Processing math: 100%

Your data matches 219 different statistics following compositions of up to 3 maps.
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Matching statistic: St001069
(load all 4 compositions to match this statistic)
Mapping
St001069: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => 0
([],2)
 => 0
([(0,1)],2)
 => 0
([],3)
 => 0
([(1,2)],3)
 => 0
([(0,2),(1,2)],3)
 => 0
([(0,1),(0,2),(1,2)],3)
 => 0
([],4)
 => 0
([(2,3)],4)
 => 0
([(1,3),(2,3)],4)
 => 0
([(0,3),(1,3),(2,3)],4)
 => 0
([(0,3),(1,2)],4)
 => 0
([(0,3),(1,2),(2,3)],4)
 => 0
([(1,2),(1,3),(2,3)],4)
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => 0
([(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)
 => 4
([],5)
 => 0
([(3,4)],5)
 => 0
([(2,4),(3,4)],5)
 => 0
([(1,4),(2,4),(3,4)],5)
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
([(1,4),(2,3)],5)
 => 0
([(1,4),(2,3),(3,4)],5)
 => 0
([(0,1),(2,4),(3,4)],5)
 => 0
([(2,3),(2,4),(3,4)],5)
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 0
([(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
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 7
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 9
Description
The coefficient of the monomial xy of the Tutte polynomial of the graph.
Matching statistic: St001310
Mapping
Mp00156: Graphs —line graph⟶ Graphs
Mp00318: Graphs —dual on components⟶ Graphs
St001310: Graphs ⟶ ℤResult quality: 26% ●values known / values provided: 51%●distinct values known / distinct values provided: 26%
Values
([],1)
 => ([],0)
 => ?
 => ? = 0
([],2)
 => ([],0)
 => ?
 => ? = 0
([(0,1)],2)
 => ([],1)
 => ([],1)
 => 0
([],3)
 => ([],0)
 => ?
 => ? = 0
([(1,2)],3)
 => ([],1)
 => ([],1)
 => 0
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([],4)
 => ([],0)
 => ?
 => ? ∊ {0,2}
([(2,3)],4)
 => ([],1)
 => ([],1)
 => 0
([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,3),(1,2)],4)
 => ([],2)
 => ([],2)
 => 0
([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 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
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(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)
 => 4
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
 => ? ∊ {0,2}
([],5)
 => ([],0)
 => ?
 => ? ∊ {0,0,1,1,2,3,7,9,13,20}
([(3,4)],5)
 => ([],1)
 => ([],1)
 => 0
([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(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
([(1,4),(2,3)],5)
 => ([],2)
 => ([],2)
 => 0
([(1,4),(2,3),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
([(0,1),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(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)
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,4),(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)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 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
([(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)
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(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),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,6),(3,6),(4,6),(5,6)],7)
 => 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)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? ∊ {0,0,1,1,2,3,7,9,13,20}
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(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)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(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
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,1,1,2,3,7,9,13,20}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
 => ? ∊ {0,0,1,1,2,3,7,9,13,20}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,7),(0,8),(0,9),(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,9),(4,8),(5,7),(5,8),(6,7),(6,9)],10)
 => ? ∊ {0,0,1,1,2,3,7,9,13,20}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,1,1,2,3,7,9,13,20}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(0,3),(0,5),(1,2),(1,3),(1,4),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,1,1,2,3,7,9,13,20}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,1,1,2,3,7,9,13,20}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? ∊ {0,0,1,1,2,3,7,9,13,20}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
 => ?
