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Your data matches 5 different statistics following compositions of up to 3 maps.
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(click to perform a complete search on your data)
Matching statistic: St000468
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(load all 2 compositions to match this statistic)
Values
([],1)
=> 1
([],2)
=> 1
([(0,1)],2)
=> 2
([],3)
=> 1
([(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> 3
([(0,1),(0,2),(1,2)],3)
=> 4
([],4)
=> 1
([(2,3)],4)
=> 2
([(1,3),(2,3)],4)
=> 3
([(0,3),(1,3),(2,3)],4)
=> 4
([(0,3),(1,2)],4)
=> 4
([(0,3),(1,2),(2,3)],4)
=> 5
([(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(1,2),(1,3),(2,3)],4)
=> 6
([(0,2),(0,3),(1,2),(1,3)],4)
=> 7
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 8
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 10
([],5)
=> 1
([(3,4)],5)
=> 2
([(2,4),(3,4)],5)
=> 3
([(1,4),(2,4),(3,4)],5)
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> 5
([(1,4),(2,3)],5)
=> 4
([(1,4),(2,3),(3,4)],5)
=> 5
([(0,1),(2,4),(3,4)],5)
=> 6
([(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,4),(2,3),(3,4)],5)
=> 7
([(1,4),(2,3),(2,4),(3,4)],5)
=> 6
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
([(1,3),(1,4),(2,3),(2,4)],5)
=> 7
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 10
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 9
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 11
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 13
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 14
([(0,4),(1,3),(2,3),(2,4)],5)
=> 8
([(0,1),(2,3),(2,4),(3,4)],5)
=> 8
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 10
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 12
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 11
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 13
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 15
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 12
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 14
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 18
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 16
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 19
Description
The Hosoya index of a graph.
This is the total number of matchings in the graph.
Matching statistic: St000300
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],0)
=> 1
([],2)
=> ([],0)
=> 1
([(0,1)],2)
=> ([],1)
=> 2
([],3)
=> ([],0)
=> 1
([(1,2)],3)
=> ([],1)
=> 2
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 3
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([],4)
=> ([],0)
=> 1
([(2,3)],4)
=> ([],1)
=> 2
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,3),(1,2)],4)
=> ([],2)
=> 4
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 5
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 7
([(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)
=> 8
([(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)
=> 10
([],5)
=> ([],0)
=> 1
([(3,4)],5)
=> ([],1)
=> 2
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
([(1,4),(2,3)],5)
=> ([],2)
=> 4
([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 5
([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 6
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 7
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
([(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)
=> 8
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 7
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 10
([(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)
=> 8
([(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)
=> 9
([(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)
=> 11
([(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)
=> 13
([(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)
=> 14
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 8
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 8
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 10
([(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)
=> 12
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 11
([(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)
=> 13
([(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)
=> 15
([(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)
=> 12
([(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)
=> 10
([(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)
=> 14
([(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)
=> 18
([(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)
=> 16
([(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)
=> 19
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? ∊ {36,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ?
=> ? ∊ {36,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? ∊ {36,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? ∊ {36,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? ∊ {36,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? ∊ {36,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? ∊ {36,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? ∊ {36,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(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)
=> ?
=> ? ∊ {36,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(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)
=> ?
=> ? ∊ {36,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ?
=> ? ∊ {36,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ?
=> ? ∊ {36,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? ∊ {36,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ?
=> ? ∊ {36,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(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)
=> ?
=> ? ∊ {36,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? ∊ {36,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(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)
=> ?
=> ? ∊ {36,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(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)
=> ?
=> ? ∊ {36,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
Description
The number of independent sets of vertices of a graph.
An independent set of vertices of a graph $G$ is a subset $U \subset V(G)$ such that no two vertices in $U$ are adjacent.
This is also the number of vertex covers of $G$ as the map $U \mapsto V(G)\setminus U$ is a bijection between independent sets of vertices and vertex covers.
The size of the largest independent set, also called independence number of $G$, is [[St000093]]
Matching statistic: St000454
Values
([],1)
=> ([],0)
=> ([],1)
=> 0 = 1 - 1
([],2)
=> ([],0)
=> ([],1)
=> 0 = 1 - 1
([(0,1)],2)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 2 - 1
([],3)
=> ([],0)
=> ([],1)
=> 0 = 1 - 1
([(1,2)],3)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([],4)
=> ([],0)
=> ([],1)
=> 0 = 1 - 1
([(2,3)],4)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> ? ∊ {4,5,6,7,8,10} - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {4,5,6,7,8,10} - 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,5,6,7,8,10} - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {4,5,6,7,8,10} - 1
([(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),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,5,6,7,8,10} - 1
([(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)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? ∊ {4,5,6,7,8,10} - 1
([],5)
=> ([],0)
=> ([],1)
=> 0 = 1 - 1
([(3,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 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),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(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),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(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),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(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),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(0,5),(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)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(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),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(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),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 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),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 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),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(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),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(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),(0,7),(1,2),(1,4),(1,5),(1,7),(2,3),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(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),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(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)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(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)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(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)
=> ([(0,3),(0,4),(0,5),(0,6),(0,8),(1,2),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,7),(2,8),(3,4),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(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)
=> ([(0,1),(0,5),(0,6),(0,7),(1,3),(1,4),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(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)
=> ([(0,2),(0,3),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,8),(2,5),(2,7),(2,8),(3,4),(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)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(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)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(0,9),(1,2),(1,3),(1,7),(1,8),(1,9),(2,3),(2,5),(2,6),(2,8),(2,9),(3,4),(3,6),(3,7),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(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)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,8),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,10),(4,5),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,7),(6,8),(6,10),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([],6)
=> ([],0)
=> ([],1)
=> 0 = 1 - 1
([(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(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),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 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),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
([(1,2),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 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),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 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)
=> ([(0,3),(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)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
([(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)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 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)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
([(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)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
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: St001645
Values
([],1)
=> ([],0)
=> ([],0)
=> ? = 1
([],2)
=> ([],0)
=> ([],0)
=> ? = 2
([(0,1)],2)
=> ([],1)
=> ([],1)
=> 1
([],3)
=> ([],0)
=> ([],0)
=> ? ∊ {2,3,4}
([(1,2)],3)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,3,4}
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,3,4}
([],4)
=> ([],0)
=> ([],0)
=> ? ∊ {3,4,4,4,5,6,7,8,10}
([(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {3,4,4,4,5,6,7,8,10}
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {3,4,4,4,5,6,7,8,10}
([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,1)],2)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {3,4,4,4,5,6,7,8,10}
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {3,4,4,4,5,6,7,8,10}
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {3,4,4,4,5,6,7,8,10}
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {3,4,4,4,5,6,7,8,10}
([(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)
=> ([(1,4),(2,3)],5)
=> ? ∊ {3,4,4,4,5,6,7,8,10}
([(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),(1,4),(2,3)],6)
=> ? ∊ {3,4,4,4,5,6,7,8,10}
([],5)
=> ([],0)
=> ([],0)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 4
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(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)
=> ([(2,4),(3,4)],5)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 16
([(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)
=> ([(1,4),(2,3)],5)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(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)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(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)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 8
([(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)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 8
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(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),(1,4),(2,3),(3,4)],5)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(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)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 5
([(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)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(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)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(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)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(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),(1,4),(2,3)],6)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(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)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(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)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(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)
=> ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(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)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,7),(5,6)],8)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(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)
=> ([(0,7),(0,8),(1,5),(1,6),(2,3),(2,4),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([(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)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ? ∊ {3,4,4,5,6,6,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,18,19,22,26}
([],6)
=> ([],0)
=> ([],0)
=> ? ∊ {4,5,6,7,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(4,5)],6)
=> ([],1)
=> ([],1)
=> 1
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {4,5,6,7,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {4,5,6,7,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {4,5,6,7,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(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)
=> ([],5)
=> ? ∊ {4,5,6,7,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 2
([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {4,5,6,7,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(1,2),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 4
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {4,5,6,7,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {4,5,6,7,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {4,5,6,7,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76}
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 16
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 8
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 8
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 5
([(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)
=> ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> 7
([(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)
=> 6
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6
Description
The pebbling number of a connected graph.
Matching statistic: St001232
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 12%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 12%
Values
([],1)
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
([],2)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(0,2),(1,2)],3)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 4 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
([(1,3),(2,3)],4)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {4,4,5,6,7,8,10} - 1
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,5,6,7,8,10} - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {4,4,5,6,7,8,10} - 1
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,5,6,7,8,10} - 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {4,4,5,6,7,8,10} - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {4,4,5,6,7,8,10} - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {4,4,5,6,7,8,10} - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,9,10,10,10,11,11,12,12,13,13,14,14,15,16,18,19,22,26} - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
([],6)
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 3 - 1
([(3,5),(4,5)],6)
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
([(2,5),(3,5),(4,5)],6)
=> [2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
([(2,5),(3,4),(4,5)],6)
=> [2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
([(1,2),(3,5),(4,5)],6)
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {4,4,5,6,6,7,7,8,8,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,16,16,16,16,16,16,16,17,17,17,18,18,18,18,18,18,18,19,19,19,20,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,24,24,24,24,24,24,25,25,26,26,26,26,26,26,27,27,27,28,28,28,28,29,29,30,30,30,30,30,31,31,32,32,32,34,34,34,34,35,35,36,36,36,36,37,38,38,40,40,42,42,42,43,44,46,46,48,50,51,56,58,66,76} - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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