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Your data matches 51 different statistics following compositions of up to 3 maps.
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Matching statistic: St001095
Values
([],2)
=> 1
([(0,1)],2)
=> 0
([],3)
=> 1
([(1,2)],3)
=> 3
([(0,1),(0,2)],3)
=> 0
([(0,2),(2,1)],3)
=> 0
([(0,2),(1,2)],3)
=> 0
([],4)
=> 1
([(2,3)],4)
=> 4
([(1,2),(1,3)],4)
=> 4
([(0,1),(0,2),(0,3)],4)
=> 0
([(0,2),(0,3),(3,1)],4)
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(1,2),(2,3)],4)
=> 5
([(0,3),(3,1),(3,2)],4)
=> 0
([(1,3),(2,3)],4)
=> 4
([(0,3),(1,3),(3,2)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2)],4)
=> 4
([(0,3),(1,2),(1,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> 0
([(0,3),(2,1),(3,2)],4)
=> 0
([(0,3),(1,2),(2,3)],4)
=> 1
([],5)
=> 1
([(3,4)],5)
=> 4
([(2,3),(2,4)],5)
=> 5
([(1,2),(1,3),(1,4)],5)
=> 4
([(0,1),(0,2),(0,3),(0,4)],5)
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 0
([(1,3),(1,4),(4,2)],5)
=> 9
([(0,3),(0,4),(4,1),(4,2)],5)
=> 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> 6
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 0
([(2,3),(3,4)],5)
=> 6
([(1,4),(4,2),(4,3)],5)
=> 6
([(0,4),(4,1),(4,2),(4,3)],5)
=> 0
([(2,4),(3,4)],5)
=> 5
([(1,4),(2,4),(4,3)],5)
=> 6
([(0,4),(1,4),(4,2),(4,3)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,3)],5)
=> 8
([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
Description
The number of non-isomorphic posets with precisely one further covering relation.
Matching statistic: St000456
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],2)
=> ([(0,1)],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1}
([(0,1)],2)
=> ([],2)
=> ([],0)
=> ([],0)
=> ? ∊ {0,1}
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {0,0,0,1,3}
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {0,0,0,1,3}
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,3}
([(0,2),(2,1)],3)
=> ([],3)
=> ([],0)
=> ([],0)
=> ? ∊ {0,0,0,1,3}
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,3}
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,4,4,4,4,5}
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,4,4,4,4,5}
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,4,4,4,4,5}
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,4,4,4,4,5}
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,4,4,4,4,5}
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,4,4,4,4,5}
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,4,4,4,4,5}
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,4,4,4,4,5}
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,4,4,4,4,5}
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,4,4,4,4,5}
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,4,4,4,4,5}
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,4,4,4,4,5}
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,4,4,4,4,5}
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,1)],2)
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],0)
=> ([],0)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,4,4,4,4,5}
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,4,4,4,4,5}
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,7),(0,8),(1,5),(1,6),(2,3),(2,4),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,1)],2)
=> 1
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,1)],2)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,7),(5,6)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> 2
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,1)],2)
=> 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 3
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(0,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 3
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> 2
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 1
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 3
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> 2
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(1,5),(2,5),(5,3),(5,4)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1
([(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)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(4,3)],6)
=> ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(1,4),(1,5),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> 5
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St001440
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001440: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 39%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001440: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 39%
Values
([],2)
=> [2]
=> []
=> []
=> ? ∊ {0,1}
([(0,1)],2)
=> [1]
=> []
=> []
=> ? ∊ {0,1}
([],3)
=> [3,3]
=> [3]
=> [1,1,1]
=> 0
([(1,2)],3)
=> [3]
=> []
=> []
=> ? ∊ {0,0,1,3}
([(0,1),(0,2)],3)
=> [2]
=> []
=> []
=> ? ∊ {0,0,1,3}
([(0,2),(2,1)],3)
=> [1]
=> []
=> []
=> ? ∊ {0,0,1,3}
([(0,2),(1,2)],3)
=> [2]
=> []
=> []
=> ? ∊ {0,0,1,3}
([],4)
=> [4,4,4,4,4,4]
=> [4,4,4,4,4]
=> [5,5,5,5]
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(2,3)],4)
=> [4,4,4]
=> [4,4]
=> [2,2,2,2]
=> 1
([(1,2),(1,3)],4)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,1),(0,2),(0,3)],4)
=> [3,3]
