Your data matches 317 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000097
(load all 5 compositions to match this statistic)
Mapping
Mp00250: Graphs —clique graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 1
([],2)
 => ([],2)
 => 1
([(0,1)],2)
 => ([],1)
 => 1
([],3)
 => ([],3)
 => 1
([(1,2)],3)
 => ([],2)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 1
([],4)
 => ([],4)
 => 1
([(2,3)],4)
 => ([],3)
 => 1
([(0,3),(1,2)],4)
 => ([],2)
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1
([],5)
 => ([],5)
 => 1
([(3,4)],5)
 => ([],4)
 => 1
([(1,4),(2,3)],5)
 => ([],3)
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
([],6)
 => ([],6)
 => 1
([(4,5)],6)
 => ([],5)
 => 1
([(2,5),(3,4)],6)
 => ([],4)
 => 1
([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => 1
([(0,5),(1,4),(2,3)],6)
 => ([],3)
 => 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 1
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([],2)
 => 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => 1
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => 1
([(5,6)],7)
 => ([],6)
 => 1
([(3,6),(4,5)],7)
 => ([],5)
 => 1
([(4,5),(4,6),(5,6)],7)
 => ([],5)
 => 1
([(1,6),(2,5),(3,4)],7)
 => ([],4)
 => 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => 1
([(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
 => ([],3)
 => 1
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => 1
([(0,6),(1,2),(1,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => 2
([(0,1),(0,5),(1,4),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 2
([(0,4),(0,6),(1,3),(1,5),(2,5),(2,6),(3,4),(3,5),(4,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(0,4),(0,5),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(0,1),(0,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
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: St001716
(load all 11 compositions to match this statistic)
Mapping
Mp00250: Graphs —clique graph⟶ Graphs
St001716: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 1
([],2)
 => ([],2)
 => 1
([(0,1)],2)
 => ([],1)
 => 1
([],3)
 => ([],3)
 => 1
([(1,2)],3)
 => ([],2)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 1
([],4)
 => ([],4)
 => 1
([(2,3)],4)
 => ([],3)
 => 1
([(0,3),(1,2)],4)
 => ([],2)
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1
([],5)
 => ([],5)
 => 1
([(3,4)],5)
 => ([],4)
 => 1
([(1,4),(2,3)],5)
 => ([],3)
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
([],6)
 => ([],6)
 => 1
([(4,5)],6)
 => ([],5)
 => 1
([(2,5),(3,4)],6)
 => ([],4)
 => 1
([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => 1
([(0,5),(1,4),(2,3)],6)
 => ([],3)
 => 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 1
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([],2)
 => 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => 1
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => 1
([(5,6)],7)
 => ([],6)
 => 1
([(3,6),(4,5)],7)
 => ([],5)
 => 1
([(4,5),(4,6),(5,6)],7)
 => ([],5)
 => 1
([(1,6),(2,5),(3,4)],7)
 => ([],4)
 => 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => 1
([(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
 => ([],3)
 => 1
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => 1
([(0,6),(1,2),(1,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => 2
([(0,1),(0,5),(1,4),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 2
([(0,4),(0,6),(1,3),(1,5),(2,5),(2,6),(3,4),(3,5),(4,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(0,4),(0,5),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(0,1),(0,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
Description
The 1-improper chromatic number of a graph. This is the least number of colours in a vertex-colouring, such that each vertex has at most one neighbour with the same colour.
