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Your data matches 346 different statistics following compositions of up to 3 maps.
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Matching statistic: St000439
(load all 53 compositions to match this statistic)
(load all 53 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 2 = 1 + 1
[1,0,1,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000010
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
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%
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%
Values
[1,0]
=> [1] => [1]
=> 1
[1,0,1,0]
=> [1,1] => [1,1]
=> 2
[1,1,0,0]
=> [2] => [2]
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> 3
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> 2
[1,1,0,1,0,0]
=> [3] => [3]
=> 1
[1,1,1,0,0,0]
=> [3] => [3]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,3] => [3,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [3,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [4] => [4]
=> 1
[1,1,0,1,1,0,0,0]
=> [4] => [4]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [4] => [4]
=> 1
[1,1,1,0,1,0,0,0]
=> [4] => [4]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [4,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [3,2]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [5]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [5]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [5]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [5]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [5]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [5]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [5]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [5]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [5]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [5]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [4,1]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [5]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [5]
=> 1
Description
The length of the partition.
Matching statistic: St000097
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> 1
[1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 2
[1,1,0,0]
=> [2] => ([],2)
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [3] => ([],3)
=> 1
[1,1,1,0,0,0]
=> [3] => ([],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [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
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [5] => ([],5)
=> 1
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 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> 1
[1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 2
[1,1,0,0]
=> [2] => ([],2)
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [3] => ([],3)
=> 1
[1,1,1,0,0,0]
=> [3] => ([],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [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
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [5] => ([],5)
=> 1
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 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
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%
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%
Values
[1,0]
=> [1] => 1 => 1
[1,0,1,0]
=> [1,1] => 11 => 2
[1,1,0,0]
=> [2] => 10 => 1
[1,0,1,0,1,0]
=> [1,1,1] => 111 => 3
[1,0,1,1,0,0]
=> [1,2] => 110 => 2
[1,1,0,0,1,0]
=> [2,1] => 101 => 2
[1,1,0,1,0,0]
=> [3] => 100 => 1
[1,1,1,0,0,0]
=> [3] => 100 => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 3
[1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 2
[1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 2
[1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 2
[1,1,0,1,0,0,1,0]
=> [3,1] => 1001 => 2
[1,1,0,1,0,1,0,0]
=> [4] => 1000 => 1
[1,1,0,1,1,0,0,0]
=> [4] => 1000 => 1
[1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 2
[1,1,1,0,0,1,0,0]
=> [4] => 1000 => 1
[1,1,1,0,1,0,0,0]
=> [4] => 1000 => 1
[1,1,1,1,0,0,0,0]
=> [4] => 1000 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 11100 => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 11000 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 11000 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 11000 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 11000 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 10100 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 10010 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 10001 => 2
[1,1,0,1,0,1,0,1,0,0]
=> [5] => 10000 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => 10000 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 10001 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [5] => 10000 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => 10000 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => 10000 => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 10010 => 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => 10001 => 2
[1,1,1,0,0,1,0,1,0,0]
=> [5] => 10000 => 1
[1,1,1,0,0,1,1,0,0,0]
=> [5] => 10000 => 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => 10001 => 2
[1,1,1,0,1,0,0,1,0,0]
=> [5] => 10000 => 1
[1,1,1,0,1,0,1,0,0,0]
=> [5] => 10000 => 1
[1,1,1,0,1,1,0,0,0,0]
=> [5] => 10000 => 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 10001 => 2
[1,1,1,1,0,0,0,1,0,0]
=> [5] => 10000 => 1
[1,1,1,1,0,0,1,0,0,0]
=> [5] => 10000 => 1
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 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [1,1] => [2] => 2
[1,1,0,0]
=> [2] => [1,1] => 1
[1,0,1,0,1,0]
=> [1,1,1] => [3] => 3
[1,0,1,1,0,0]
=> [1,2] => [1,2] => 2
[1,1,0,0,1,0]
=> [2,1] => [2,1] => 2
[1,1,0,1,0,0]
=> [3] => [1,1,1] => 1
[1,1,1,0,0,0]
=> [3] => [1,1,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,3] => 3
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1,2] => 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,2,1] => 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [2,1,1] => 2
[1,1,0,1,0,1,0,0]
=> [4] => [1,1,1,1] => 1
[1,1,0,1,1,0,0,0]
=> [4] => [1,1,1,1] => 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [2,1,1] => 2
[1,1,1,0,0,1,0,0]
=> [4] => [1,1,1,1] => 1
[1,1,1,0,1,0,0,0]
=> [4] => [1,1,1,1] => 1
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,4] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [1,1,3] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,1,3] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1,1,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1,1,2] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1,1,2] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1,1,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1,1,2] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1,2,1] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,2,1] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,2,1,1] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [2,1,1,1] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [1,1,1,1,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [1,1,1,1,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [2,1,1,1] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [1,1,1,1,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [1,1,1,1,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [1,1,1,1,1] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,2,1,1] => 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [2,1,1,1] => 2
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [1,1,1,1,1] => 1
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [1,1,1,1,1] => 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [2,1,1,1] => 2
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [1,1,1,1,1] => 1
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [1,1,1,1,1] => 1
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [1,1,1,1,1] => 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [2,1,1,1] => 2
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [1,1,1,1,1] => 1
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [1,1,1,1,1] => 1
Description
The largest part of an integer composition.