=> [3]
=> [1,1,1]
=> 0
([(0,2),(0,3),(3,1)],4)
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> []
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(1,2),(2,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,3),(3,1),(3,2)],4)
=> [2]
=> []
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(1,3),(2,3)],4)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,3),(1,3),(3,2)],4)
=> [2]
=> []
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,3),(1,3),(2,3)],4)
=> [3,3]
=> [3]
=> [1,1,1]
=> 0
([(0,3),(1,2)],4)
=> [4,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(2,1),(3,2)],4)
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,3),(1,2),(2,3)],4)
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
=> [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
=> [23,23,23,23,23]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> [5,5,5,5,5,5,5,5,5,5,5]
=> [11,11,11,11,11]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(2,3),(2,4)],5)
=> [10,10,10,10]
=> [10,10,10]
=> [3,3,3,3,3,3,3,3,3,3]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,2),(1,3),(1,4)],5)
=> [15,15]
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(0,3),(0,4)],5)
=> [4,4,4,4,4,4]
=> [4,4,4,4,4]
=> [5,5,5,5]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> [4,4]
=> [2,2,2,2]
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [8]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [3]
=> [1,1,1]
=> 0
([(1,3),(1,4),(4,2)],5)
=> [15]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(4,1),(4,2)],5)
=> [8]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,2),(1,3),(2,4),(3,4)],5)
=> [5,5]
=> [5]
=> [1,1,1,1,1]
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,2]
=> [2]
=> [1,1]
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(2,3),(3,4)],5)
=> [5,5,5,5]
=> [5,5,5]
=> [3,3,3,3,3]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> [5]
=> [1,1,1,1,1]
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,3]
=> [3]
=> [1,1,1]
=> 0
([(2,4),(3,4)],5)
=> [10,10,10,10]
=> [10,10,10]
=> [3,3,3,3,3,3,3,3,3,3]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> [5]
=> [1,1,1,1,1]
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [15,15]
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,3]
=> [3]
=> [1,1,1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,4,4,4,4,4]
=> [4,4,4,4,4]
=> [5,5,5,5]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3)],5)
=> [10,10]
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [12,4]
=> [4]
=> [1,1,1,1]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [14]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6,6]
=> [6]
=> [1,1,1,1,1,1]
=> 0
([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3),(2,4)],5)
=> [10,4,4]
=> [4,4]
=> [2,2,2,2]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> [4,4]
=> [2,2,2,2]
=> 1
([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> [5,5,5,5,5]
=> [5,5,5,5,5]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> [5,5]
=> [2,2,2,2,2]
=> 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> [4]
=> [1,1,1,1]
=> 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [5,5,5,5]
=> [5,5,5]
=> [3,3,3,3,3]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,3)],5)
=> [10,10]
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> [10,4,4]
=> [4,4]
=> [2,2,2,2]
=> 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> [4,3]
=> [2,2,2,1]
=> 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(1,4)],5)
=> [12,4]
=> [4]
=> [1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [14]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [6,6]
=> [6]
=> [1,1,1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [5,3]
=> [3]
=> [1,1,1]
=> 0
([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> [4]
=> [1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(3,2),(4,3)],5)
=> [5]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(2,3),(3,4)],5)
=> [15]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(1,4),(4,2)],5)
=> [5,5]
=> [5]
=> [1,1,1,1,1]
=> 0
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [2]
=> [1,1]
=> 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [3,3]
=> [3]
=> [1,1,1]
=> 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> [10,10]
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> [12,4]
=> [4]
=> [1,1,1,1]
=> 0
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6,6]
=> [6]
=> [1,1,1,1,1,1]
=> 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [10,4,4]
=> [4,4]
=> [2,2,2,2]
=> 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,5]
=> [5]
=> [1,1,1,1,1]
=> 0
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> [15,5,5]
=> [5,5]
=> [2,2,2,2,2]
=> 3
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> [10,10]
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> 0
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> [10,4,4]
=> [4,4]
=> [2,2,2,2]
=> 1
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> [12,4]
=> [4]
=> [1,1,1,1]
=> 0
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [6,6]
=> [6]
=> [1,1,1,1,1,1]
=> 0
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [3,3]
=> [3]
=> [1,1,1]
=> 0
Description
The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition.
Matching statistic: St001876
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1}
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1}
([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,3}
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,3}
([(0,1),(0,2)],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,3}
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,3}
([],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,4,4,4,4,5}
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,1,4,4,4,4,5}
([(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,4,4,4,4,5}
([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,4,4,4,4,5}
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,1,4,4,4,4,5}
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,4,4,4,4,5}
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,4,4,4,4,5}
([(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,4,4,4,4,5}
([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,4,4,4,4,5}
([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,4,4,4,4,5}
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,1,4,4,4,4,5}
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,1,4,4,4,4,5}
([],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(1,4),(4,2),(4,3)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(1,4),(1,5),(4,3),(5,2)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(2,3),(3,5),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,5),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,5),(1,5),(2,3),(3,4)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,5),(1,5),(2,3),(3,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(2,5),(3,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,3)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St000046
Mp00074: Posets —to graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000046: Integer partitions ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 28%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000046: Integer partitions ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 28%
Values
([],2)
=> ([],2)
=> [1,1]
=> [1]
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 0
([],3)
=> ([],3)
=> [1,1,1]
=> [1,1]
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,3}
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,3}
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,3}
([],4)
=> ([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 0
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,4,4,4,5}
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,4,4,4,5}
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,4,4,4,5}
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,4,4,4,5}
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,4,4,4,5}
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,4,4,4,5}
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 4
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,4,4,4,5}
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,4,4,4,5}
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,4,4,4,5}
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,4,4,4,5}
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(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)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 4
([(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 4
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 4
([],6)
=> ([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
([(4,5)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,4),(3,5)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(2,3),(2,4),(2,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(1,4),(1,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(2,3),(2,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,4),(1,5),(5,2),(5,3)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,4),(1,5),(4,3),(5,2)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(3,4),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(2,3),(3,4),(3,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,5),(5,2),(5,3),(5,4)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(2,3),(3,5),(5,4)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,4),(4,5),(5,2),(5,3)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(2,5),(3,5),(5,4)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
Description
The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition.
Matching statistic: St000137
Mp00074: Posets —to graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000137: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 27%●distinct values known / distinct values provided: 17%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000137: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 27%●distinct values known / distinct values provided: 17%
Values
([],2)
=> ([],2)
=> [1,1]
=> [1]
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 0
([],3)
=> ([],3)
=> [1,1,1]
=> [1,1]
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,3}
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,3}
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,3}
([],4)
=> ([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 0
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 0
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(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)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0
([],6)
=> ([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
([(4,5)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,4),(3,5)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(2,3),(2,4),(2,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(1,4),(1,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(2,3),(2,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,4),(1,5),(5,2),(5,3)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,4),(1,5),(4,3),(5,2)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(3,4),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(2,3),(3,4),(3,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,5),(5,2),(5,3),(5,4)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(2,3),(3,5),(5,4)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,4),(4,5),(5,2),(5,3)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(2,5),(3,5),(5,4)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
Description
The Grundy value of an integer partition.
Consider the two-player game on an integer partition.
In each move, a player removes either a box, or a 2x2-configuration of boxes such that the resulting diagram is still a partition.
The first player that cannot move lose. This happens exactly when the empty partition is reached.
The grundy value of an integer partition is defined as the grundy value of this two-player game as defined in [1].
This game was described to me during Norcom 2013, by Urban Larsson, and it seems to be quite difficult to give a good description of the partitions with Grundy value 0.
Matching statistic: St001122
Mp00074: Posets —to graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001122: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 27%●distinct values known / distinct values provided: 11%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001122: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 27%●distinct values known / distinct values provided: 11%
Values
([],2)
=> ([],2)
=> [1,1]
=> [1]
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 0
([],3)
=> ([],3)
=> [1,1,1]
=> [1,1]
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,3}
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,3}
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,3}
([],4)
=> ([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 0
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,4,4,4,4,5}
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,4,4,4,4,5}
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,4,4,4,4,5}
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,4,4,4,4,5}
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,4,4,4,4,5}
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,4,4,4,4,5}
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 0
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,4,4,4,4,5}
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,4,4,4,4,5}
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,4,4,4,4,5}
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,4,4,4,4,5}
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(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)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0
([],6)
=> ([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(4,5)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,4),(3,5)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(2,4),(2,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(1,4),(1,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(2,3),(2,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,4),(1,5),(5,2),(5,3)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,4),(1,5),(4,3),(5,2)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(3,4),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(3,4),(3,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,5),(5,2),(5,3),(5,4)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(2,3),(3,5),(5,4)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,4),(4,5),(5,2),(5,3)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(2,5),(3,5),(5,4)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
Description
The multiplicity of the sign representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity gλμ,ν of the Specht module Sλ in Sμ⊗Sν:
Sμ⊗Sν=⨁λgλμ,νSλ
This statistic records the Kronecker coefficient g1nλ,λ, for λ⊢n. It equals 1 if and only if λ is self-conjugate.
Matching statistic: St001525
Mp00074: Posets —to graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001525: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 27%●distinct values known / distinct values provided: 17%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001525: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 27%●distinct values known / distinct values provided: 17%
Values
([],2)
=> ([],2)
=> [1,1]
=> [1]
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 0
([],3)
=> ([],3)
=> [1,1,1]
=> [1,1]
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,3}
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,3}
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,3}
([],4)
=> ([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 0
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,4,4,4,4,5}
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,4,4,4,4,5}
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,4,4,4,4,5}
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,4,4,4,4,5}
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,4,4,4,4,5}
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,4,4,4,4,5}
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 0
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,4,4,4,4,5}
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,4,4,4,4,5}
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,4,4,4,4,5}
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,4,4,4,4,5}
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(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)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0
([],6)
=> ([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(4,5)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,4),(3,5)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(2,4),(2,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(1,4),(1,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(2,3),(2,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,4),(1,5),(5,2),(5,3)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,4),(1,5),(4,3),(5,2)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(3,4),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(3,4),(3,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,5),(5,2),(5,3),(5,4)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(2,3),(3,5),(5,4)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,4),(4,5),(5,2),(5,3)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(2,5),(3,5),(5,4)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
Description
The number of symmetric hooks on the diagonal of a partition.
Matching statistic: St001593
Mp00074: Posets —to graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001593: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 27%●distinct values known / distinct values provided: 11%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001593: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 27%●distinct values known / distinct values provided: 11%
Values
([],2)
=> ([],2)
=> [1,1]
=> [1]
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 0
([],3)
=> ([],3)
=> [1,1,1]
=> [1,1]
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,3}
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,3}
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,3}
([],4)
=> ([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 0
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(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)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
([],6)
=> ([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(4,5)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,4),(3,5)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(2,4),(2,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(1,4),(1,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(2,3),(2,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,4),(1,5),(5,2),(5,3)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,4),(1,5),(4,3),(5,2)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(3,4),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(3,4),(3,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,5),(5,2),(5,3),(5,4)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(2,3),(3,5),(5,4)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,4),(4,5),(5,2),(5,3)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(2,5),(3,5),(5,4)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
Description
This is the number of standard Young tableaux of the given shifted shape.
For an integer partition λ=(λ1,…,λk), the shifted diagram is obtained by moving the i-th row in the diagram i−1 boxes to the right, i.e.,
λ∗={(i,j)|1≤i≤k,i≤j≤λi+i−1}.
In particular, this statistic is zero if and only if λi+1=λi for some i.
Matching statistic: St001628
Mp00074: Posets —to graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001628: Integer partitions ⟶ ℤResult quality: 22% ●values known / values provided: 27%●distinct values known / distinct values provided: 22%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001628: Integer partitions ⟶ ℤResult quality: 22% ●values known / values provided: 27%●distinct values known / distinct values provided: 22%
Values
([],2)
=> ([],2)
=> [1,1]
=> [1]
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 0
([],3)
=> ([],3)
=> [1,1,1]
=> [1,1]
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> [1]
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,3}
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,3}
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,3}
([],4)
=> ([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 0
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,4,4,4,4,5}
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(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)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,6,7,7,8,8,9,9,9,11}
([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
([],6)
=> ([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 5
([(4,5)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,4),(3,5)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(2,4),(2,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(1,4),(1,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(2,3),(2,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,4),(1,5),(5,2),(5,3)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,4),(1,5),(4,3),(5,2)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(3,4),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(3,4),(3,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,5),(5,2),(5,3),(5,4)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(2,3),(3,5),(5,4)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,4),(4,5),(5,2),(5,3)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(2,5),(3,5),(5,4)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs.
The following 41 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000567The sum of the products of all pairs of parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000937The number of positive values of the symmetric group character corresponding to the partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000929The constant term of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001632The number of indecomposable injective modules I with dimExt1(I,A)=1 for the incidence algebra A of a poset. St001498The normalised height of a Nakayama algebra with magnitude 1. St001877Number of indecomposable injective modules with projective dimension 2. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St000455The second largest eigenvalue of a graph if it is integral. St001651The Frankl number of a lattice. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001118The acyclic chromatic index of a graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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