- FindStat

Your data matches 379 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001578
(load all 6 compositions to match this statistic)

Mapping

St001578: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => 0 = 1 - 1
([],2)
 => 0 = 1 - 1
([(0,1)],2)
 => 0 = 1 - 1
([],3)
 => 0 = 1 - 1
([(1,2)],3)
 => 0 = 1 - 1
([(0,2),(1,2)],3)
 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => 0 = 1 - 1
([],4)
 => 0 = 1 - 1
([(2,3)],4)
 => 0 = 1 - 1
([(1,3),(2,3)],4)
 => 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
([],5)
 => 0 = 1 - 1
([(3,4)],5)
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(1,4),(2,3)],5)
 => 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
 => 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
 => 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1

Description

The minimal number of edges to add or remove to make a graph a line graph.

Matching statistic: St001774
(load all 7 compositions to match this statistic)

Mapping

Mp00264: Graphs —delete endpoints⟶ Graphs
St001774: 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
([(0,2),(1,2)],3)
 => ([],1)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([],4)
 => ([],4)
 => 1
([(2,3)],4)
 => ([],3)
 => 1
([(1,3),(2,3)],4)
 => ([],2)
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => 1
([(0,3),(1,2)],4)
 => ([],2)
 => 1
([(0,3),(1,2),(2,3)],4)
 => ([],1)
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([],5)
 => ([],5)
 => 1
([(3,4)],5)
 => ([],4)
 => 1
([(2,4),(3,4)],5)
 => ([],3)
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([],2)
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
([(1,4),(2,3)],5)
 => ([],3)
 => 1
([(1,4),(2,3),(3,4)],5)
 => ([],2)
 => 1
([(0,1),(2,4),(3,4)],5)
 => ([],2)
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([],1)
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
([(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),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1

Description

The degree of the minimal polynomial of the smallest eigenvalue of a graph.

Matching statistic: St000388
(load all 3 compositions to match this statistic)

Mapping

Mp00250: Graphs —clique graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000388: 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
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => 1
([],4)
 => ([],4)
 => ([],1)
 => 1
([(2,3)],4)
 => ([],3)
 => ([],1)
 => 1
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,3),(1,2)],4)
 => ([],2)
 => ([],1)
 => 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ([],1)
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 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
([(2,4),(3,4)],5)
 => ([(2,3)],4)
 => ([(1,2)],3)
 => 2
([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(1,4),(2,3)],5)
 => ([],3)
 => ([],1)
 => 1
([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 2
([(0,1),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([],1)
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ([],1)
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 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)
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ([],1)
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1

Description

The number of orbits of vertices of a graph under automorphisms.

Matching statistic: St001111
(load all 4 compositions to match this statistic)

Mapping

Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
St001111: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => ([],1)
 => 1
([],2)
 => ([],1)
 => ([],1)
 => 1
([(0,1)],2)
 => ([(0,1)],2)
 => ([],1)
 => 1
([],3)
 => ([],1)
 => ([],1)
 => 1
([(1,2)],3)
 => ([(1,2)],3)
 => ([],2)
 => 1
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 1
([],4)
 => ([],1)
 => ([],1)
 => 1
([(2,3)],4)
 => ([(1,2)],3)
 => ([],2)
 => 1
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ([],2)
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([],2)
 => 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 2
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1
([],5)
 => ([],1)
 => ([],1)
 => 1
([(3,4)],5)
 => ([(1,2)],3)
 => ([],2)
 => 1
([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([],2)
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([],2)
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([],3)
 => 1
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 2
([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ([],2)
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => ([],2)
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 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)
 => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 1

Description

The weak 2-dynamic chromatic number of a graph. A

$k$ -weak-dynamic coloring of a graph

$G$ is a (non-proper) coloring of

$G$ in such a way that each vertex

$v$ sees at least

$\min\{d(v), k\}$ colors in its neighborhood. The

$k$ -weak-dynamic number of a graph is the smallest number of colors needed to find an

$k$ -dynamic coloring. This statistic records the

$2$ -weak-dynamic number of a graph.

