searching the database
Your data matches 82 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001475
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],1)
=> 1
([],2)
=> 1
([(0,1)],2)
=> 1
([],3)
=> 1
([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(1,2)],3)
=> 2
([],4)
=> 1
([(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> 1
([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
([],5)
=> 1
([(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> 1
([(1,4),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3)],5)
=> 1
([(1,4),(2,3),(3,4)],5)
=> 1
([(0,1),(2,4),(3,4)],5)
=> 1
([(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 7
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 6
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 12
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 10
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 14
Description
The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0).
Matching statistic: St000771
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
([],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> ([],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {1,2}
([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {1,2}
([],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {1,1,1,2,3,4,6}
([(0,3),(1,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {1,1,1,2,3,4,6}
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {1,1,1,2,3,4,6}
([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,3,4,6}
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {1,1,1,2,3,4,6}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {1,1,1,2,3,4,6}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {1,1,1,2,3,4,6}
([],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {1,1,1,2,2,2,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,4),(2,3)],5)
=> ([(0,4),(1,4),(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)
=> 3
([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {1,1,1,2,2,2,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {1,1,1,2,2,2,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(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)
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,2,2,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {1,1,1,2,2,2,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([(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)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,2,2,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {1,1,1,2,2,2,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {1,1,1,2,2,2,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,2,2,2,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? ∊ {1,1,1,2,2,2,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ? ∊ {1,1,1,2,2,2,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? ∊ {1,1,1,2,2,2,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? ∊ {1,1,1,2,2,2,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {1,1,1,2,2,2,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {1,1,1,2,2,2,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,2,2,2,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(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)
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(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)
=> ? ∊ {1,1,1,2,2,2,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ? ∊ {1,1,1,2,2,2,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,2,2,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {1,1,1,2,2,2,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {1,1,1,2,2,2,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? ∊ {1,1,1,2,2,2,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(2,5),(3,4)],6)
=> ([(0,4),(1,4),(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)
=> 3
([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
([(1,2),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(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)
=> 3
([(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(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)
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(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)
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,2),(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),(3,5),(4,5)],6)
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(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)
=> 1
([(0,5),(1,5),(2,4),(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)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([(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)
=> ([(0,4),(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)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,4),(2,3)],6)
=> ([],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)
=> ? ∊ {1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4)],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),(4,5)],6)
=> ([(2,5),(3,4)],6)
=> ? ∊ {1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4)],6)
=> ([(0,1),(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)],6)
=> ? ∊ {1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(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)
=> ([(3,5),(4,5)],6)
=> ? ∊ {1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4)],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)
=> 2
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000772
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 27%●distinct values known / distinct values provided: 14%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 27%●distinct values known / distinct values provided: 14%
Values
([],1)
=> [1] => [1] => ([],1)
=> 1
([],2)
=> [2] => [1,1] => ([(0,1)],2)
=> 1
([(0,1)],2)
=> [1,1] => [2] => ([],2)
=> ? = 1
([],3)
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,2)],3)
=> [1,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1}
([(0,2),(1,2)],3)
=> [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1}
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
([],4)
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(2,3)],4)
=> [1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,4,6}
([(1,3),(2,3)],4)
=> [1,1,2] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,2,4,6}
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,4,6}
([(0,3),(1,2)],4)
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [4] => ([],4)
=> ? ∊ {1,1,1,2,4,6}
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [4] => ([],4)
=> ? ∊ {1,1,1,2,4,6}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,4,6}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([],5)
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(3,4)],5)
=> [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,2,2,2,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(2,4),(3,4)],5)
=> [1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,2,2,2,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,2,2,2,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,2,2,2,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,4),(2,3)],5)
=> [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,1,2,2,2,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,1,2,2,2,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {1,1,1,2,2,2,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,1,2,2,2,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,2,2,2,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,2,2,2,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {1,1,1,2,2,2,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {1,1,1,2,2,2,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {1,1,1,2,2,2,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,2,2,2,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {1,1,1,2,2,2,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {1,1,1,2,2,2,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,2,2,2,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {1,1,1,2,2,2,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {1,1,1,2,2,2,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {1,1,1,2,2,2,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,2,2,2,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {1,1,1,2,2,2,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([],6)
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(4,5)],6)
=> [1,5] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(3,5),(4,5)],6)
=> [1,1,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(2,5),(3,4)],6)
=> [2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1,3] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(1,2),(3,5),(4,5)],6)
=> [1,1,1,3] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(3,4),(3,5),(4,5)],6)
=> [2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,2] => [1,5] => ([(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,3] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,3,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [1,5] => ([(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1,1] => [6] => ([],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [1,5] => ([(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [6] => ([],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,3,1] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(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] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(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)
