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Your data matches 71 different statistics following compositions of up to 3 maps.
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Matching statistic: St001574
Values
([],1)
=> 0
([],2)
=> 0
([(0,1)],2)
=> 0
([],3)
=> 0
([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(1,2)],3)
=> 0
([],4)
=> 0
([(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2)],4)
=> 0
([(0,3),(1,2),(2,3)],4)
=> 1
([(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([],5)
=> 0
([(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> 2
([(1,4),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(1,4),(2,3)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> 2
([(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
Description
The minimal number of edges to add or remove to make a graph regular.
Matching statistic: St001576
Values
([],1)
=> 0
([],2)
=> 0
([(0,1)],2)
=> 0
([],3)
=> 0
([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(1,2)],3)
=> 0
([],4)
=> 0
([(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2)],4)
=> 0
([(0,3),(1,2),(2,3)],4)
=> 1
([(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([],5)
=> 0
([(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> 2
([(1,4),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(1,4),(2,3)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> 2
([(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
Description
The minimal number of edges to add or remove to make a graph vertex transitive.
A graph is vertex transitive if for any two edges there is an automorphism that maps one vertex to the other.
Matching statistic: St001060
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 50%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 50%
Values
([],1)
=> [1] => [1] => ([],1)
=> ? = 0
([],2)
=> [2] => [1] => ([],1)
=> ? ∊ {0,0}
([(0,1)],2)
=> [1,1] => [2] => ([],2)
=> ? ∊ {0,0}
([],3)
=> [3] => [1] => ([],1)
=> ? ∊ {0,0,1,1}
([(1,2)],3)
=> [1,2] => [1,1] => ([(0,1)],2)
=> ? ∊ {0,0,1,1}
([(0,2),(1,2)],3)
=> [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0,1,1}
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,1] => ([(0,1)],2)
=> ? ∊ {0,0,1,1}
([],4)
=> [4] => [1] => ([],1)
=> ? ∊ {0,0,0,0,1,1,1,2}
([(2,3)],4)
=> [1,3] => [1,1] => ([(0,1)],2)
=> ? ∊ {0,0,0,0,1,1,1,2}
([(1,3),(2,3)],4)
=> [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(1,2)],4)
=> [2,2] => [2] => ([],2)
=> ? ∊ {0,0,0,0,1,1,1,2}
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2] => ([],2)
=> ? ∊ {0,0,0,0,1,1,1,2}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,1,1,1,2}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,1] => ([(0,1)],2)
=> ? ∊ {0,0,0,0,1,1,1,2}
([],5)
=> [5] => [1] => ([],1)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4}
([(3,4)],5)
=> [1,4] => [1,1] => ([(0,1)],2)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4}
([(2,4),(3,4)],5)
=> [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 2
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,4),(2,3)],5)
=> [2,3] => [1,1] => ([(0,1)],2)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4}
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [1,1] => ([(0,1)],2)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [1,1] => ([(0,1)],2)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => [1,1] => ([(0,1)],2)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4}
([],6)
=> [6] => [1] => ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(4,5)],6)
=> [1,5] => [1,1] => ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(3,5),(4,5)],6)
=> [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 2
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
([(2,5),(3,4)],6)
=> [2,4] => [1,1] => ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(2,5),(3,4),(4,5)],6)
=> [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
([(1,2),(3,5),(4,5)],6)
=> [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
([(3,4),(3,5),(4,5)],6)
=> [2,4] => [1,1] => ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1,1] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,1,1,1,1] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,2,1,2] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,1,2] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St000259
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
([],1)
=> ([],0)
=> ([],0)
=> ? = 0
([],2)
=> ([],0)
=> ([],0)
=> ? = 0
([(0,1)],2)
=> ([],1)
=> ([],1)
=> 0
([],3)
=> ([],0)
=> ([],0)
=> ? ∊ {0,1,1}
([(1,2)],3)
=> ([],1)
=> ([],1)
=> 0
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {0,1,1}
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {0,1,1}
([],4)
=> ([],0)
=> ([],0)
=> ? ∊ {0,0,0,1,1,2,2,3,3}
([(2,3)],4)
=> ([],1)
=> ([],1)
=> 0
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {0,0,0,1,1,2,2,3,3}
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {0,0,0,1,1,2,2,3,3}
([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,2,2,3,3}
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {0,0,0,1,1,2,2,3,3}
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {0,0,0,1,1,2,2,3,3}
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,1,1,2,2,3,3}
([(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)
=> ? ∊ {0,0,0,1,1,2,2,3,3}
([(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)
=> ? ∊ {0,0,0,1,1,2,2,3,3}
([],5)
=> ([],0)
=> ([],0)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
([(3,4)],5)
=> ([],1)
=> ([],1)
=> 0
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,1)],2)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
([(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)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
([(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)
=> 4
([(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)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
([(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)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
([(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)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> 3
([(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)
=> 4
([(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)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
([(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)
=> 4
([(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)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
([(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
([(0,2),(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)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
([(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)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
([(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)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
([],6)
=> ([],0)
=> ([],0)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(4,5)],6)
=> ([],1)
=> ([],1)
=> 0
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(1,2),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> 3
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(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)
=> 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
([(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)
=> 3
([(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)
=> 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> 3
([(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)
=> 3