Matching statistic: St001112
(load all 3 compositions to match this statistic)

Mapping

Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
St001112: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => ([],1)
 => 1
([],2)
 => ([],1)
 => ([],1)
 => 1
([(0,1)],2)
 => ([(0,1)],2)
 => ([],1)
 => 1
([],3)
 => ([],1)
 => ([],1)
 => 1
([(1,2)],3)
 => ([(1,2)],3)
 => ([],2)
 => 1
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 1
([],4)
 => ([],1)
 => ([],1)
 => 1
([(2,3)],4)
 => ([(1,2)],3)
 => ([],2)
 => 1
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ([],2)
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([],2)
 => 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 2
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1
([],5)
 => ([],1)
 => ([],1)
 => 1
([(3,4)],5)
 => ([(1,2)],3)
 => ([],2)
 => 1
([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([],2)
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([],2)
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([],3)
 => 1
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 2
([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ([],2)
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => ([],2)
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 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)
 => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 1

Description

The 3-weak dynamic number of a graph. A

$k$ -weak-dynamic coloring of a graph

$G$ is a (non-proper) coloring of

$G$ in such a way that each vertex

$v$ sees at least

$\min\{d(v), k\}$ colors in its neighborhood. The

$k$ -weak-dynamic number of a graph is the smallest number of colors needed to find an

$k$ -dynamic coloring. This statistic records the

$3$ -weak-dynamic number of a graph.

Matching statistic: St001336

Mapping

Mp00250: Graphs —clique graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001336: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
([],2)
 => ([],2)
 => ([],1)
 => 0 = 1 - 1
([(0,1)],2)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
([],3)
 => ([],3)
 => ([],1)
 => 0 = 1 - 1
([(1,2)],3)
 => ([],2)
 => ([],1)
 => 0 = 1 - 1
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
([],4)
 => ([],4)
 => ([],1)
 => 0 = 1 - 1
([(2,3)],4)
 => ([],3)
 => ([],1)
 => 0 = 1 - 1
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => ([],2)
 => ([],1)
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ([],1)
 => 0 = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
([],5)
 => ([],5)
 => ([],1)
 => 0 = 1 - 1
([(3,4)],5)
 => ([],4)
 => ([],1)
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(1,4),(2,3)],5)
 => ([],3)
 => ([],1)
 => 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([],1)
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ([],1)
 => 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],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)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ([],1)
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1

Description

The minimal number of vertices in a graph whose complement is triangle-free.

Matching statistic: St001573

Mapping

Mp00250: Graphs —clique graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001573: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
([],2)
 => ([],2)
 => ([],1)
 => 0 = 1 - 1
([(0,1)],2)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
([],3)
 => ([],3)
 => ([],1)
 => 0 = 1 - 1
([(1,2)],3)
 => ([],2)
 => ([],1)
 => 0 = 1 - 1
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
([],4)
 => ([],4)
 => ([],1)
 => 0 = 1 - 1
([(2,3)],4)
 => ([],3)
 => ([],1)
 => 0 = 1 - 1
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => ([],2)
 => ([],1)
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ([],1)
 => 0 = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
([],5)
 => ([],5)
 => ([],1)
 => 0 = 1 - 1
([(3,4)],5)
 => ([],4)
 => ([],1)
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(1,4),(2,3)],5)
 => ([],3)
 => ([],1)
 => 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([],1)
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ([],1)
 => 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],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)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ([],1)
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1

Description

The minimal number of edges to remove to make a graph triangle-free.

Matching statistic: St001742

Mapping

Mp00250: Graphs —clique graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001742: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
([],2)
 => ([],2)
 => ([],1)
 => 0 = 1 - 1
([(0,1)],2)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
([],3)
 => ([],3)
 => ([],1)
 => 0 = 1 - 1
([(1,2)],3)
 => ([],2)
 => ([],1)
 => 0 = 1 - 1
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
([],4)
 => ([],4)
 => ([],1)
 => 0 = 1 - 1
([(2,3)],4)
 => ([],3)
 => ([],1)
 => 0 = 1 - 1
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => ([],2)
 => ([],1)
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ([],1)
 => 0 = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
([],5)
 => ([],5)
 => ([],1)
 => 0 = 1 - 1
([(3,4)],5)
 => ([],4)
 => ([],1)
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => ([(2,3)],4)
 => ([(1,2)],3)
 => 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => ([],3)
 => ([],1)
 => 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([],1)
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ([],1)
 => 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],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)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ([],1)
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1

Description

The difference of the maximal and the minimal degree in a graph. The graph is regular if and only if this statistic is zero.