Matching statistic: St000382
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => 1
[1,0,1,0]
=> [1,1,0,0]
=> [2] => 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => 3
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1] => 2
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1] => 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [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] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 1
Description
The first part of an integer composition.
Matching statistic: St000808
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000808: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000808: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [1,1] => [2] => 2
[1,1,0,0]
=> [2] => [1,1] => 1
[1,0,1,0,1,0]
=> [1,1,1] => [3] => 3
[1,0,1,1,0,0]
=> [1,2] => [1,2] => 2
[1,1,0,0,1,0]
=> [2,1] => [2,1] => 2
[1,1,0,1,0,0]
=> [3] => [1,1,1] => 1
[1,1,1,0,0,0]
=> [3] => [1,1,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,3] => 3
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1,2] => 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,2,1] => 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [2,1,1] => 2
[1,1,0,1,0,1,0,0]
=> [4] => [1,1,1,1] => 1
[1,1,0,1,1,0,0,0]
=> [4] => [1,1,1,1] => 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [2,1,1] => 2
[1,1,1,0,0,1,0,0]
=> [4] => [1,1,1,1] => 1
[1,1,1,0,1,0,0,0]
=> [4] => [1,1,1,1] => 1
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,4] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [1,1,3] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,1,3] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1,1,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1,1,2] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1,1,2] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1,1,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1,1,2] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1,2,1] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,2,1] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,2,1,1] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [2,1,1,1] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [1,1,1,1,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [1,1,1,1,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [2,1,1,1] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [1,1,1,1,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [1,1,1,1,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [1,1,1,1,1] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,2,1,1] => 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [2,1,1,1] => 2
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [1,1,1,1,1] => 1
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [1,1,1,1,1] => 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [2,1,1,1] => 2
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [1,1,1,1,1] => 1
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [1,1,1,1,1] => 1
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [1,1,1,1,1] => 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [2,1,1,1] => 2
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [1,1,1,1,1] => 1
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [1,1,1,1,1] => 1
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: St001330
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> 1
[1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 2
[1,1,0,0]
=> [2] => ([],2)
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [3] => ([],3)
=> 1
[1,1,1,0,0,0]
=> [3] => ([],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [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
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [5] => ([],5)
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [5] => ([],5)
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [5] => ([],5)
=> 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000011
(load all 106 compositions to match this statistic)
(load all 106 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00314: Integer compositions —Foata bijection⟶ Integer compositions
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%
Mp00314: Integer compositions —Foata bijection⟶ Integer compositions
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%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 1
[1,0,1,0]
=> [1,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,0]
=> [2] => [2] => [1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> [1,2] => [1,2] => [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [3] => [3] => [1,1,1,0,0,0]
=> 1
[1,1,1,0,0,0]
=> [3] => [3] => [1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
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.