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [4,2] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(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)
=> [2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St000777
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([],4)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,3,4,6}
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,3,4,6}
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,3,4,6}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,3,4,6}
([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {2,2,3,4,6}
([],5)
=> ([],5)
=> ([],1)
=> ([],1)
=> 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(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)
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(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)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(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),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(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)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ([],4)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(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)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([],6)
=> ([],6)
=> ([],1)
=> ([],1)
=> 1
([(4,5)],6)
=> ([],5)
=> ([],1)
=> ([],1)
=> 1
([(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 3
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(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)
=> 3
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000259
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([],2)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([],3)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([],4)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,3,4,6} - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,3,4,6} - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,3,4,6} - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,3,4,6} - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {2,2,3,4,6} - 1
([],5)
=> ([],5)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 3 - 1
([(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),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(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)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ([],4)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(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)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([],6)
=> ([],6)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(4,5)],6)
=> ([],5)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 3 - 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 3 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(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)
=> 2 = 3 - 1
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000260
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([],2)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([],3)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([],4)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,3,4,6} - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,3,4,6} - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,3,4,6} - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,3,4,6} - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {2,2,3,4,6} - 1
([],5)
=> ([],5)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 3 - 1
([(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),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(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)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ([],4)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([(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)
=> ? ∊ {2,2,2,2,2,2,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24} - 1
([],6)
=> ([],6)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(4,5)],6)
=> ([],5)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 3 - 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 3 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(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)
=> 2 = 3 - 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000302
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([],2)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([],3)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([],4)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,3,4,6} - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,3,4,6} - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,3,4,6} - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,3,4,6} - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {2,2,3,4,6} - 1
([],5)
=> ([],5)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,8,8,10,12,14,18,24} - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,8,8,10,12,14,18,24} - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,8,8,10,12,14,18,24} - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,8,8,10,12,14,18,24} - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,8,8,10,12,14,18,24} - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,8,8,10,12,14,18,24} - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,8,8,10,12,14,18,24} - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,8,8,10,12,14,18,24} - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,8,8,10,12,14,18,24} - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,8,8,10,12,14,18,24} - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,8,8,10,12,14,18,24} - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,8,8,10,12,14,18,24} - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,8,8,10,12,14,18,24} - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 6 = 7 - 1
([(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),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,8,8,10,12,14,18,24} - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,8,8,10,12,14,18,24} - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,8,8,10,12,14,18,24} - 1
([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {2,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,8,8,10,12,14,18,24} - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {2,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,8,8,10,12,14,18,24} - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {2,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,8,8,10,12,14,18,24} - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,8,8,10,12,14,18,24} - 1
([(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)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,8,8,10,12,14,18,24} - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ([],4)
=> ? ∊ {2,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,8,8,10,12,14,18,24} - 1
([(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)
=> ? ∊ {2,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,8,8,10,12,14,18,24} - 1
([],6)
=> ([],6)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(4,5)],6)
=> ([],5)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120} - 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 6 = 7 - 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 6 = 7 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(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)
=> 6 = 7 - 1
Description
The determinant of the distance matrix of a connected graph.
Matching statistic: St000284
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000284: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 16%●distinct values known / distinct values provided: 8%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000284: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 16%●distinct values known / distinct values provided: 8%
Values
([],1)
=> [1]
=> []
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> ? ∊ {1,1}
([(0,1)],2)
=> [2]
=> []
=> ? ∊ {1,1}
([],3)
=> [1,1,1]
=> [1,1]
=> 1
([(1,2)],3)
=> [2,1]
=> [1]
=> ? ∊ {1,1,2}
([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {1,1,2}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {1,1,2}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,2,2,3,4,6}
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,3,4,6}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,3,4,6}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,2,2,3,4,6}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,3,4,6}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,3,4,6}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,3,4,6}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,3,4,6}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 4
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 9
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 4
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 4
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 4
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2]
=> [2]
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [3]
=> 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1
Description
The Plancherel distribution on integer partitions.