Matching statistic: St001776
(load all 18 compositions to match this statistic)
Mapping
Mp00250: Graphs —clique graph⟶ Graphs
St001776: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 1
([],2)
 => ([],2)
 => 1
([(0,1)],2)
 => ([],1)
 => 1
([],3)
 => ([],3)
 => 1
([(1,2)],3)
 => ([],2)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 1
([],4)
 => ([],4)
 => 1
([(2,3)],4)
 => ([],3)
 => 1
([(0,3),(1,2)],4)
 => ([],2)
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1
([],5)
 => ([],5)
 => 1
([(3,4)],5)
 => ([],4)
 => 1
([(1,4),(2,3)],5)
 => ([],3)
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
([],6)
 => ([],6)
 => 1
([(4,5)],6)
 => ([],5)
 => 1
([(2,5),(3,4)],6)
 => ([],4)
 => 1
([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => 1
([(0,5),(1,4),(2,3)],6)
 => ([],3)
 => 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 1
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([],2)
 => 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => 1
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => 1
([(5,6)],7)
 => ([],6)
 => 1
([(3,6),(4,5)],7)
 => ([],5)
 => 1
([(4,5),(4,6),(5,6)],7)
 => ([],5)
 => 1
([(1,6),(2,5),(3,4)],7)
 => ([],4)
 => 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => 1
([(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
 => ([],3)
 => 1
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => 1
([(0,6),(1,2),(1,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => 2
([(0,1),(0,5),(1,4),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 2
([(0,4),(0,6),(1,3),(1,5),(2,5),(2,6),(3,4),(3,5),(4,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(0,4),(0,5),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(0,1),(0,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
Description
The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph.
Matching statistic: St001325
(load all 48 compositions to match this statistic)
Mapping
Mp00250: Graphs —clique graph⟶ Graphs
St001325: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 0 = 1 - 1
([],2)
 => ([],2)
 => 0 = 1 - 1
([(0,1)],2)
 => ([],1)
 => 0 = 1 - 1
([],3)
 => ([],3)
 => 0 = 1 - 1
([(1,2)],3)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0 = 1 - 1
([],4)
 => ([],4)
 => 0 = 1 - 1
([(2,3)],4)
 => ([],3)
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => ([],2)
 => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0 = 1 - 1
([],5)
 => ([],5)
 => 0 = 1 - 1
([(3,4)],5)
 => ([],4)
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => ([],3)
 => 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
([],6)
 => ([],6)
 => 0 = 1 - 1
([(4,5)],6)
 => ([],5)
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => ([],4)
 => 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
 => ([],3)
 => 0 = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 0 = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1 = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 0 = 1 - 1
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => 0 = 1 - 1
([(5,6)],7)
 => ([],6)
 => 0 = 1 - 1
([(3,6),(4,5)],7)
 => ([],5)
 => 0 = 1 - 1
([(4,5),(4,6),(5,6)],7)
 => ([],5)
 => 0 = 1 - 1
([(1,6),(2,5),(3,4)],7)
 => ([],4)
 => 0 = 1 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => 0 = 1 - 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => 0 = 1 - 1
([(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1 = 2 - 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => 0 = 1 - 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1 = 2 - 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
 => ([],3)
 => 0 = 1 - 1
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => 0 = 1 - 1
([(0,6),(1,2),(1,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 2 - 1
([(0,1),(0,5),(1,4),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1 = 2 - 1
([(0,4),(0,6),(1,3),(1,5),(2,5),(2,6),(3,4),(3,5),(4,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(0,4),(0,5),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(0,1),(0,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
Description
The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. A graph is a comparability graph if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ and $(b,c)$ are edges and $(a,c)$ is not an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001328
(load all 7 compositions to match this statistic)
Mapping
Mp00250: Graphs —clique graph⟶ Graphs
St001328: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 0 = 1 - 1
([],2)
 => ([],2)
 => 0 = 1 - 1