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6)
=> 2
([(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)
=> 4
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> 3
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6)],7)
=> 3
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(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)
=> 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001875
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ? = 0
([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0}
([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {0,0}
([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,1}
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1}
([(0,2),(1,2)],3)
=> ([],2)
=> ([],1)
=> ? ∊ {0,0,1,1}
([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {0,0,1,1}
([],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,1,2,2,3}
([(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,1,1,1,2,2,3}
([(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,1,1,1,2,2,3}
([(0,3),(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,1,2,2,3}
([(0,3),(1,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,1,2,2,3}
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,1,1,1,2,2,3}
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,1,1,1,2,2,3}
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],2)
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,1,2,2,3}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],3)
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,1,2,2,3}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,1,2,2,3}
([],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([(1,4),(2,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
([(0,1),(2,4),(3,4)],5)
=> ([],4)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([],5)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],5)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],4)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ([],1)
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
([],6)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(2,5),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(2,5),(3,4)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 4
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(2,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(0,2),(2,1)],3)
=> 3
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(2,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,5),(2,3),(5,4)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St000939
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 50%
Values
([],1)
=> []
=> ?
=> ? = 0
([],2)
=> []
=> ?
=> ? ∊ {0,0}
([(0,1)],2)
=> [1]
=> []
=> ? ∊ {0,0}
([],3)
=> []
=> ?
=> ? ∊ {0,0,1,1}
([(1,2)],3)
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([(0,2),(1,2)],3)
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,1,1}
([],4)
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(2,3)],4)
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(1,3),(2,3)],4)
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> 2
([(0,3),(1,2)],4)
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3]
=> []
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> []
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([],5)
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(3,4)],5)
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(2,4),(3,4)],5)
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3]
=> []
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> []
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> []
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [3]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> []
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> []
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> []
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> []
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> []
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([],6)
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
([(4,5)],6)
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
([(3,5),(4,5)],6)
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 5
([(2,5),(3,4)],6)
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 2
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 2
([(3,4),(3,5),(4,5)],6)
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 5
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 5
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 5
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> 2
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1]
=> 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 5
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,3]
=> [3]
=> 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,3,1]
=> [3,1]
=> 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 3
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> 2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 5
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,1]
=> [1,1]
=> 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 3
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [4,3]
=> [3]
=> 2
Description
The number of characters of the symmetric group whose value on the partition is positive.
Matching statistic: St000993
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 67%
Values
([],1)
=> []
=> ?
=> ? = 0
([],2)
=> []
=> ?
=> ? ∊ {0,0}
([(0,1)],2)
=> [1]
=> []
=> ? ∊ {0,0}
([],3)
=> []
=> ?
=> ? ∊ {0,0,1,1}
([(1,2)],3)
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([(0,2),(1,2)],3)
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,1,1}
([],4)
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(2,3)],4)
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(1,3),(2,3)],4)
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> 2
([(0,3),(1,2)],4)
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3]
=> []
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> []
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([],5)
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(3,4)],5)
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(2,4),(3,4)],5)
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3]
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([],6)
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(4,5)],6)
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(3,5),(4,5)],6)
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(2,5),(3,4)],6)
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 2
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> 2
([(3,4),(3,5),(4,5)],6)
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> 2
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1]
=> 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,3]
=> [3]
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,3,1]
=> [3,1]
=> 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 3
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> 2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,1]
=> [1,1]
=> 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 3
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [4,3]
=> [3]
=> 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000477
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000477: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 83%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000477: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 83%
Values
([],1)
=> []
=> ?