Matching statistic: St000278
(load all 5 compositions to match this statistic)

Mapping

Mp00117: Graphs —Ore closure⟶ Graphs
Mp00157: Graphs —connected complement⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000278: 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)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => [1]
 => 1
([],3)
 => ([],3)
 => ([],3)
 => []
 => 1
([(1,2)],3)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => [1]
 => 1
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [1,1]
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
([],4)
 => ([],4)
 => ([],4)
 => []
 => 1
([(2,3)],4)
 => ([(2,3)],4)
 => ([(2,3)],4)
 => [1]
 => 1
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [1,1]
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => 1
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => [1,1]
 => 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 1
([],5)
 => ([],5)
 => ([],5)
 => []
 => 1
([(3,4)],5)
 => ([(3,4)],5)
 => ([(3,4)],5)
 => [1]
 => 1
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [1,1]
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => 1
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => [1,1]
 => 1
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => 1
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 3
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => 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)
 => [5]
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => 1

Description

The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. This is the multinomial of the multiplicities of the parts, see [1]. This is the same as

$m_\lambda(x_1,\dotsc,x_k)$ evaluated at

$x_1=\dotsb=x_k=1$ , where

$k$ is the number of parts of

$\lambda$ . An explicit formula is

$\frac{k!}{m_1(\lambda)! m_2(\lambda)! \dotsb m_k(\lambda) !}$ where

$m_i(\lambda)$ is the number of parts of

$\lambda$ equal to

$i$ .

Matching statistic: St001775
(load all 2 compositions to match this statistic)

Mapping

Mp00250: Graphs —clique graph⟶ Graphs
Mp00157: Graphs —connected complement⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St001775: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([],2)
 => ([],2)
 => ([],2)
 => ([(0,1)],2)
 => 1
([(0,1)],2)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([],3)
 => ([],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(1,2)],3)
 => ([],2)
 => ([],2)
 => ([(0,1)],2)
 => 1
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([],4)
 => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(2,3)],4)
 => ([],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 1
([(0,3),(1,2)],4)
 => ([],2)
 => ([],2)
 => ([(0,1)],2)
 => 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ([],2)
 => ([(0,1)],2)
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([],5)
 => ([],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(3,4)],5)
 => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(2,4),(3,4)],5)
 => ([(2,3)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 1
([(1,4),(2,3)],5)
 => ([],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,1),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ([],2)
 => ([(0,1)],2)
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 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)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ([],2)
 => ([(0,1)],2)
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 1

Description

The degree of the minimal polynomial of the largest eigenvalue of a graph.

The following 369 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001689The number of celebrities in a graph. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001256Number of simple reflexive modules that are 2-stable reflexive. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001140Number of indecomposable modules with projective and injective dimension at least two 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. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of

$Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra

$A$ such that

$eA$ is a minimal faithful projective-injective module. 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

$eA$ . St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001924The number of cells in an integer partition whose arm and leg length coincide. St001964The interval resolution global dimension of a poset. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000480The number of lower covers of a partition in dominance order. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001732The number of peaks visible from the left. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000553The number of blocks of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000552The number of cut vertices of a graph. St000916The packing number of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000785The number of distinct colouring schemes of a graph. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001829The common independence number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000442The maximal area to the right of an up step of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St001196The global dimension of

$A$ minus the global dimension of

$eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001316The domatic number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001716The 1-improper chromatic number of a graph. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001393The induced matching number of a graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001261The Castelnuovo-Mumford regularity of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001353The number of prime nodes in the modular decomposition of a graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001128The exponens consonantiae of a partition. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001272The number of graphs with the same degree sequence. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000287The number of connected components of a graph. St000053The number of valleys of the Dyck path. St000286The number of connected components of the complement of a graph. St000335The difference of lower and upper interactions. St000443The number of long tunnels of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000781The number of proper colouring schemes of a Ferrers diagram. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000955Number of times one has

$Ext^i(D(A),A)>0$ for

$i>0$ for the corresponding LNakayama algebra. 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. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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