([(0,1)],2)
 => ([],1)
 => 0 = 1 - 1
([],3)
 => ([],3)
 => 0 = 1 - 1
([(1,2)],3)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0 = 1 - 1
([],4)
 => ([],4)
 => 0 = 1 - 1
([(2,3)],4)
 => ([],3)
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => ([],2)
 => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0 = 1 - 1
([],5)
 => ([],5)
 => 0 = 1 - 1
([(3,4)],5)
 => ([],4)
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => ([],3)
 => 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
([],6)
 => ([],6)
 => 0 = 1 - 1
([(4,5)],6)
 => ([],5)
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => ([],4)
 => 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
 => ([],3)
 => 0 = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 0 = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1 = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 0 = 1 - 1
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => 0 = 1 - 1
([(5,6)],7)
 => ([],6)
 => 0 = 1 - 1
([(3,6),(4,5)],7)
 => ([],5)
 => 0 = 1 - 1
([(4,5),(4,6),(5,6)],7)
 => ([],5)
 => 0 = 1 - 1
([(1,6),(2,5),(3,4)],7)
 => ([],4)
 => 0 = 1 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => 0 = 1 - 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => 0 = 1 - 1
([(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1 = 2 - 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => 0 = 1 - 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1 = 2 - 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
 => ([],3)
 => 0 = 1 - 1
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => 0 = 1 - 1
([(0,6),(1,2),(1,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 2 - 1
([(0,1),(0,5),(1,4),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1 = 2 - 1
([(0,4),(0,6),(1,3),(1,5),(2,5),(2,6),(3,4),(3,5),(4,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(0,4),(0,5),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(0,1),(0,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
Description
The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. A graph is bipartite if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ and $(b,c)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001331
(load all 7 compositions to match this statistic)
Mapping
Mp00250: Graphs —clique graph⟶ Graphs
St001331: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 0 = 1 - 1
([],2)
 => ([],2)
 => 0 = 1 - 1
([(0,1)],2)
 => ([],1)
 => 0 = 1 - 1
([],3)
 => ([],3)
 => 0 = 1 - 1
([(1,2)],3)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0 = 1 - 1
([],4)
 => ([],4)
 => 0 = 1 - 1
([(2,3)],4)
 => ([],3)
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => ([],2)
 => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0 = 1 - 1
([],5)
 => ([],5)
 => 0 = 1 - 1
([(3,4)],5)
 => ([],4)
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => ([],3)
 => 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
([],6)
 => ([],6)
 => 0 = 1 - 1
([(4,5)],6)
 => ([],5)
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => ([],4)
 => 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
 => ([],3)
 => 0 = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 0 = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1 = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 0 = 1 - 1
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => 0 = 1 - 1
([(5,6)],7)
 => ([],6)
 => 0 = 1 - 1
([(3,6),(4,5)],7)
 => ([],5)
 => 0 = 1 - 1
([(4,5),(4,6),(5,6)],7)
 => ([],5)
 => 0 = 1 - 1
([(1,6),(2,5),(3,4)],7)
 => ([],4)
 => 0 = 1 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => 0 = 1 - 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => 0 = 1 - 1
([(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1 = 2 - 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => 0 = 1 - 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1 = 2 - 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
 => ([],3)
 => 0 = 1 - 1
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => 0 = 1 - 1
([(0,6),(1,2),(1,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 2 - 1
([(0,1),(0,5),(1,4),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1 = 2 - 1
([(0,4),(0,6),(1,3),(1,5),(2,5),(2,6),(3,4),(3,5),(4,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(0,4),(0,5),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(0,1),(0,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
Description
The size of the minimal feedback vertex set. A feedback vertex set is a set of vertices whose removal results in an acyclic graph.