The following 336 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000013The height of a Dyck path. St000147The largest part of an integer partition. 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. 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. St000160The multiplicity of the smallest part of a partition. 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. St000519The largest length of a factor maximising the subword complexity. St000548The number of different non-empty partial sums of an integer partition. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001581The achromatic number of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000395The sum of the heights of the peaks of a Dyck path. St000668The least common multiple of the parts of the partition. St000678The number of up steps after the last double rise of a Dyck path. 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. 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. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001672The restrained domination number of a 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. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000144The pyramid weight of the Dyck path. St000444The length of the maximal rise of a Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St000306The bounce count of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. 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 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. St000326The position of the first one in a binary word after appending a 1 at the end. St000383The last part of an integer composition. St000297The number of leading ones in a binary word. St000054The first entry of the permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000675The number of centered multitunnels of a Dyck path. St001462The number of factors of a standard tableaux under concatenation. St000806The semiperimeter of the associated bargraph. St001933The largest multiplicity of a part in an integer partition. St001176The size of a partition minus its first part. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000068The number of minimal elements in a poset. St001118The acyclic chromatic index of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000925The number of topologically connected components of a set partition. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001780The order of promotion on the set of standard tableaux of given shape. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001175The size of a partition minus the hook length of the base cell. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000007The number of saliances of the permutation. St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000567The sum of the products of all pairs of parts. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000237The number of small exceedances. St000546The number of global descents of a permutation. St000759The smallest missing part in an integer partition. St000843The decomposition number of a perfect matching. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000504The cardinality of the first block of a set partition. St000971The smallest closer of a set partition. St001733The number of weak left to right maxima of a Dyck path. St000617The number of global maxima of a Dyck path. St001050The number of terminal closers of a set partition. St000069The number of maximal elements of a poset. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St001644The dimension of a graph. St000025The number of initial rises of a Dyck path. St000172The Grundy number of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001029The size of the core of a graph. St001116The game chromatic number of a graph. St001322The size of a minimal independent dominating set in a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000234The number of global ascents of a permutation. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000502The number of successions of a set partitions. St000536The pathwidth of a graph. St001971The number of negative eigenvalues of the adjacency matrix of the graph. St000026The position of the first return of a Dyck path. St000363The number of minimal vertex covers of a graph. St000469The distinguishing number of a graph. St000686The finitistic dominant dimension of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. 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:
St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001829The common independence number of a graph. St001883The mutual visibility number of a graph. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000171The degree of the graph. St000454The largest eigenvalue of a graph if it is integral. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001120The length of a longest path in a graph. St001189The number of simple modules with dominant and codominant dimension equal to zero 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. St001777The number of weak descents in an integer composition. St001949The rigidity index of a graph. St001479The number of bridges of a graph. 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. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000822The Hadwiger number of the graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001725The harmonious chromatic number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001270The bandwidth of a graph. St001826The maximal number of leaves on a vertex of a graph. St001962The proper pathwidth of a graph. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001316The domatic number of a graph. St001812The biclique partition number of a graph. St000031The number of cycles in the cycle decomposition of a permutation. St000286The number of connected components of the complement of a graph. St000052The number of valleys of a Dyck path not on the x-axis. St000654The first descent of a permutation. St000475The number of parts equal to 1 in a partition. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001461The number of topologically connected components of the chord diagram of a permutation. St000989The number of final rises of a permutation. St000883The number of longest increasing subsequences of a permutation. St000676The number of odd rises of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000740The last entry of a permutation. St000214The number of adjacencies of a permutation. St000553The number of blocks of a graph. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000203The number of external nodes of a binary tree. St000738The first entry in the last row of a standard tableau. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000741The Colin de Verdière graph invariant. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. 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. St000287The number of connected components of a graph. St000314The number of left-to-right-maxima of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000542The number of left-to-right-minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000087The number of induced subgraphs. St000335The difference of lower and upper interactions. St000352The Elizalde-Pak rank of a permutation. St000443The number of long tunnels of a Dyck path. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000926The clique-coclique number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001342The number of vertices in the center of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001481The minimal height of a peak of a Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001530The depth of a Dyck path. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. 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. St001746The coalition number of a graph. St001828The Euler characteristic of a graph. St000051The size of the left subtree of a binary tree. St000133The "bounce" 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. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000734The last entry in the first row of a standard tableau. St001119The length of a shortest maximal path in a graph. 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 $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. 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. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001702The absolute value of the determinant of the adjacency matrix of a graph. St000061The number of nodes on the left branch of a binary tree. St000990The first ascent of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001959The product of the heights of the peaks of a Dyck path. St001480The number of simple summands of the module J^2/J^3. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000648The number of 2-excedences of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000456The monochromatic index of a connected graph. 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 $eA$. St000924The number of topologically connected components of a perfect matching. St000022The number of fixed points of a permutation. St001545The second Elser number of a connected graph. St001651The Frankl number of a lattice. 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. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000241The number of cyclical small excedances. St000338The number of pixed points of a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001552The number of inversions between excedances and fixed points of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000386The number of factors DDU in a Dyck path. St000742The number of big ascents of a permutation after prepending zero. St000871The number of very big ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001862The number of crossings of a signed permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001889The size of the connectivity set of a signed permutation. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000649The number of 3-excedences of a permutation. St000942The number of critical left to right maxima of the parking functions. 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. St001621The number of atoms of a lattice. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St001267The length of the Lyndon factorization of the binary word. St001557The number of inversions of the second entry of a permutation. 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.
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