $c_{n-1}=1$ and $c_i -1 \leq c_{i+1}$ for all $i$. The number of such algebras is then the $n-1$-st Catalan number $C_{n-1}$.
One can get also an interpretation with Dyck paths by associating the top boundary of the Auslander-Reiten quiver (which is a Dyck path) to those algebras. Example: [3,4,3,3,2,1] corresponds to the Dyck path [1,1,0,1,1,0,0,1,0,0].
Conjecture: that there is an explicit bijection between $n$-LNakayama algebras with global dimension bounded by $m$ and Dyck paths with height at most $m$.
Examples:
* For $m=2$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1,2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192.
* For $m=3$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418.
Matching statistic: St000808
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000808: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000808: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,1] => [2] => 2
[2] => [1,1] => 1
[1,1,1] => [3] => 3
[1,2] => [1,2] => 2
[2,1] => [2,1] => 2
[3] => [1,1,1] => 1
[1,1,1,1] => [4] => 4
[1,1,2] => [1,3] => 3
[1,3] => [1,1,2] => 2
[2,1,1] => [3,1] => 3
[2,2] => [1,2,1] => 2
[3,1] => [2,1,1] => 2
[4] => [1,1,1,1] => 1
[1,1,1,1,1] => [5] => 5
[1,1,1,2] => [1,4] => 4
[1,1,3] => [1,1,3] => 3
[1,4] => [1,1,1,2] => 2
[2,1,1,1] => [4,1] => 4
[2,1,2] => [1,3,1] => 3
[2,3] => [1,1,2,1] => 2
[3,1,1] => [3,1,1] => 3
[3,2] => [1,2,1,1] => 2
[4,1] => [2,1,1,1] => 2
[5] => [1,1,1,1,1] => 1
[1,1,1,1,1,1] => [6] => 6
[1,1,1,1,2] => [1,5] => 5
[1,1,1,3] => [1,1,4] => 4
[1,1,4] => [1,1,1,3] => 3
[1,5] => [1,1,1,1,2] => 2
[2,1,1,1,1] => [5,1] => 5
[2,1,1,2] => [1,4,1] => 4
[2,1,3] => [1,1,3,1] => 3
[2,4] => [1,1,1,2,1] => 2
[3,1,1,1] => [4,1,1] => 4
[3,1,2] => [1,3,1,1] => 3
[3,3] => [1,1,2,1,1] => 2
[4,1,1] => [3,1,1,1] => 3
[4,2] => [1,2,1,1,1] => 2
[5,1] => [2,1,1,1,1] => 2
[6] => [1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1] => [7] => 7
[1,1,1,1,1,2] => [1,6] => 6
[1,1,1,1,3] => [1,1,5] => 5
[1,1,1,4] => [1,1,1,4] => 4
[1,1,5] => [1,1,1,1,3] => 3
[1,6] => [1,1,1,1,1,2] => 2
[2,1,1,1,1,1] => [6,1] => 6
[2,1,1,1,2] => [1,5,1] => 5
[2,1,1,3] => [1,1,4,1] => 4
Description
The number of up steps of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of up steps.
Matching statistic: St000013
(load all 41 compositions to match this statistic)
(load all 41 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 1
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[2] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 4
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 3
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 4
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 3
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 4
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 3
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 2
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 3
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 2
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 7
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> 5
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> 4
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 3
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 2
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 6
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> 5
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> 4
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000147
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(load all 7 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ 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
[2] => [2]
=> [1,1]
=> 1
[1,1,1] => [1,1,1]
=> [3]
=> 3
[1,2] => [2,1]
=> [2,1]
=> 2
[2,1] => [2,1]
=> [2,1]
=> 2
[3] => [3]
=> [1,1,1]
=> 1
[1,1,1,1] => [1,1,1,1]
=> [4]
=> 4
[1,1,2] => [2,1,1]
=> [3,1]
=> 3
[1,3] => [3,1]
=> [2,1,1]
=> 2
[2,1,1] => [2,1,1]
=> [3,1]
=> 3
[2,2] => [2,2]
=> [2,2]
=> 2
[3,1] => [3,1]
=> [2,1,1]
=> 2
[4] => [4]
=> [1,1,1,1]
=> 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 5
[1,1,1,2] => [2,1,1,1]
=> [4,1]
=> 4
[1,1,3] => [3,1,1]
=> [3,1,1]
=> 3
[1,4] => [4,1]
=> [2,1,1,1]
=> 2
[2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 4
[2,1,2] => [2,2,1]
=> [3,2]
=> 3
[2,3] => [3,2]
=> [2,2,1]
=> 2
[3,1,1] => [3,1,1]
=> [3,1,1]
=> 3
[3,2] => [3,2]
=> [2,2,1]
=> 2
[4,1] => [4,1]
=> [2,1,1,1]
=> 2
[5] => [5]
=> [1,1,1,1,1]
=> 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [6]
=> 6
[1,1,1,1,2] => [2,1,1,1,1]
=> [5,1]
=> 5
[1,1,1,3] => [3,1,1,1]
=> [4,1,1]
=> 4
[1,1,4] => [4,1,1]
=> [3,1,1,1]
=> 3
[1,5] => [5,1]
=> [2,1,1,1,1]
=> 2
[2,1,1,1,1] => [2,1,1,1,1]
=> [5,1]
=> 5
[2,1,1,2] => [2,2,1,1]
=> [4,2]
=> 4
[2,1,3] => [3,2,1]
=> [3,2,1]
=> 3
[2,4] => [4,2]
=> [2,2,1,1]
=> 2
[3,1,1,1] => [3,1,1,1]
=> [4,1,1]
=> 4
[3,1,2] => [3,2,1]
=> [3,2,1]
=> 3
[3,3] => [3,3]
=> [2,2,2]
=> 2
[4,1,1] => [4,1,1]
=> [3,1,1,1]
=> 3
[4,2] => [4,2]
=> [2,2,1,1]
=> 2
[5,1] => [5,1]
=> [2,1,1,1,1]
=> 2
[6] => [6]
=> [1,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [7]
=> 7
[1,1,1,1,1,2] => [2,1,1,1,1,1]
=> [6,1]
=> 6
[1,1,1,1,3] => [3,1,1,1,1]
=> [5,1,1]
=> 5
[1,1,1,4] => [4,1,1,1]
=> [4,1,1,1]
=> 4
[1,1,5] => [5,1,1]
=> [3,1,1,1,1]
=> 3
[1,6] => [6,1]
=> [2,1,1,1,1,1]
=> 2
[2,1,1,1,1,1] => [2,1,1,1,1,1]
=> [6,1]
=> 6
[2,1,1,1,2] => [2,2,1,1,1]
=> [5,2]
=> 5
[2,1,1,3] => [3,2,1,1]
=> [4,2,1]
=> 4
Description
The largest part of an integer partition.
The following 656 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000228The size of a partition. St000378The diagonal inversion number of an integer partition. St000392The length of the longest run of ones in a binary word. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000733The row containing the largest entry of a standard tableau. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000982The length of the longest constant subword. St001330The hat guessing number of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001389The number of partitions of the same length below the given integer partition. St001419The length of the longest palindromic factor beginning with a one of a binary word. St000157The number of descents of a standard tableau. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000996The number of exclusive left-to-right maxima of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001644The dimension of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000007The number of saliances of the permutation. St000167The number of leaves of an ordered tree. St000184The size of the centralizer of any permutation of given cycle type. St000293The number of inversions of a binary word. St000297The number of leading ones in a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000384The maximal part of the shifted composition of an integer partition. St000395The sum of the heights of the peaks of a Dyck path. St000507The number of ascents of a standard tableau. St000676The number of odd rises of a Dyck path. St000734The last entry in the first row of a standard tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000784The maximum of the length and the largest part of the integer partition. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000983The length of the longest alternating subword. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001034The area of the parallelogram polyomino associated with the Dyck path. St001058The breadth of the ordered tree. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001415The length of the longest palindromic prefix of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001733The number of weak left to right maxima of a Dyck path. St001777The number of weak descents in an integer composition. St000008The major index of the composition. St000340The number of non-final maximal constant sub-paths of length greater than one. St000377The dinv defect of an integer partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000439The position of the first down step of a Dyck path. St000546The number of global descents of a permutation. St000548The number of different non-empty partial sums of an integer partition. St000691The number of changes of a binary word. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000877The depth of the binary word interpreted as a path. St000921The number of internal inversions of a binary word. St001176The size of a partition minus its first part. St000806The semiperimeter of the associated bargraph. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000093The cardinality of a maximal independent set of vertices of a graph. St000172The Grundy number of a graph. St000363The number of minimal vertex covers of a graph. St000469The distinguishing number of a graph. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001883The mutual visibility number of a graph. St000012The area of a Dyck path. St000171The degree of the graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000454The largest eigenvalue of a graph if it is integral. St000536The pathwidth of a graph. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001120The length of a longest path in a graph. St001949The rigidity index of a graph. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000451The length of the longest pattern of the form k 1 2. St000636The hull number of a graph. St000678The number of up steps after the last double rise of a Dyck path. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000971The smallest closer of a set partition. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001286The annihilation number of a graph. St001315The dissociation number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000160The multiplicity of the smallest part of a partition. St000374The number of exclusive right-to-left minima of a permutation. St000445The number of rises of length 1 of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000519The largest length of a factor maximising the subword complexity. St000521The number of distinct subtrees of an ordered tree. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000932The number of occurrences of the pattern UDU in a Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001484The number of singletons of an integer partition. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St000686The finitistic dominant dimension of a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St000053The number of valleys of the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000025The number of initial rises of a Dyck path. St000058The order of a permutation. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000105The number of blocks in the set partition. St000144The pyramid weight of the Dyck path. St000444The length of the maximal rise of a Dyck path. St000527The width of the poset. St000909The number of maximal chains of maximal size in a poset. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001779The order of promotion on the set of linear extensions of a poset. St001809The index of the step at the first peak of maximal height in a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000234The number of global ascents of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000632The jump number of the poset. St000984The number of boxes below precisely one peak. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000069The number of maximal elements of a poset. St000189The number of elements in the poset. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000273The domination number of a graph. St000336The leg major index of a standard tableau. St000382The first part of an integer composition. St000383The last part of an integer composition. St000468The Hosoya index of a graph. St000504The cardinality of the first block of a set partition. St000544The cop number of a graph. St000553The number of blocks of a graph. St000675The number of centered multitunnels of a Dyck path. St000722The number of different neighbourhoods in a graph. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000883The number of longest increasing subsequences of a permutation. St000916The packing number of a graph. St000926The clique-coclique number of a graph. St001050The number of terminal closers of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001062The maximal size of a block of a set partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001342The number of vertices in the center of a graph. St001363The Euler characteristic of a graph according to Knill. St001368The number of vertices of maximal degree in a graph. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001674The number of vertices of the largest induced star graph in the graph. St001725The harmonious chromatic number of a graph. St000026The position of the first return of a Dyck path. St000081The number of edges of a graph. St000211The rank of the set partition. St000502The number of successions of a set partitions. St000503The maximal difference between two elements in a common block. St000662The staircase size of the code of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000728The dimension of a set partition. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001479The number of bridges of a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001746The coalition number of a graph. St001826The maximal number of leaves on a vertex of a graph. St000822The Hadwiger number of the graph. St000925The number of topologically connected components of a set partition. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St000874The position of the last double rise in a Dyck path. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001962The proper pathwidth of a graph. St000087The number of induced subgraphs. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St000730The maximal arc length of a set partition. St000947The major index east count of a Dyck path. St000528The height of a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St000054The first entry of the permutation. St000485The length of the longest cycle of a permutation. St001829The common independence number of a graph. St000028The number of stack-sorts needed to sort a permutation. St000141The maximum drop size of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St000809The reduced reflection length of the permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001925The minimal number of zeros in a row of an alternating sign matrix. St001812The biclique partition number of a graph. St000029The depth of a permutation. St000197The number of entries equal to positive one in the alternating sign matrix. St000209Maximum difference of elements in cycles. St000210Minimum over maximum difference of elements in cycles. St000216The absolute length of a permutation. St000924The number of topologically connected components of a perfect matching. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001316The domatic number of a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001480The number of simple summands of the module J^2/J^3. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001958The degree of the polynomial interpolating the values of a permutation. St000245The number of ascents of a permutation. St000306The bounce count of a Dyck path. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000673The number of non-fixed points of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001468The smallest fixpoint of a permutation. St001717The largest size of an interval in a poset. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000532The total number of rook placements on a Ferrers board. St000738The first entry in the last row of a standard tableau. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001400The total number of Littlewood-Richardson tableaux of given shape. St001584The area statistic between a Dyck path and its bounce path. St001933The largest multiplicity of a part in an integer partition. St000148The number of odd parts of a partition. St000475The number of parts equal to 1 in a partition. St000867The sum of the hook lengths in the first row of an integer partition. St001127The sum of the squares of the parts of a partition. St000031The number of cycles in the cycle decomposition of a permutation. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001726The number of visible inversions of a permutation. St000006The dinv of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000910The number of maximal chains of minimal length in a poset. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000956The maximal displacement of a permutation. St000656The number of cuts of a poset. St000308The height of the tree associated to a permutation. St000653The last descent of a permutation. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001118The acyclic chromatic index of a graph. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000470The number of runs in a permutation. St000041The number of nestings of a perfect matching. St000159The number of distinct parts of the integer partition. St000237The number of small exceedances. St000868The aid statistic in the sense of Shareshian-Wachs. St000989The number of final rises of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000153The number of adjacent cycles of a permutation. St000110The number of permutations less than or equal to a permutation in left weak order. St000890The number of nonzero entries in an alternating sign matrix. St001461The number of topologically connected components of the chord diagram of a permutation. St000214The number of adjacencies of a permutation. St000354The number of recoils of a permutation. St000446The disorder of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000833The comajor index of a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000044The number of vertices of the unicellular map given by a perfect matching. St000327The number of cover relations in a poset. St000539The number of odd inversions of a permutation. St000795The mad of a permutation. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000292The number of ascents of a binary word. St000015The number of peaks of a Dyck path. St000100The number of linear extensions of a poset. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001530The depth of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000331The number of upper interactions of a Dyck path. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000441The number of successions of a permutation. St000993The multiplicity of the largest part of an integer partition. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000225Difference between largest and smallest parts in a partition. St000651The maximal size of a rise in a permutation. St000908The length of the shortest maximal antichain in a poset. St000203The number of external nodes of a binary tree. St000356The number of occurrences of the pattern 13-2. St001083The number of boxed occurrences of 132 in a permutation. St000914The sum of the values of the Möbius function of a poset. St001432The order dimension of the partition. St000741The Colin de Verdière graph invariant. St000740The last entry of a permutation. St000161The sum of the sizes of the right subtrees of a binary tree. St000648The number of 2-excedences of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000654The first descent of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000702The number of weak deficiencies of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000246The number of non-inversions of a permutation. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000794The mak of a permutation. St001427The number of descents of a signed permutation. St000164The number of short pairs. St000843The decomposition number of a perfect matching. St000066The column of the unique '1' in the first row of the alternating sign matrix. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000628The balance of a binary word. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000332The positive inversions of an alternating sign matrix. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001397Number of pairs of incomparable elements in a finite poset. St000067The inversion number of the alternating sign matrix. St000542The number of left-to-right-minima of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001645The pebbling number of a connected graph. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000084The number of subtrees. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000335The difference of lower and upper interactions. St000443The number of long tunnels of a Dyck path. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St000021The number of descents of a permutation. St000030The sum of the descent differences of a permutations. St000080The rank of the poset. St000224The sorting index of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001119The length of a shortest maximal path in a graph. St001180Number of indecomposable injective modules with projective dimension at most 1. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St000035The number of left outer peaks of a permutation. St000086The number of subgraphs. St000166The depth minus 1 of an ordered tree. St000240The number of indices that are not small excedances. St000258The burning number of a graph. St000287The number of connected components of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000389The number of runs of ones of odd length in a binary word. St000479The Ramsey number of a graph. St000482The (zero)-forcing number of a graph. St000531The leading coefficient of the rook polynomial of an integer partition. St000667The greatest common divisor of the parts of the partition. St000742The number of big ascents of a permutation after prepending zero. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000846The maximal number of elements covering an element of a poset. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000990The first ascent of a permutation. St000992The alternating sum of the parts of an integer partition. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001312Number of parabolic noncrossing partitions indexed by the composition. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001360The number of covering relations in Young's lattice below a partition. St001441The number of non-empty connected induced subgraphs of a graph. St001463The number of distinct columns in the nullspace of a graph. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001672The restrained domination number of a graph. St001675The number of parts equal to the part in the reversed composition. St001691The number of kings in a graph. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001828The Euler characteristic of a graph. St000004The major index of a permutation. St000005The bounce statistic of a Dyck path. St000057The Shynar inversion number of a standard tableau. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000091The descent variation of a composition. St000094The depth of an ordered tree. St000120The number of left tunnels of a Dyck path. St000145The Dyson rank of a partition. St000168The number of internal nodes of an ordered tree. St000204The number of internal nodes of a binary tree. St000238The number of indices that are not small weak excedances. St000305The inverse major index of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001117The game chromatic index of a graph. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001340The cardinality of a minimal non-edge isolating set of a graph. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001428The number of B-inversions of a signed permutation. St001814The number of partitions interlacing the given partition. St001843The Z-index of a set partition. St001869The maximum cut size of a graph. St000061The number of nodes on the left branch of a binary tree. St001959The product of the heights of the peaks of a Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000082The number of elements smaller than a binary tree in Tamari order. St001346The number of parking functions that give the same permutation. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001834The number of non-isomorphic minors of a graph. St001706The number of closed sets in a graph. St000731The number of double exceedences of a permutation. St000619The number of cyclic descents of a permutation. St000652The maximal difference between successive positions of a permutation. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000294The number of distinct factors of a binary word. St000518The number of distinct subsequences in a binary word. St000290The major index of a binary word. St000296The length of the symmetric border of a binary word. St000393The number of strictly increasing runs in a binary word. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000538The number of even inversions of a permutation. St000627The exponent of a binary word. St000711The number of big exceedences of a permutation. St000922The minimal number such that all substrings of this length are unique. St001267The length of the Lyndon factorization of the binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001437The flex of a binary word. St001485The modular major index of a binary word. St001884The number of borders of a binary word. St000295The length of the border of a binary word. St000477The weight of a partition according to Alladi. St000770The major index of an integer partition when read from bottom to top. St000359The number of occurrences of the pattern 23-1. St000681The Grundy value of Chomp on Ferrers diagrams. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001323The independence gap of a graph. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001060The distinguishing index of a graph. St001152The number of pairs with even minimum in a perfect matching. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001651The Frankl number of a lattice. St000355The number of occurrences of the pattern 21-3. St000317The cycle descent number of a permutation. St000358The number of occurrences of the pattern 31-2. St000732The number of double deficiencies of a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001684The reduced word complexity of a permutation. St001727The number of invisible inversions of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001497The position of the largest weak excedence of a permutation. St001555The order of a signed permutation. St001589The nesting number of a perfect matching. St001671Haglund's hag of a permutation. St000039The number of crossings of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000236The number of cyclical small weak excedances. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000710The number of big deficiencies of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001864The number of excedances of a signed permutation. St001894The depth of a signed permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001863The number of weak excedances of a signed permutation. St001889The size of the connectivity set of a signed permutation. St000455The second largest eigenvalue of a graph if it is integral. St001896The number of right descents of a signed permutations. St000782The indicator function of whether a given perfect matching is an L & P matching. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001712The number of natural descents of a standard Young tableau. St000670The reversal length of a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001811The Castelnuovo-Mumford regularity of a permutation. St001821The sorting index of a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. 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 $eA$. St001875The number of simple modules with projective dimension at most 1. 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 $eA$. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000023The number of inner peaks of a permutation. St000353The number of inner valleys of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St001388The number of non-attacking neighbors of a permutation. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000456The monochromatic index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000259The diameter of a connected graph. St000260The radius 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.
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