Matching statistic: St001335
(load all 16 compositions to match this statistic)
Mapping
Mp00250: Graphs —clique graph⟶ Graphs
St001335: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 0 = 1 - 1
([],2)
 => ([],2)
 => 0 = 1 - 1
([(0,1)],2)
 => ([],1)
 => 0 = 1 - 1
([],3)
 => ([],3)
 => 0 = 1 - 1
([(1,2)],3)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0 = 1 - 1
([],4)
 => ([],4)
 => 0 = 1 - 1
([(2,3)],4)
 => ([],3)
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => ([],2)
 => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0 = 1 - 1
([],5)
 => ([],5)
 => 0 = 1 - 1
([(3,4)],5)
 => ([],4)
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => ([],3)
 => 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
([],6)
 => ([],6)
 => 0 = 1 - 1
([(4,5)],6)
 => ([],5)
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => ([],4)
 => 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
 => ([],3)
 => 0 = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 0 = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1 = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 0 = 1 - 1
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => 0 = 1 - 1
([(5,6)],7)
 => ([],6)
 => 0 = 1 - 1
([(3,6),(4,5)],7)
 => ([],5)
 => 0 = 1 - 1
([(4,5),(4,6),(5,6)],7)
 => ([],5)
 => 0 = 1 - 1
([(1,6),(2,5),(3,4)],7)
 => ([],4)
 => 0 = 1 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => 0 = 1 - 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => 0 = 1 - 1
([(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1 = 2 - 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => 0 = 1 - 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1 = 2 - 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
 => ([],3)
 => 0 = 1 - 1
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => 0 = 1 - 1
([(0,6),(1,2),(1,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 2 - 1
([(0,1),(0,5),(1,4),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1 = 2 - 1
([(0,4),(0,6),(1,3),(1,5),(2,5),(2,6),(3,4),(3,5),(4,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(0,4),(0,5),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(0,1),(0,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
Description
The cardinality of a minimal cycle-isolating set of a graph. Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$. This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains all cycles.
Matching statistic: St001572
(load all 11 compositions to match this statistic)
Mapping
Mp00250: Graphs —clique graph⟶ Graphs
St001572: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 0 = 1 - 1
([],2)
 => ([],2)
 => 0 = 1 - 1
([(0,1)],2)
 => ([],1)
 => 0 = 1 - 1
([],3)
 => ([],3)
 => 0 = 1 - 1
([(1,2)],3)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0 = 1 - 1
([],4)
 => ([],4)
 => 0 = 1 - 1
([(2,3)],4)
 => ([],3)
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => ([],2)
 => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0 = 1 - 1
([],5)
 => ([],5)
 => 0 = 1 - 1
([(3,4)],5)
 => ([],4)
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => ([],3)
 => 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
([],6)
 => ([],6)
 => 0 = 1 - 1
([(4,5)],6)
 => ([],5)
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => ([],4)
 => 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
 => ([],3)
 => 0 = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 0 = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1 = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 0 = 1 - 1
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => 0 = 1 - 1
([(5,6)],7)
 => ([],6)
 => 0 = 1 - 1
([(3,6),(4,5)],7)
 => ([],5)
 => 0 = 1 - 1
([(4,5),(4,6),(5,6)],7)
 => ([],5)
 => 0 = 1 - 1
([(1,6),(2,5),(3,4)],7)
 => ([],4)
 => 0 = 1 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => 0 = 1 - 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => 0 = 1 - 1
([(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1 = 2 - 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => 0 = 1 - 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1 = 2 - 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
 => ([],3)
 => 0 = 1 - 1
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => 0 = 1 - 1
([(0,6),(1,2),(1,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 2 - 1
([(0,1),(0,5),(1,4),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1 = 2 - 1
([(0,4),(0,6),(1,3),(1,5),(2,5),(2,6),(3,4),(3,5),(4,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(0,4),(0,5),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(0,1),(0,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
Description
The minimal number of edges to remove to make a graph bipartite.
