Your data matches 174 different statistics following compositions of up to 3 maps.
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Mp00324: Graphs chromatic difference sequenceInteger compositions
Mp00040: Integer compositions to partitionInteger partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1]
=> 1
([],2)
=> [2] => [2]
=> 1
([(0,1)],2)
=> [1,1] => [1,1]
=> 2
([],3)
=> [3] => [3]
=> 1
([(1,2)],3)
=> [2,1] => [2,1]
=> 2
([(0,2),(1,2)],3)
=> [2,1] => [2,1]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1]
=> 3
([],4)
=> [4] => [4]
=> 1
([(2,3)],4)
=> [3,1] => [3,1]
=> 2
([(1,3),(2,3)],4)
=> [3,1] => [3,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [3,1]
=> 2
([(0,3),(1,2)],4)
=> [2,2] => [2,2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [2,2]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [2,1,1]
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [2,1,1]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [2,2]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [2,1,1]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1]
=> 4
([],5)
=> [5] => [5]
=> 1
([(3,4)],5)
=> [4,1] => [4,1]
=> 2
([(2,4),(3,4)],5)
=> [4,1] => [4,1]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [4,1]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [4,1]
=> 2
([(1,4),(2,3)],5)
=> [3,2] => [3,2]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [3,2]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [3,2]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [3,2]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [3,2]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [3,2]
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [3,2]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [3,2]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,2,1]
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [2,1,1,1]
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [2,1,1,1]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [2,1,1,1]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [2,2,1]
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 3
Description
The length of the partition.
Mp00324: Graphs chromatic difference sequenceInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1,0]
=> 1
([],2)
=> [2] => [1,1,0,0]
=> 1
([(0,1)],2)
=> [1,1] => [1,0,1,0]
=> 2
([],3)
=> [3] => [1,1,1,0,0,0]
=> 1
([(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> 2
([(0,2),(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> 3
([],4)
=> [4] => [1,1,1,1,0,0,0,0]
=> 1
([(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 2
([(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 2
([(0,3),(1,2)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
([],5)
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
([(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
([(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
([(1,4),(2,3)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
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.
Mp00324: Graphs chromatic difference sequenceInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000097: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => ([],1)
=> 1
([],2)
=> [2] => ([],2)
=> 1
([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 2
([],3)
=> [3] => ([],3)
=> 1
([(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
([],4)
=> [4] => ([],4)
=> 1
([(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
([(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],5)
=> [5] => ([],5)
=> 1
([(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
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.
Mp00324: Graphs chromatic difference sequenceInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000098: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => ([],1)
=> 1
([],2)
=> [2] => ([],2)
=> 1
([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 2
([],3)
=> [3] => ([],3)
=> 1
([(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
([],4)
=> [4] => ([],4)
=> 1
([(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
([(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],5)
=> [5] => ([],5)
=> 1
([(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
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.
Mp00324: Graphs chromatic difference sequenceInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000172: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => ([],1)
=> 1
([],2)
=> [2] => ([],2)
=> 1
([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 2
([],3)
=> [3] => ([],3)
=> 1
([(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
([],4)
=> [4] => ([],4)
=> 1
([(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
([(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],5)
=> [5] => ([],5)
=> 1
([(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
Description
The Grundy number of a graph. The Grundy number $\Gamma(G)$ is defined to be the largest $k$ such that $G$ admits a greedy $k$-coloring. Any order of the vertices of $G$ induces a greedy coloring by assigning to the $i$-th vertex in this order the smallest positive integer such that the partial coloring remains a proper coloring. In particular, we have that $\chi(G) \leq \Gamma(G) \leq \Delta(G) + 1$, where $\chi(G)$ is the chromatic number of $G$ ([[St000098]]), and where $\Delta(G)$ is the maximal degree of a vertex of $G$ ([[St000171]]).
Mp00324: Graphs chromatic difference sequenceInteger compositions
Mp00094: Integer compositions to binary wordBinary words
St000288: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => 1 => 1
([],2)
=> [2] => 10 => 1
([(0,1)],2)
=> [1,1] => 11 => 2
([],3)
=> [3] => 100 => 1
([(1,2)],3)
=> [2,1] => 101 => 2
([(0,2),(1,2)],3)
=> [2,1] => 101 => 2
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => 111 => 3
([],4)
=> [4] => 1000 => 1
([(2,3)],4)
=> [3,1] => 1001 => 2
([(1,3),(2,3)],4)
=> [3,1] => 1001 => 2
([(0,3),(1,3),(2,3)],4)
=> [3,1] => 1001 => 2
([(0,3),(1,2)],4)
=> [2,2] => 1010 => 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => 1010 => 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 1011 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 1011 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => 1010 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 1011 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 4
([],5)
=> [5] => 10000 => 1
([(3,4)],5)
=> [4,1] => 10001 => 2
([(2,4),(3,4)],5)
=> [4,1] => 10001 => 2
([(1,4),(2,4),(3,4)],5)
=> [4,1] => 10001 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => 10001 => 2
([(1,4),(2,3)],5)
=> [3,2] => 10010 => 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => 10010 => 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => 10010 => 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 10011 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => 10010 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 10011 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 10011 => 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 10010 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => 10010 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 10011 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 10011 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 10011 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 10010 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 10011 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => 10010 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 10101 => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 10101 => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 10101 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => 10101 => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 10101 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 10101 => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 10101 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 10111 => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 10111 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 10111 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => 10101 => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 10101 => 3
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Mp00324: Graphs chromatic difference sequenceInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St001029: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => ([],1)
=> 1
([],2)
=> [2] => ([],2)
=> 1
([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 2
([],3)
=> [3] => ([],3)
=> 1
([(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
([],4)
=> [4] => ([],4)
=> 1
([(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
([(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],5)
=> [5] => ([],5)
=> 1
([(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
Description
The size of the core of a graph. The core of the graph $G$ is the smallest graph $C$ such that there is a graph homomorphism from $G$ to $C$ and a graph homomorphism from $C$ to $G$.
Mp00324: Graphs chromatic difference sequenceInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1,0]
=> 1
([],2)
=> [2] => [1,1,0,0]
=> 1
([(0,1)],2)
=> [1,1] => [1,0,1,0]
=> 2
([],3)
=> [3] => [1,1,1,0,0,0]
=> 1
([(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> 2
([(0,2),(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> 3
([],4)
=> [4] => [1,1,1,1,0,0,0,0]
=> 1
([(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 2
([(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 2
([(0,3),(1,2)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
([],5)
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
([(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
([(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
([(1,4),(2,3)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Mp00324: Graphs chromatic difference sequenceInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
St001203: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1,0]
=> 1
([],2)
=> [2] => [1,1,0,0]
=> 1
([(0,1)],2)
=> [1,1] => [1,0,1,0]
=> 2
([],3)
=> [3] => [1,1,1,0,0,0]
=> 1
([(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> 2
([(0,2),(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> 3
([],4)
=> [4] => [1,1,1,1,0,0,0,0]
=> 1
([(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 2
([(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 2
([(0,3),(1,2)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
([],5)
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
([(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
([(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
([(1,4),(2,3)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
Description
We 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: In the list $L$ delete the first entry $c_0$ and substract from all other entries $n-1$ and then append the last element 1 (this was suggested by Christian Stump). The result is a Kupisch series of an LNakayama algebra. Example: [5,6,6,6,6] goes into [2,2,2,2,1]. Now associate to the CNakayama algebra with the above properties the Dyck path corresponding to the Kupisch series of the LNakayama algebra. The statistic return the global dimension of the CNakayama algebra divided by 2.
Mp00324: Graphs chromatic difference sequenceInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St001494: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => ([],1)
=> 1
([],2)
=> [2] => ([],2)
=> 1
([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 2
([],3)
=> [3] => ([],3)
=> 1
([(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
([],4)
=> [4] => ([],4)
=> 1
([(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
([(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],5)
=> [5] => ([],5)
=> 1
([(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
Description
The Alon-Tarsi number of a graph. Let $G$ be a graph with vertices $\{1,\dots,n\}$ and edge set $E$. Let $P_G=\prod_{i < j, (i,j)\in E} x_i-x_j$ be its graph polynomial. Then the Alon-Tarsi number is the smallest number $k$ such that $P_G$ contains a monomial with exponents strictly less than $k$.
The following 164 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St000053The number of valleys of the Dyck path. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. 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. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000228The size of a partition. St000733The row containing the largest entry of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001007Number of simple modules with projective dimension 1 in 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. St001116The game chromatic number of a graph. St001437The flex of a binary word. St000024The number of double up and double down steps of a Dyck path. St000157The number of descents of a standard tableau. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. 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. 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. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000306The bounce count of a Dyck path. St000264The girth of a graph, which is not a tree. St000259The diameter of a connected graph. St001875The number of simple modules with projective dimension at most 1. St001060The distinguishing index of a graph. St000741The Colin de Verdière graph invariant. St001644The dimension of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001480The number of simple summands of the module J^2/J^3. St000144The pyramid weight of the Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000822The Hadwiger number of the graph. St000260The radius of a connected graph. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000015The number of peaks of a Dyck path. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001530The depth of a Dyck path. St000443The number of long tunnels of a Dyck path. 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. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000636The hull number of a graph. St001108The 2-dynamic chromatic number of a graph. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001270The bandwidth of a graph. St001962The proper pathwidth of a graph. St000482The (zero)-forcing number of a graph. St000820The number of compositions obtained by rotating the composition. St000918The 2-limited packing number of a graph. 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. St001315The dissociation number of a graph. St001655The general position number of a graph. St001672The restrained domination number of 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. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001746The coalition number of a graph. St001777The number of weak descents in an integer composition. St000087The number of induced subgraphs. St000286The number of connected components of the complement of a graph. St000363The number of minimal vertex covers of a graph. St000384The maximal part of the shifted composition of an integer partition. St000393The number of strictly increasing runs in a binary word. St000469The distinguishing number of a graph. St000722The number of different neighbourhoods in a graph. 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. St000926The clique-coclique number of a graph. St001110The 3-dynamic chromatic number of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001286The annihilation number of a graph. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001656The monophonic position number of a graph. St001675The number of parts equal to the part in the reversed composition. 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. St001725The harmonious chromatic number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St000120The number of left tunnels of a Dyck path. St000171The degree of the graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. 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. St001321The number of vertices of the largest induced subforest of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001391The disjunction number of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001949The rigidity index of a graph. St001812The biclique partition number of a graph. St001331The size of the minimal feedback vertex set. St001742The difference of the maximal and the minimal degree in a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000479The Ramsey number of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001834The number of non-isomorphic minors of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001323The independence gap of a 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 $eA$. St001638The book thickness of a graph. St000738The first entry in the last row of a standard tableau. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. 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 $eA$. St001570The minimal number of edges to add to make a graph Hamiltonian. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St000474Dyson's crank of a partition. St001118The acyclic chromatic index of a graph. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000933The number of multipartitions of sizes given by an integer partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000455The second largest eigenvalue of a graph if it is integral. St001621The number of atoms of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001624The breadth of a lattice. St000422The energy of a graph, if it is integral. St001336The minimal number of vertices in a graph whose complement is triangle-free. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000776The maximal multiplicity of an eigenvalue in a graph. St000910The number of maximal chains of minimal length in a poset. St001668The number of points of the poset minus the width of the poset.