This is defined as the distribution induced by the RSK shape of the uniform distribution on permutations. In other words, this is the size of the preimage of the map 'Robinson-Schensted tableau shape' from permutations to integer partitions.
Equivalently, this is given by the square of the number of standard Young tableaux of the given shape.
Matching statistic: St000668
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 16%●distinct values known / distinct values provided: 8%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 16%●distinct values known / distinct values provided: 8%
Values
([],1)
=> [1]
=> []
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> ? ∊ {1,1}
([(0,1)],2)
=> [2]
=> []
=> ? ∊ {1,1}
([],3)
=> [1,1,1]
=> [1,1]
=> 1
([(1,2)],3)
=> [2,1]
=> [1]
=> ? ∊ {1,1,2}
([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {1,1,2}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {1,1,2}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,2,3,4,6}
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,1,1,2,3,4,6}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,1,1,2,3,4,6}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,2,3,4,6}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,1,1,2,3,4,6}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,1,1,2,3,4,6}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,1,1,2,3,4,6}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,1,1,2,3,4,6}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 2
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 2
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2]
=> [2]
=> 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [3]
=> 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
Description
The least common multiple of the parts of the partition.
Matching statistic: St000704
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 16%●distinct values known / distinct values provided: 8%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 16%●distinct values known / distinct values provided: 8%
Values
([],1)
=> [1]
=> []
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> ? ∊ {1,1}
([(0,1)],2)
=> [2]
=> []
=> ? ∊ {1,1}
([],3)
=> [1,1,1]
=> [1,1]
=> 1
([(1,2)],3)
=> [2,1]
=> [1]
=> ? ∊ {1,1,2}
([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {1,1,2}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {1,1,2}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,2,2,3,4,6}
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,3,4,6}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,3,4,6}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,2,2,3,4,6}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,3,4,6}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,3,4,6}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,3,4,6}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,3,4,6}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,6,6,6,7,8,8,10,12,14,18,24}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 3
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 2
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,12,12,12,12,12,12,12,12,12,14,14,14,14,14,15,16,16,16,16,16,16,17,18,18,18,18,20,20,20,22,22,24,24,24,24,24,24,24,26,28,28,30,30,31,32,32,34,36,36,36,38,42,42,46,46,48,48,54,54,60,64,72,78,96,120}
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2]
=> [2]
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [3]
=> 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1
Description
The number of semistandard tableaux on a given integer partition with minimal maximal entry.
This is, for an integer partition $\lambda = (\lambda_1 > \cdots > \lambda_k > 0)$, the number of [[SemistandardTableaux|semistandard tableaux]] of shape $\lambda$ with maximal entry $k$.
Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $k$ variables, or, explicitly,
$$ \prod_{(i,j) \in L} \frac{k + j - i}{ \operatorname{hook}(i,j) }$$
where the product is over all cells $(i,j) \in L$ and $\operatorname{hook}(i,j)$ is the hook length of a cell.
See [Theorem 6.3, 1] for details.
The following 72 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000418The number of Dyck paths that are weakly below a Dyck path. St000444The length of the maximal rise of a Dyck path. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000744The length of the path to the largest entry in a standard Young tableau. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001531Number of partial orders contained in the poset determined by the Dyck path. St001808The box weight or horizontal decoration of a Dyck path. St001959The product of the heights of the peaks of a Dyck path. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001845The number of join irreducibles minus the rank of a lattice. St001626The number of maximal proper sublattices of a lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001877Number of indecomposable injective modules with projective dimension 2. St000455The second largest eigenvalue of a graph if it is integral. St001396Number of triples of incomparable elements in a finite poset. St001964The interval resolution global dimension of a poset. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000544The cop number of a graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!