Matching statistic: St001793
(load all 18 compositions to match this statistic)
Mapping
Mp00250: Graphs —clique graph⟶ Graphs
St001793: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 0 = 1 - 1
([],2)
 => ([],2)
 => 0 = 1 - 1
([(0,1)],2)
 => ([],1)
 => 0 = 1 - 1
([],3)
 => ([],3)
 => 0 = 1 - 1
([(1,2)],3)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0 = 1 - 1
([],4)
 => ([],4)
 => 0 = 1 - 1
([(2,3)],4)
 => ([],3)
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => ([],2)
 => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0 = 1 - 1
([],5)
 => ([],5)
 => 0 = 1 - 1
([(3,4)],5)
 => ([],4)
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => ([],3)
 => 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
([],6)
 => ([],6)
 => 0 = 1 - 1
([(4,5)],6)
 => ([],5)
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => ([],4)
 => 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
 => ([],3)
 => 0 = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 0 = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1 = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 0 = 1 - 1
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => 0 = 1 - 1
([(5,6)],7)
 => ([],6)
 => 0 = 1 - 1
([(3,6),(4,5)],7)
 => ([],5)
 => 0 = 1 - 1
([(4,5),(4,6),(5,6)],7)
 => ([],5)
 => 0 = 1 - 1
([(1,6),(2,5),(3,4)],7)
 => ([],4)
 => 0 = 1 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => 0 = 1 - 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => 0 = 1 - 1
([(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1 = 2 - 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => 0 = 1 - 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1 = 2 - 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
 => ([],3)
 => 0 = 1 - 1
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => 0 = 1 - 1
([(0,6),(1,2),(1,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 2 - 1
([(0,1),(0,5),(1,4),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1 = 2 - 1
([(0,4),(0,6),(1,3),(1,5),(2,5),(2,6),(3,4),(3,5),(4,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(0,4),(0,5),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
([(0,1),(0,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1 = 2 - 1
Description
The difference between the clique number and the chromatic number of a graph. A graph whose clique number and chromatic number coincide are called weakly perfect.
Matching statistic: St000093
(load all 2 compositions to match this statistic)
Mapping
Mp00250: Graphs —clique graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => ([],1)
 => 1
([],2)
 => ([],2)
 => ([],1)
 => 1
([(0,1)],2)
 => ([],1)
 => ([],1)
 => 1
([],3)
 => ([],3)
 => ([],1)
 => 1
([(1,2)],3)
 => ([],2)
 => ([],1)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => 1
([],4)
 => ([],4)
 => ([],1)
 => 1
([(2,3)],4)
 => ([],3)
 => ([],1)
 => 1
([(0,3),(1,2)],4)
 => ([],2)
 => ([],1)
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 1
([],5)
 => ([],5)
 => ([],1)
 => 1
([(3,4)],5)
 => ([],4)
 => ([],1)
 => 1
([(1,4),(2,3)],5)
 => ([],3)
 => ([],1)
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([],1)
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ([],1)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => 1
([],6)
 => ([],6)
 => ([],1)
 => 1
([(4,5)],6)
 => ([],5)
 => ([],1)
 => 1
([(2,5),(3,4)],6)
 => ([],4)
 => ([],1)
 => 1
([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => ([],1)
 => 1
([(0,5),(1,4),(2,3)],6)
 => ([],3)
 => ([],1)
 => 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => ([],1)
 => 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => ([],1)
 => 1
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([],2)
 => ([],1)
 => 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => ([],1)
 => 1
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ([],1)
 => 1
([(5,6)],7)
 => ([],6)
 => ([],1)
 => 1
([(3,6),(4,5)],7)
 => ([],5)
 => ([],1)
 => 1
([(4,5),(4,6),(5,6)],7)
 => ([],5)
 => ([],1)
 => 1
([(1,6),(2,5),(3,4)],7)
 => ([],4)
 => ([],1)
 => 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => ([],1)
 => 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => ([],1)
 => 1
([(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => ([],1)
 => 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
 => ([],3)
 => ([],1)
 => 1
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => ([],1)
 => 1
([(0,6),(1,2),(1,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(0,1),(0,5),(1,4),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(0,4),(0,6),(1,3),(1,5),(2,5),(2,6),(3,4),(3,5),(4,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(0,4),(0,5),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(0,1),(0,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
Description
The cardinality of a maximal independent set of vertices of a graph. An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
The following 307 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000268The number of strongly connected orientations of a graph. St000273The domination number of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000482The (zero)-forcing number of a graph. St000535The rank-width of a graph. St000544The cop number of a graph. St000758The length of the longest staircase fitting into an integer composition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000917The open packing number of a graph. St001073The number of nowhere zero 3-flows of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001286The annihilation number of a graph. St001316The domatic number of a graph. St001322The size of a minimal independent dominating set in a graph. St001330The hat guessing number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001358The largest degree of a regular subgraph of a graph. St001654The monophonic hull number of a graph. St001656The monophonic position number of a graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001829The common independence number of a graph. St001957The number of Hasse diagrams with a given underlying undirected graph. St000256The number of parts from which one can substract 2 and still get an integer partition. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000671The maximin edge-connectivity for choosing a subgraph. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001071The beta invariant of the graph. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001261The Castelnuovo-Mumford regularity of a graph. St001280The number of parts of an integer partition that are at least two. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001349The number of different graphs obtained from the given graph by removing an edge. St001353The number of prime nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001638The book thickness of a graph. St001673The degree of asymmetry of an integer composition. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001736The total number of cycles in a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001797The number of overfull subgraphs of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St000143The largest repeated part of a partition. St000257The number of distinct parts of a partition that occur at least twice. St000481The number of upper covers of a partition in dominance order. St001568The smallest positive integer that does not appear twice in the partition. St001871The number of triconnected components of a graph. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000422The energy of a graph, if it is integral. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001271The competition number of a graph. St000741The Colin de Verdière graph invariant. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001256Number of simple reflexive modules that are 2-stable reflexive. 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 $eA$. St001204Call 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. 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 $eA$. St001651The Frankl number of a lattice. St000455The second largest eigenvalue of a graph if it is integral. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. St000056The decomposition (or block) number of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001461The number of topologically connected components of the chord diagram of a permutation. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000623The number of occurrences of the pattern 52341 in a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000787The number of flips required to make a perfect matching noncrossing. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001381The fertility of a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001720The minimal length of a chain of small intervals in a lattice. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St000475The number of parts equal to 1 in a partition. St000929The constant term of the character polynomial of an integer partition. St000069The number of maximal elements of a poset. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001578The minimal number of edges to add or remove to make a graph a line graph. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000264The girth of a graph, which is not a tree. St000781The number of proper colouring schemes of a Ferrers diagram. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. 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. St001175The size of a partition minus the hook length of the base cell. St001570The minimal number of edges to add to make a graph Hamiltonian. St000068The number of minimal elements in a poset. 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 $eA$. St000181The number of connected components of the Hasse diagram for the poset. St000908The length of the shortest maximal antichain in a poset. St001301The first Betti number of the order complex associated with the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001626The number of maximal proper sublattices of a lattice. St001856The number of edges in the reduced word graph of a permutation. St001118The acyclic chromatic index of a graph. St000456The monochromatic index of a connected graph. St001281The normalized isoperimetric number of a graph. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001592The maximal number of simple paths between any two different vertices of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000379The number of Hamiltonian cycles in a graph. St001060The distinguishing index of a graph. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001093The detour number of a graph. St000699The toughness times the least common multiple of 1,. St001545The second Elser number of a connected graph. St000464The Schultz index of a connected graph. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001960The number of descents of a permutation minus one if its first entry is not one. St001890The maximum magnitude of the Möbius function of a poset. St001703The villainy of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001363The Euler characteristic of a graph according to Knill. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in a graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001356The number of vertices in prime modules of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001479The number of bridges of a graph. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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 $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. 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. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001722The number of minimal chains with small intervals between a binary word and the top element. St000015The number of peaks of a Dyck path. St000117The number of centered tunnels of a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 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. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001530The depth of a Dyck path. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000553The number of blocks of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000914The sum of the values of the Möbius function of a poset. 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. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St001095The number of non-isomorphic posets with precisely one further covering relation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001272The number of graphs with the same degree sequence. St001307The number of induced stars on four vertices in a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001765The number of connected components of the friends and strangers graph. St000447The number of pairs of vertices of a graph with distance 3. St001877Number of indecomposable injective modules with projective dimension 2. 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. St001875The number of simple modules with projective dimension at most 1. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000286The number of connected components of the complement of a graph. St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St000096The number of spanning trees of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St001577The minimal number of edges to add or remove to make a graph a cograph. St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001645The pebbling number of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. 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. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000939The number of characters of the symmetric group whose value on the partition is positive. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001571The Cartan determinant of the integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer 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. St000937The number of positive values of the symmetric group character corresponding to the partition. St000944The 3-degree of an integer partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition.