Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000144
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000144: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
([],2)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
([(0,1)],2)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
([],3)
 => [2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3
([(1,2)],3)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 5
([(0,1),(0,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 4
([(0,2),(2,1)],3)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3
([(0,2),(1,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 4
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 4
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 4
([(1,2),(2,3)],4)
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 4
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 4
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 4
([(0,3),(1,2)],4)
 => [3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 5
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 6
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 5
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 5
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 5
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 5
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 5
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 5
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 5
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 6
([(0,4),(1,2),(2,3),(3,4)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 5
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 5
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
Description
The pyramid weight of the Dyck path. The pyramid weight of a Dyck path is the sum of the lengths of the maximal pyramids (maximal sequences of the form $1^h0^h$) in the path. Maximal pyramids are called lower interactions by Le Borgne [2], see [[St000331]] and [[St000335]] for related statistics.
Matching statistic: St001880
(load all 2 compositions to match this statistic)
Mapping
St001880: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 80%
Values
([],1)
 => ? = 2
([],2)
 => ? = 3
([(0,1)],2)
 => ? = 2
([],3)
 => ? = 3
([(1,2)],3)
 => ? = 5
([(0,1),(0,2)],3)
 => ? = 4
([(0,2),(2,1)],3)
 => 3
([(0,2),(1,2)],3)
 => ? = 4
([(0,1),(0,2),(0,3)],4)
 => ? = 4
([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
([(1,2),(2,3)],4)
 => ? = 5
([(0,3),(3,1),(3,2)],4)
 => ? = 4
([(0,3),(1,3),(3,2)],4)
 => ? = 4
([(0,3),(1,3),(2,3)],4)
 => ? = 4
([(0,3),(1,2)],4)
 => ? = 5
([(0,3),(1,2),(1,3)],4)
 => ? = 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => ? = 5
([(0,3),(2,1),(3,2)],4)
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => ? = 6
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ? = 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 5
([(0,4),(1,4),(4,2),(4,3)],5)
 => ? = 5
([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 5
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ? = 5
([(0,2),(0,4),(3,1),(4,3)],5)
 => ? = 5
([(0,3),(3,4),(4,1),(4,2)],5)
 => ? = 5
([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 6
([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 5
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 5
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ? = 6
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 6
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ? = 6
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000464
Mapping
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000464: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 20%
Values
([],1)
 => ([],1)
 => ([],0)
 => ([],0)
 => ? = 2 - 3
([],2)
 => ([(0,1)],2)
 => ([],1)
 => ([],1)
 => ? = 3 - 3
([(0,1)],2)
 => ([],2)
 => ([],0)
 => ([],0)
 => ? = 2 - 3
([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? = 3 - 3
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([],2)
 => ? = 5 - 3
([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => ([],1)
 => ([],1)
 => ? = 4 - 3
([(0,2),(2,1)],3)
 => ([],3)
 => ([],0)
 => ([],0)
 => ? = 3 - 3
([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([],1)
 => ([],1)
 => ? = 4 - 3
([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? = 4 - 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([],1)
 => ([],1)
 => ? = 4 - 3
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? = 5 - 3
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => ([],1)
 => ([],1)
 => ? = 4 - 3
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([],1)
 => ([],1)
 => ? = 4 - 3
([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? = 4 - 3
([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => ? = 5 - 3
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ? = 4 - 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => ([],2)
 => ([(0,1)],2)
 => 2 = 5 - 3
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],0)
 => ([],0)
 => ? = 4 - 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([],1)
 => ([],1)
 => ? = 5 - 3
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => ? = 6 - 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ? = 5 - 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3)],5)
 => ([],2)
 => ([(0,1)],2)
 => 2 = 5 - 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,3)],5)
 => ([],2)
 => ([(0,1)],2)
 => 2 = 5 - 3
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([],1)
 => ([],1)
 => ? = 5 - 3
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ? = 5 - 3
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(1,4),(2,3)],5)
 => ([],2)
 => ([(0,1)],2)
 => 2 = 5 - 3
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? = 5 - 3
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(3,4)],5)
 => ([],1)
 => ([],1)
 => ? = 5 - 3
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],0)
 => ([],0)
 => ? = 5 - 3
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => ? = 6 - 3
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? = 5 - 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([],1)
 => ([],1)
 => ? = 5 - 3
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ? = 6 - 3
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ? = 6 - 3
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ? = 6 - 3
Description
The Schultz index of a connected graph. This is $$\sum_{\{u,v\}\subseteq V} (d(u)+d(v))d(u,v)$$ where $d(u)$ is the degree of vertex $u$ and $d(u,v)$ is the distance between vertices $u$ and $v$. For trees on $n$ vertices, the Schultz index is related to the Wiener index via $S(T)=4W(T)-n(n-1)$ [2].
Matching statistic: St001545
Mapping
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St001545: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 20%
Values
([],1)
 => ([],1)
 => ([],0)
 => ([],0)
 => ? = 2 - 3
([],2)
 => ([(0,1)],2)
 => ([],1)
 => ([],1)
 => ? = 3 - 3
([(0,1)],2)
 => ([],2)
 => ([],0)
 => ([],0)
 => ? = 2 - 3
([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? = 3 - 3
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([],2)
 => ? = 5 - 3
([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => ([],1)
 => ([],1)
 => ? = 4 - 3
([(0,2),(2,1)],3)
 => ([],3)
 => ([],0)
 => ([],0)
 => ? = 3 - 3
([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([],1)
 => ([],1)
 => ? = 4 - 3
([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? = 4 - 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([],1)
 => ([],1)
 => ? = 4 - 3
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? = 5 - 3
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => ([],1)
 => ([],1)
 => ? = 4 - 3
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([],1)
 => ([],1)
 => ? = 4 - 3
([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? = 4 - 3
([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => ? = 5 - 3
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ? = 4 - 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => ([],2)
 => ([(0,1)],2)
 => 2 = 5 - 3
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],0)
 => ([],0)
 => ? = 4 - 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([],1)
 => ([],1)
 => ? = 5 - 3
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => ? = 6 - 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ? = 5 - 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3)],5)
 => ([],2)
 => ([(0,1)],2)
 => 2 = 5 - 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,3)],5)
 => ([],2)
 => ([(0,1)],2)
 => 2 = 5 - 3
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([],1)
 => ([],1)
 => ? = 5 - 3
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ? = 5 - 3
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(1,4),(2,3)],5)
 => ([],2)
 => ([(0,1)],2)
 => 2 = 5 - 3
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? = 5 - 3
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(3,4)],5)
 => ([],1)
 => ([],1)
 => ? = 5 - 3
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],0)
 => ([],0)
 => ? = 5 - 3
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => ? = 6 - 3
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? = 5 - 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([],1)
 => ([],1)
 => ? = 5 - 3
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ? = 6 - 3
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ? = 6 - 3
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ? = 6 - 3
Description
The second Elser number of a connected graph. For a connected graph $G$ the $k$-th Elser number is $$ els_k(G) = (-1)^{|V(G)|+1} \sum_N (-1)^{|E(N)|} |V(N)|^k $$ where the sum is over all nuclei of $G$, that is, the connected subgraphs of $G$ whose vertex set is a vertex cover of $G$. It is clear that this number is even. It was shown in [1] that it is non-negative.
Matching statistic: St000456
Mapping
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 20%
Values
([],1)
 => ([],1)
 => ([],0)
 => ([],0)
 => ? = 2 - 4
([],2)
 => ([(0,1)],2)
 => ([],1)
 => ([],1)
 => ? = 3 - 4
([(0,1)],2)
 => ([],2)
 => ([],0)
 => ([],0)
 => ? = 2 - 4
([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? = 3 - 4
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([],2)
 => ? = 5 - 4
([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => ([],1)
 => ([],1)
 => ? = 4 - 4
([(0,2),(2,1)],3)
 => ([],3)
 => ([],0)
 => ([],0)
 => ? = 3 - 4
([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([],1)
 => ([],1)
 => ? = 4 - 4
([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? = 4 - 4
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([],1)
 => ([],1)
 => ? = 4 - 4
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? = 5 - 4
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => ([],1)
 => ([],1)
 => ? = 4 - 4
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([],1)
 => ([],1)
 => ? = 4 - 4
([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? = 4 - 4
([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => ? = 5 - 4
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ? = 4 - 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => ([],2)
 => ([(0,1)],2)
 => 1 = 5 - 4
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],0)
 => ([],0)
 => ? = 4 - 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([],1)
 => ([],1)
 => ? = 5 - 4
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => ? = 6 - 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ? = 5 - 4
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3)],5)
 => ([],2)
 => ([(0,1)],2)
 => 1 = 5 - 4
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,3)],5)
 => ([],2)
 => ([(0,1)],2)
 => 1 = 5 - 4
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([],1)
 => ([],1)
 => ? = 5 - 4
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ? = 5 - 4
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(1,4),(2,3)],5)
 => ([],2)
 => ([(0,1)],2)
 => 1 = 5 - 4
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? = 5 - 4
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(3,4)],5)
 => ([],1)
 => ([],1)
 => ? = 5 - 4
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],0)
 => ([],0)
 => ? = 5 - 4
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => ? = 6 - 4
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? = 5 - 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([],1)
 => ([],1)
 => ? = 5 - 4
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ? = 6 - 4
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ? = 6 - 4
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ? = 6 - 4
Description
The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
Mapping
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
Mp00157: Graphs —connected complement⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 20%
Values
([],1)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 2 - 5
([],2)
 => ([(0,1)],2)
 => ([],1)
 => ([],1)
 => ? = 3 - 5
([(0,1)],2)
 => ([],2)
 => ([],2)
 => ([],2)
 => ? = 2 - 5
([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => ? = 3 - 5
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0 = 5 - 5
([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => ([],2)
 => ([],2)
 => ? = 4 - 5
([(0,2),(2,1)],3)
 => ([],3)
 => ([],3)
 => ([],3)
 => ? = 3 - 5
([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([],2)
 => ([],2)
 => ? = 4 - 5
([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ([],2)
 => ? = 4 - 5
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([],3)
 => ([],3)
 => ? = 4 - 5
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 0 = 5 - 5
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => ([],3)
 => ([],3)
 => ? = 4 - 5
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([],3)
 => ([],3)
 => ? = 4 - 5
([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ([],2)
 => ? = 4 - 5
([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => ([],1)
 => ? = 5 - 5
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 5
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => ([],2)
 => ([],2)
 => ? = 5 - 5
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],4)
 => ([],4)
 => ? = 4 - 5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([],4)
 => ([],4)
 => ? = 5 - 5
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => ([],2)
 => ? = 6 - 5
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3)],5)
 => ([],3)
 => ([],3)
 => ? = 5 - 5
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,3)],5)
 => ([],3)
 => ([],3)
 => ? = 5 - 5
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([],4)
 => ([],4)
 => ? = 5 - 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 5
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(1,4),(2,3)],5)
 => ([],3)
 => ([],3)
 => ? = 5 - 5
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0 = 5 - 5
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(3,4)],5)
 => ([],4)
 => ([],4)
 => ? = 5 - 5
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],5)
 => ([],5)
 => ? = 5 - 5
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => ([],2)
 => ? = 6 - 5
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0 = 5 - 5
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([],4)
 => ([],4)
 => ? = 5 - 5
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(2,6),(3,5),(4,5),(4,6)],7)
 => ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 6 - 5
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(2,6),(3,5),(4,5),(4,6)],7)
 => ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 6 - 5
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(2,6),(3,5),(4,5),(4,6)],7)
 => ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 6 - 5
Description
The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.
Matching statistic: St000264
Mapping
Mp00074: Posets —to graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 20%
Values
([],1)
 => ([],1)
 => ([],1)
 => ? = 2 - 2
([],2)
 => ([],2)
 => ([],1)
 => ? = 3 - 2
([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 2 - 2
([],3)
 => ([],3)
 => ([],1)
 => ? = 3 - 2
([(1,2)],3)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 5 - 2
([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ? = 4 - 2
([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ? = 3 - 2
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ? = 4 - 2
([(0,1),(0,2),(0,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? = 4 - 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ? = 4 - 2
([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 5 - 2
([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? = 4 - 2
([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? = 4 - 2
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? = 4 - 2
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? = 5 - 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 4 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ? = 5 - 2
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 4 - 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 5 - 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 6 - 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 5 - 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ? = 5 - 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 5 - 2
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 5 - 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 5 - 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ? = 5 - 2
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 5 - 2
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 5 - 2
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 5 - 2
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 6 - 2
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 5 - 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 5 - 2
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 6 - 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => 4 = 6 - 2
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 6 - 2
Description
The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.