 => ? ∊ {0,0,1,1,2,3,7,9,13,20}
([],6)
 => ([],0)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(4,5)],6)
 => ([],1)
 => ([],1)
 => 0
([(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(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
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(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)
 => 0
([(2,5),(3,4)],6)
 => ([],2)
 => ([],2)
 => 0
([(2,5),(3,4),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
([(1,2),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(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
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,4),(2,6),(3,5),(3,7),(4,8),(5,8),(6,8),(7,8)],9)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,5),(0,7),(1,2),(1,4),(1,6),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,2),(1,4),(1,7),(2,3),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,7),(0,8),(0,9),(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,9),(4,8),(5,7),(5,8),(6,7),(6,9)],10)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(0,6),(0,7),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,2),(1,4),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,6),(0,7),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,2),(1,4),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,8),(2,3),(2,6),(2,8),(3,4),(3,5),(3,7),(3,8),(4,5),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(0,5),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,4),(3,7),(4,6),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,7),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(0,3),(0,5),(1,2),(1,3),(1,4),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(2,3),(2,5),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,7),(2,4),(2,6),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,7),(1,2),(1,3),(1,6),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1),(0,3),(0,7),(1,2),(1,6),(2,4),(2,6),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(0,5),(0,8),(1,2),(1,4),(1,7),(2,3),(2,6),(2,7),(3,6),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,7),(1,8),(2,3),(2,6),(2,8),(3,4),(3,6),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,6),(0,7),(1,4),(1,5),(2,3),(2,5),(2,7),(2,8),(3,4),(3,6),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,7,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
Description
The number of induced diamond graphs in a graph. A diamond graph is a cycle on four vertices, with an additional edge connecting two of the non-adjacent vertices.
Matching statistic: St001329
Mapping
Mp00156: Graphs —line graph⟶ Graphs
Mp00318: Graphs —dual on components⟶ Graphs
St001329: Graphs ⟶ ℤResult quality: 21% ●values known / values provided: 51%●distinct values known / distinct values provided: 21%
Values
([],1)
 => ([],0)
 => ?
 => ? = 0
([],2)
 => ([],0)
 => ?
 => ? = 0
([(0,1)],2)
 => ([],1)
 => ([],1)
 => 0
([],3)
 => ([],0)
 => ?
 => ? = 0
([(1,2)],3)
 => ([],1)
 => ([],1)
 => 0
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([],4)
 => ([],0)
 => ?
 => ? ∊ {1,4}
([(2,3)],4)
 => ([],1)
 => ([],1)
 => 0
([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,3),(1,2)],4)
 => ([],2)
 => ([],2)
 => 0
([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(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
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(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)
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
 => ? ∊ {1,4}
([],5)
 => ([],0)
 => ?
 => ? ∊ {1,1,2,4,4,5,7,9,13,20}
([(3,4)],5)
 => ([],1)
 => ([],1)
 => 0
([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(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)
 => ([],2)
 => ([],2)
 => 0
([(1,4),(2,3),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
([(0,1),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(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)
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,4),(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)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(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
([(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)
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(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
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,6),(3,6),(4,6),(5,6)],7)
 => 3
([(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)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? ∊ {1,1,2,4,4,5,7,9,13,20}
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(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
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 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
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,2,4,4,5,7,9,13,20}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
 => ? ∊ {1,1,2,4,4,5,7,9,13,20}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,7),(0,8),(0,9),(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,9),(4,8),(5,7),(5,8),(6,7),(6,9)],10)
 => ? ∊ {1,1,2,4,4,5,7,9,13,20}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {1,1,2,4,4,5,7,9,13,20}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(0,3),(0,5),(1,2),(1,3),(1,4),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,1,2,4,4,5,7,9,13,20}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {1,1,2,4,4,5,7,9,13,20}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? ∊ {1,1,2,4,4,5,7,9,13,20}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
 => ?
 => ? ∊ {1,1,2,4,4,5,7,9,13,20}
([],6)
 => ([],0)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(4,5)],6)
 => ([],1)
 => ([],1)
 => 0
([(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(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
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(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)
 => 5
([(2,5),(3,4)],6)
 => ([],2)
 => ([],2)
 => 0
([(2,5),(3,4),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
([(1,2),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(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)
 => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,4),(2,6),(3,5),(3,7),(4,8),(5,8),(6,8),(7,8)],9)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,5),(0,7),(1,2),(1,4),(1,6),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,2),(1,4),(1,7),(2,3),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,7),(0,8),(0,9),(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,9),(4,8),(5,7),(5,8),(6,7),(6,9)],10)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(0,6),(0,7),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,2),(1,4),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,6),(0,7),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,2),(1,4),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,8),(2,3),(2,6),(2,8),(3,4),(3,5),(3,7),(3,8),(4,5),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(0,5),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,4),(3,7),(4,6),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,7),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
 => ([(0,2),(0,3),(0,5),(1,2),(1,3),(1,4),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(2,3),(2,5),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,7),(2,4),(2,6),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,7),(1,2),(1,3),(1,6),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1),(0,3),(0,7),(1,2),(1,6),(2,4),(2,6),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(0,5),(0,8),(1,2),(1,4),(1,7),(2,3),(2,6),(2,7),(3,6),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,7),(1,8),(2,3),(2,6),(2,8),(3,4),(3,6),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,6),(0,7),(1,4),(1,5),(2,3),(2,5),(2,7),(2,8),(3,4),(3,6),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,8,8,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
Description
The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. A graph is outerplanar if and only if in any linear ordering of its vertices, there are no four vertices a<b<c<d such that (a,c) and (b,d) are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001140
(load all 3 compositions to match this statistic)
Mapping
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001140: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 47%●distinct values known / distinct values provided: 9%
Values
([],1)
 => []
 => []
 => []
 => ? = 0
([],2)
 => []
 => []
 => []
 => ? = 0
([(0,1)],2)
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
([],3)
 => []
 => []
 => []
 => ? = 0
([(1,2)],3)
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
([(0,2),(1,2)],3)
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 0
([],4)
 => []
 => []
 => []
 => ? ∊ {0,4}
([(2,3)],4)
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
([(1,3),(2,3)],4)
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
([(0,3),(1,2)],4)
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,4}
([],5)
 => []
 => []
 => []
 => ? ∊ {0,0,0,3,3,4,4,5,7,9,13,20}
([(3,4)],5)
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
([(2,4),(3,4)],5)
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
([(1,4),(2,3)],5)
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,3,3,4,4,5,7,9,13,20}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,3,3,4,4,5,7,9,13,20}
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,3,3,4,4,5,7,9,13,20}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,3,3,4,4,5,7,9,13,20}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,3,3,4,4,5,7,9,13,20}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? ∊ {0,0,0,3,3,4,4,5,7,9,13,20}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,3,3,4,4,5,7,9,13,20}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,3,3,4,4,5,7,9,13,20}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,3,3,4,4,5,7,9,13,20}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,3,3,4,4,5,7,9,13,20}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,3,3,4,4,5,7,9,13,20}
([],6)
 => []
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(4,5)],6)
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
([(3,5),(4,5)],6)
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 2
([(2,5),(3,4)],6)
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
Description
Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001265
Mapping
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001265: Dyck paths ⟶ ℤResult quality: 15% ●values known / values provided: 47%●distinct values known / distinct values provided: 15%
Values
([],1)
 => []
 => []
 => []
 => ? = 0
([],2)
 => []
 => []
 => []
 => ? = 0
([(0,1)],2)
 => [1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 0
([],3)
 => []
 => []
 => []
 => ? = 0
([(1,2)],3)
 => [1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 0
([(0,2),(1,2)],3)
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 0
([],4)
 => []
 => []
 => []
 => ? ∊ {1,4}
([(2,3)],4)
 => [1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 0
([(1,3),(2,3)],4)
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 0
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
([(0,3),(1,2)],4)
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? ∊ {1,4}
([],5)
 => []
 => []
 => []
 => ? ∊ {1,1,1,1,2,2,3,5,7,9,13,20}
([(3,4)],5)
 => [1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 0
([(2,4),(3,4)],5)
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 0
([(1,4),(2,3)],5)
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? ∊ {1,1,1,1,2,2,3,5,7,9,13,20}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? ∊ {1,1,1,1,2,2,3,5,7,9,13,20}
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? ∊ {1,1,1,1,2,2,3,5,7,9,13,20}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? ∊ {1,1,1,1,2,2,3,5,7,9,13,20}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? ∊ {1,1,1,1,2,2,3,5,7,9,13,20}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? ∊ {1,1,1,1,2,2,3,5,7,9,13,20}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => ? ∊ {1,1,1,1,2,2,3,5,7,9,13,20}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? ∊ {1,1,1,1,2,2,3,5,7,9,13,20}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => ? ∊ {1,1,1,1,2,2,3,5,7,9,13,20}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => ? ∊ {1,1,1,1,2,2,3,5,7,9,13,20}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {1,1,1,1,2,2,3,5,7,9,13,20}
([],6)
 => []
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(4,5)],6)
 => [1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 0
([(3,5),(4,5)],6)
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 0
([(2,5),(3,4)],6)
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
Description
The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra.