=> ?
=> ? = 0
([],2)
=> []
=> ?
=> ?
=> ? ∊ {0,0}
([(0,1)],2)
=> [1]
=> []
=> []
=> ? ∊ {0,0}
([],3)
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,1}
([(1,2)],3)
=> [1]
=> []
=> []
=> ? ∊ {0,0,1,1}
([(0,2),(1,2)],3)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> []
=> ? ∊ {0,0,1,1}
([],4)
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(2,3)],4)
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(1,3),(2,3)],4)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(0,3),(1,2)],4)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([],5)
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(3,4)],5)
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(2,4),(3,4)],5)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [3]
=> [1,1,1]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([],6)
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(4,5)],6)
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(3,5),(4,5)],6)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
([(2,5),(3,4)],6)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(3,4),(3,5),(4,5)],6)
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> [2]
=> 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> [2]
=> 2
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,3]
=> [3]
=> [1,1,1]
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,3,1]
=> [3,1]
=> [2,1,1]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> [2]
=> 2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [4,3]
=> [3]
=> [1,1,1]
=> 1
Description
The weight of a partition according to Alladi.
Matching statistic: St000668
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 67%
Values
([],1)
=> []
=> ?
=> ?
=> ? = 0
([],2)
=> []
=> ?
=> ?
=> ? ∊ {0,0}
([(0,1)],2)
=> [1]
=> []
=> []
=> ? ∊ {0,0}
([],3)
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,1}
([(1,2)],3)
=> [1]
=> []
=> []
=> ? ∊ {0,0,1,1}
([(0,2),(1,2)],3)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> []
=> ? ∊ {0,0,1,1}
([],4)
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(2,3)],4)
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(1,3),(2,3)],4)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(0,3),(1,2)],4)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([],5)
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(3,4)],5)
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(2,4),(3,4)],5)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [3]
=> [1,1,1]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([],6)
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(4,5)],6)
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(3,5),(4,5)],6)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
([(2,5),(3,4)],6)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(3,4),(3,5),(4,5)],6)
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> [2]
=> 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> [2]
=> 2
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,3]
=> [3]
=> [1,1,1]
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,3,1]
=> [3,1]
=> [2,1,1]
=> 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> [2]
=> 2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [4,3]
=> [3]
=> [1,1,1]
=> 1
Description
The least common multiple of the parts of the partition.
Matching statistic: St000708
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 67%
Values
([],1)
=> []
=> ?
=> ?
=> ? = 0
([],2)
=> []
=> ?
=> ?
=> ? ∊ {0,0}
([(0,1)],2)
=> [1]
=> []
=> []
=> ? ∊ {0,0}
([],3)
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,1}
([(1,2)],3)
=> [1]
=> []
=> []
=> ? ∊ {0,0,1,1}
([(0,2),(1,2)],3)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> []
=> ? ∊ {0,0,1,1}
([],4)
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(2,3)],4)
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(1,3),(2,3)],4)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(0,3),(1,2)],4)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,3,3}
([],5)
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(3,4)],5)
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(2,4),(3,4)],5)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [3]
=> [1,1,1]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
([],6)
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(4,5)],6)
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(3,5),(4,5)],6)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
([(2,5),(3,4)],6)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(3,4),(3,5),(4,5)],6)
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5}
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> [2]
=> 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> [2]
=> 2
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,3]
=> [3]
=> [1,1,1]
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,3,1]
=> [3,1]
=> [2,1,1]
=> 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [1,1]
=> [2]
=> 2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [4,3]
=> [3]
=> [1,1,1]
=> 1
Description
The product of the parts of an integer partition.
The following 61 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000260The radius of a connected graph. St000422The energy of a graph, if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000681The Grundy value of Chomp on Ferrers diagrams. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000678The number of up steps after the last double rise of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000264The girth of a graph, which is not a tree. St001626The number of maximal proper sublattices of a lattice. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St001624The breadth of a lattice. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001718The number of non-empty open intervals in a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. 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. St000388The number of orbits of vertices of a graph under automorphisms. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph.
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