$g$ . St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . 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. St001282The number of graphs with the same chromatic polynomial. St001286The annihilation number of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001481The minimal height of a peak of a Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001518The number of graphs with the same ordinary spectrum as the given graph. St001642The Prague dimension of a graph. St001734The lettericity of a graph. St000015The number of peaks of a Dyck path. St000026The position of the first return of a Dyck path. St000075The orbit size of a standard tableau under promotion. St000172The Grundy number of a graph. St000268The number of strongly connected orientations of a graph. St000340The number of non-final maximal constant sub-paths of length greater than one. St000450The number of edges minus the number of vertices plus 2 of a graph. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000482The (zero)-forcing number of a graph. St000678The number of up steps after the last double rise of a Dyck path. 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. St000822The Hadwiger number of the graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000946The sum of the skew hook positions in a Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001073The number of nowhere zero 3-flows of a graph. St001116The game chromatic number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. 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. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001330The hat guessing number of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001432The order dimension of the partition. St001478The number of nowhere zero 4-flows of a graph. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001494The Alon-Tarsi number of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001530The depth of a Dyck path. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001883The mutual visibility number of a graph. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001957The number of Hasse diagrams with a given underlying undirected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001363The Euler characteristic of a graph according to Knill. St001765The number of connected components of the friends and strangers graph. St000322The skewness of a graph. St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St001307The number of induced stars on four vertices in a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001577The minimal number of edges to add or remove to make a graph a cograph. St000762The sum of the positions of the weak records of an integer composition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000256The number of parts from which one can substract 2 and still get an integer partition. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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. 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

$eA$ . St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. 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. 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. St001531Number of partial orders contained in the poset determined by the Dyck path. St001959The product of the heights of the peaks of a Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000775The multiplicity of the largest eigenvalue in a graph. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n−1}]$ by adding

$c_0$ to

$c_{n−1}$ . St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. 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. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between

$e_i J$ and

$e_j J$ (the radical of the indecomposable projective modules). St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. 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,. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. 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. St001846The number of elements which do not have a complement in the lattice. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000069The number of maximal elements of a poset. St000079The number of alternating sign matrices for a given Dyck path. St000120The number of left tunnels of a Dyck path. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001273The projective dimension of the first term in an injective coresolution of the regular module. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001480The number of simple summands of the module J^2/J^3. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000881The number of short braid edges in the graph of braid moves of a permutation. St000570The Edelman-Greene number of a permutation. St000618The number of self-evacuating tableaux of given shape. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000667The greatest common divisor of the parts of the partition. St000668The least common multiple of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000770The major index of an integer partition when read from bottom to top. 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

$K[x]/(x^n)$ . St001389The number of partitions of the same length below the given integer partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001571The Cartan determinant of the integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. 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. St001901The largest multiplicity of an irreducible representation 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. St000407The number of occurrences of the pattern 2143 in a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001856The number of edges in the reduced word graph of a permutation. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001845The number of join irreducibles minus the rank of a lattice. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000284The Plancherel distribution on integer partitions. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000456The monochromatic index of a connected graph. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001118The acyclic chromatic index of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St000068The number of minimal elements in a poset. St001933The largest multiplicity of a part in an integer partition. 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. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St000455The second largest eigenvalue of a graph if it is integral. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000003The number of standard Young tableaux of the partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000147The largest part of an integer partition. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000182The number of permutations whose cycle type is the given integer partition. St000183The side length of the Durfee square of an integer partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000321The number of integer partitions of n that are dominated by an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000378The diagonal inversion number of an integer partition. St000390The number of runs of ones in a binary word. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000517The Kreweras number of an integer partition. St000529The number of permutations whose descent word is the given binary word. St000531The leading coefficient of the rook polynomial of an integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000628The balance of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000655The length of the minimal rise of a Dyck path. St000691The number of changes of a binary word. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000734The last entry in the first row of a standard tableau. St000783The side length of the largest staircase partition fitting into a partition. St000847The number of standard Young tableaux whose descent set is the binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000897The number of different multiplicities of parts of an integer partition. St000913The number of ways to refine the partition into singletons. St000935The number of ordered refinements of an integer partition. St000947The major index east count of a Dyck path. St000983The length of the longest alternating subword. St001129The product of the squares of the parts of a partition. St001161The major index north count of a Dyck path. St001313The number of Dyck paths above the lattice path given by a binary word. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001595The number of standard Young tableaux of the skew partition. St001597The Frobenius rank of a skew partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001838The number of nonempty primitive factors of a binary word. St001884The number of borders of a binary word. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001739The number of graphs with the same edge polytope as the given graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St001793The difference between the clique number and the chromatic number of a graph. 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. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001527The cyclic permutation representation number of an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001914The size of the orbit of an integer partition in Bulgarian solitaire. 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. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001877Number of indecomposable injective modules with projective dimension 2. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000914The sum of the values of the Möbius function of a poset. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. 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. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001360The number of covering relations in Young's lattice below a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001060The distinguishing index of a graph. St000782The indicator function of whether a given perfect matching is an L & P matching.