Matching statistic: St000936
(load all 9 compositions to match this statistic)
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000936: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 44%●distinct values known / distinct values provided: 15%
Values
([],1)
 => [1] => [[1],[]]
 => []
 => ? = 0
([],2)
 => [2] => [[2],[]]
 => []
 => ? ∊ {0,0}
([(0,1)],2)
 => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0}
([],3)
 => [3] => [[3],[]]
 => []
 => ? ∊ {0,0,0,0}
([(1,2)],3)
 => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,0,0,0}
([(0,2),(1,2)],3)
 => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0}
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [[2,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0}
([],4)
 => [4] => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,4}
([(2,3)],4)
 => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,4}
([(1,3),(2,3)],4)
 => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,4}
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,2,4}
([(0,3),(1,2)],4)
 => [2,2] => [[3,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,2,4}
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,4}
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [[3,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,2,4}
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,2,4}
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,2,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [[3,3],[2]]
 => [2]
 => 0
([],5)
 => [5] => [[5],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,4,5,7,9,13,20}
([(3,4)],5)
 => [1,4] => [[4,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,4,5,7,9,13,20}
([(2,4),(3,4)],5)
 => [1,1,3] => [[3,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,4,5,7,9,13,20}
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,4,5,7,9,13,20}
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 0
([(1,4),(2,3)],5)
 => [2,3] => [[4,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,4,5,7,9,13,20}
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,4,5,7,9,13,20}
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,4,5,7,9,13,20}
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [[4,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,4,5,7,9,13,20}
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,4,5,7,9,13,20}
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,4,5,7,9,13,20}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,4,5,7,9,13,20}
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,4,5,7,9,13,20}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,4,5,7,9,13,20}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,4,5,7,9,13,20}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,4,5,7,9,13,20}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,4,5,7,9,13,20}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,4,5,7,9,13,20}
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,4,5,7,9,13,20}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,4,5,7,9,13,20}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,4,5,7,9,13,20}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,4,5,7,9,13,20}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [[4,3],[2]]
 => [2]
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,3,3,4,5,7,9,13,20}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 4
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [[4,4],[3]]
 => [3]
 => 0
([],6)
 => [6] => [[6],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(4,5)],6)
 => [1,5] => [[5,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(3,5),(4,5)],6)
 => [1,1,4] => [[4,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(2,5),(3,5),(4,5)],6)
 => [1,2,3] => [[4,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,3,2] => [[4,3,1],[2]]
 => [2]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,4,1] => [[4,4,1],[3]]
 => [3]
 => 0
([(2,5),(3,4)],6)
 => [2,4] => [[5,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(2,5),(3,4),(4,5)],6)
 => [1,1,1,3] => [[3,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,2),(3,5),(4,5)],6)
 => [1,1,1,3] => [[3,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [[5,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,3] => [[3,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 0
([(0,5),(1,5),(2,4),(3,4)],6)
 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [3,3] => [[5,3],[2]]
 => [2]
 => 0
([(0,1),(2,5),(3,4),(4,5)],6)
 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => 0
([(1,2),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,3,2] => [[4,3,1],[2]]
 => [2]
 => 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 4
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 0
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 0
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 2
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 0
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3] => [[5,3],[2]]
 => [2]
 => 0
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => 0
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => 2
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 0
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 0
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 2
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 0
Description
The number of even values of the symmetric group character corresponding to the partition. For example, the character values of the irreducible representation S(2,2) are 2 on the conjugacy classes (4) and (2,2), 0 on the conjugacy classes (3,1) and (1,1,1,1), and −1 on the conjugace class (2,1,1). Therefore, the statistic on the partition (2,2) is 4. It is shown in [1] that the sum of the values of the statistic over all partitions of a given size is even.
Matching statistic: St000454
(load all 4 compositions to match this statistic)
Mapping
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 43%●distinct values known / distinct values provided: 12%
Values
([],1)
 => ([],1)
 => ([],1)
 => ([],1)
 => 0
([],2)
 => ([],1)
 => ([],1)
 => ([],1)
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 0
([],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => 0
([(1,2)],3)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ? = 0
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 0
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 0
([],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 0
([(2,3)],4)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ? ∊ {0,0,0,1,4}
([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ? ∊ {0,0,0,1,4}
([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 0
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,0,0,1,4}
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,1,4}
([(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)
 => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,1,4}
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 0
([(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)
 => ([],4)
 => 0
([],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 0
([(3,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3,3,4,4,5,7,9,13,20}
([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3,3,4,4,5,7,9,13,20}
([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3,3,4,4,5,7,9,13,20}
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 0
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3,3,4,4,5,7,9,13,20}
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3,3,4,4,5,7,9,13,20}
([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(2,3),(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)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3,3,4,4,5,7,9,13,20}
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3,3,4,4,5,7,9,13,20}
([(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)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3,3,4,4,5,7,9,13,20}
([(0,4),(1,4),(2,3),(2,4),(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)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3,3,4,4,5,7,9,13,20}
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3,3,4,4,5,7,9,13,20}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3,3,4,4,5,7,9,13,20}
([(1,3),(1,4),(2,3),(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)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3,3,4,4,5,7,9,13,20}
([(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)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3,3,4,4,5,7,9,13,20}
([(0,4),(1,3),(1,4),(2,3),(2,4),(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)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3,3,4,4,5,7,9,13,20}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 0
([(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)
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(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)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3,3,4,4,5,7,9,13,20}
([(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)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3,3,4,4,5,7,9,13,20}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(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)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3,3,4,4,5,7,9,13,20}
([(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)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => 0
([(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),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => 0
([(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)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3,3,4,4,5,7,9,13,20}
([(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)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
([(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)
 => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3,3,4,4,5,7,9,13,20}
([(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),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,3,3,4,4,5,7,9,13,20}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 0
([(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),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 0
([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => 0
([],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 0
([(4,5)],6)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 0
([(2,5),(3,4)],6)
 => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,2),(3,5),(4,5)],6)
 => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(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)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 0
([(0,5),(1,4),(2,3)],6)
 => ([(0,5),(1,4),(2,3)],6)
 => ([(0,5),(1,4),(2,3)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => ([(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)
 => ([],5)
 => 0
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => ([],5)
 => 0
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => ([],5)
 => 0
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => ([],5)
 => 0
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(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)
 => 3
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],6)
 => 0
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],6)
 => 0
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],6)
 => 0
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(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)
 => ([],5)
 => 0
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => ([],5)
 => 0
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
Description
The largest eigenvalue of a graph if it is integral. If a graph is d-regular, then its largest eigenvalue equals d. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000379
(load all 2 compositions to match this statistic)
Mapping
Mp00274: Graphs —block-cut tree⟶ Graphs
Mp00157: Graphs —connected complement⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000379: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 41%●distinct values known / distinct values provided: 9%
Values
([],1)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 0
([],2)
 => ([],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0}
([(0,1)],2)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0}
([],3)
 => ([],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,0}
([(1,2)],3)
 => ([],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,0}
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0}
([],4)
 => ([],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,2,4}
([(2,3)],4)
 => ([],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,0,0,0,2,4}
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0
([(0,3),(1,2)],4)
 => ([],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,0,0,0,2,4}
([(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)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,0,0,0,2,4}
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,2,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,2,4}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,2,4}
([],5)
 => ([],5)
 => ([],5)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,7,9,13,20}
([(3,4)],5)
 => ([],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,7,9,13,20}
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 0
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 0
([(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)
 => ([(0,1)],2)
 => 0
([(1,4),(2,3)],5)
 => ([],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,7,9,13,20}
([(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)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,7,9,13,20}
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,7,9,13,20}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,7,9,13,20}
([(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)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,7,9,13,20}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,7,9,13,20}
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,7,9,13,20}
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,7,9,13,20}
([(0,3),(1,2),(1,4),(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)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,7,9,13,20}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,7,9,13,20}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,7,9,13,20}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,7,9,13,20}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,7,9,13,20}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,7,9,13,20}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,7,9,13,20}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,7,9,13,20}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,7,9,13,20}
([],6)
 => ([],6)
 => ([],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(4,5)],6)
 => ([],5)
 => ([],5)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 0
([(2,5),(3,5),(4,5)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 0
([(2,5),(3,4)],6)
 => ([],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(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)
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,2),(3,5),(4,5)],6)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 0
([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => ([],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 0
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([(0,5),(1,5),(2,4),(3,4)],6)
 => ([(0,5),(1,5),(2,4),(3,4)],6)
 => ([(0,1)],2)
 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
 => ([(0,2),(0,3),(0,4),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,5),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(2,3)],6)
 => ([],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
 => ([(1,2),(1,4),(1,5),(1,7),(2,3),(2,5),(2,6),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => ([],3)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
Description
The number of Hamiltonian cycles in a graph. A Hamiltonian cycle in a graph G is a subgraph (this is, a subset of the edges) that is a cycle which contains every vertex of G. Since it is unclear whether the graph on one vertex is Hamiltonian, the statistic is undefined for this graph.
Matching statistic: St000455
(load all 9 compositions to match this statistic)
Mapping
Mp00274: Graphs —block-cut tree⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 40%●distinct values known / distinct values provided: 6%
Values
([],1)
 => ([],1)
 => ? = 0
([],2)
 => ([],2)
 => ? ∊ {0,0}
([(0,1)],2)
 => ([],1)
 => ? ∊ {0,0}
([],3)
 => ([],3)
 => ? ∊ {0,0,0}
([(1,2)],3)
 => ([],2)
 => ? ∊ {0,0,0}
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => ? ∊ {0,0,0}
([],4)
 => ([],4)
 => ? ∊ {0,0,0,0,0,2,4}
([(2,3)],4)
 => ([],3)
 => ? ∊ {0,0,0,0,0,2,4}
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
([(0,3),(1,2)],4)
 => ([],2)
 => ? ∊ {0,0,0,0,0,2,4}
([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ? ∊ {0,0,0,0,0,2,4}
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,2,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,2,4}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,2,4}
([],5)
 => ([],5)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,3,4,4,5,7,9,13,20}
([(3,4)],5)
 => ([],4)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,3,4,4,5,7,9,13,20}
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
([(1,4),(2,3)],5)
 => ([],3)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,3,4,4,5,7,9,13,20}
([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 1
([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 0
([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,3,4,4,5,7,9,13,20}
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,3,4,4,5,7,9,13,20}
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,3,4,4,5,7,9,13,20}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,3,4,4,5,7,9,13,20}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,3,4,4,5,7,9,13,20}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,3,4,4,5,7,9,13,20}
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,3,4,4,5,7,9,13,20}
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,3,4,4,5,7,9,13,20}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,3,4,4,5,7,9,13,20}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,3,4,4,5,7,9,13,20}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,3,4,4,5,7,9,13,20}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,3,4,4,5,7,9,13,20}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,3,4,4,5,7,9,13,20}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,3,4,4,5,7,9,13,20}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,3,4,4,5,7,9,13,20}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,3,4,4,5,7,9,13,20}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,2,2,2,2,3,3,4,4,5,7,9,13,20}
([],6)
 => ([],6)
 => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(4,5)],6)
 => ([],5)
 => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => 0
([(2,5),(3,5),(4,5)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0
([(2,5),(3,4)],6)
 => ([],4)
 => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(2,5),(3,4),(4,5)],6)
 => ([(2,6),(3,5),(4,5),(4,6)],7)
 => 1
([(1,2),(3,5),(4,5)],6)
 => ([(2,4),(3,4)],5)
 => 0
([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(1,4),(2,4),(3,4)],5)
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,4),(3,4)],5)
 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,4),(3,4)],5)
 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([],3)
 => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([(0,5),(1,5),(2,4),(3,4)],6)
 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,3),(2,3)],4)
 => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
 => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,3),(2,3)],4)
 => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([],2)
 => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(1,2)],3)
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],2)
 => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(2,3)],6)
 => ([],3)
 => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
 => ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,3),(2,3)],4)
 => 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,3),(2,3)],4)
 => 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 0
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,3),(2,3)],4)
 => 0
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 0
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
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: St001570
(load all 15 compositions to match this statistic)
Mapping
Mp00156: Graphs —line graph⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 38%●distinct values known / distinct values provided: 12%
Values
([],1)
 => ([],0)
 => ? = 0
([],2)
 => ([],0)
 => ? ∊ {0,0}
([(0,1)],2)
 => ([],1)
 => ? ∊ {0,0}
([],3)
 => ([],0)
 => ? ∊ {0,0,0}
([(1,2)],3)
 => ([],1)
 => ? ∊ {0,0,0}
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,0,0}
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([],4)
 => ([],0)
 => ? ∊ {0,0,2,4}
([(2,3)],4)
 => ([],1)
 => ? ∊ {0,0,2,4}
([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,2,4}
([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,3),(1,2)],4)
 => ([],2)
 => ? ∊ {0,0,2,4}
([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 0
([],5)
 => ([],0)
 => ? ∊ {1,2,2,3,3,4,4,5,7,9,13,20}
([(3,4)],5)
 => ([],1)
 => ? ∊ {1,2,2,3,3,4,4,5,7,9,13,20}
([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {1,2,2,3,3,4,4,5,7,9,13,20}
([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(1,4),(2,3)],5)
 => ([],2)
 => ? ∊ {1,2,2,3,3,4,4,5,7,9,13,20}
([(1,4),(2,3),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(0,1),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,4),(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
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 0
([(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
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
([(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
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? ∊ {1,2,2,3,3,4,4,5,7,9,13,20}
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,2,3,3,4,4,5,7,9,13,20}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,2,3,3,4,4,5,7,9,13,20}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,2,2,3,3,4,4,5,7,9,13,20}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
 => ? ∊ {1,2,2,3,3,4,4,5,7,9,13,20}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,2,2,3,3,4,4,5,7,9,13,20}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {1,2,2,3,3,4,4,5,7,9,13,20}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
 => ? ∊ {1,2,2,3,3,4,4,5,7,9,13,20}
([],6)
 => ([],0)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(4,5)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(2,5),(3,4)],6)
 => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(2,5),(3,4),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 1
([(1,2),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 2
([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 0
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([(0,3),(1,2)],4)
 => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,2),(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,5),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,5),(0,7),(1,2),(1,4),(1,6),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,6),(2,3),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,2),(1,4),(1,7),(2,3),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(4,5)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(0,6),(0,7),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,2),(1,4),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,6),(0,7),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,2),(1,4),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,13,13,13,14,15,15,16,16,18,18,19,19,20,20,21,24,25,27,28,29,30,33,35,41,46,52,59,78,106}
Description
The minimal number of edges to add to make a graph Hamiltonian. A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
The following 209 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001846The number of elements which do not have a complement in the lattice. 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. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St000117The number of centered tunnels of a Dyck path. St000699The toughness times the least common multiple of 1,. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000016The number of attacking pairs of a standard tableau. St000017The number of inversions of a standard tableau. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000185The weighted size of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000290The major index of a binary word. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000377The dinv defect of an integer partition. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000547The number of even non-empty partial sums of an integer partition. St000628The balance of a binary word. St000629The defect of a binary word. St000661The number of rises of length 3 of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000687The dimension of Hom(I,P) for the LNakayama algebra of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000691The number of changes of a binary word. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000877The depth of the binary word interpreted as a path. St000921The number of internal inversions of a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000995The largest even part of an integer partition. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001161The major index north count of a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001192The maximal dimension of Ext2A(S,A) for a simple module S over the corresponding Nakayama algebra A. St001193The dimension of Ext1A(A/AeA,A) in the corresponding Nakayama algebra A such that eA is a minimal faithful projective-injective module. St001214The aft of an integer partition. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by τΩ1 composed with its inverse in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001371The length of the longest Yamanouchi prefix of a binary word. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001414Half the length of the longest odd length palindromic prefix of a binary word. 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. St001423The number of distinct cubes in a binary word. St001435The number of missing boxes in the first row. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001438The number of missing boxes of a skew partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001485The modular major index of a binary word. St001524The degree of symmetry of a binary word. St001584The area statistic between a Dyck path and its bounce path. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001596The number of two-by-two squares inside a skew partition. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001721The degree of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001910The height of the middle non-run of a Dyck path. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St001961The sum of the greatest common divisors of all pairs of parts. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001881The number of factors of a lattice as a Cartesian product of lattices. St000478Another weight of a partition according to Alladi. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000567The sum of the products of all pairs of parts. St000929The constant term of the character polynomial of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001577The minimal number of edges to add or remove to make a graph a cograph. St000145The Dyson rank of a partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001195The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af. St001703The villainy of a graph. St001845The number of join irreducibles minus the rank of a lattice. St000068The number of minimal elements in a poset. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000674The number of hills of a Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series L=[c0,c1,...,cn−1] such that n=c0<ci for all i>0 a special CNakayama algebra. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St000456The monochromatic index of a connected graph. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001301The first Betti number of the order complex associated with the poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000181The number of connected components of the Hasse diagram for the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000323The minimal crossing number of a 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. St000552The number of cut vertices of a graph. St001793The difference between the clique number and the chromatic number of a graph. St000553The number of blocks of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000916The packing number of a graph. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph.