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Your data matches 79 different statistics following compositions of up to 3 maps.
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Matching statistic: St000276
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(1,2)],3)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],1)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(2,3)],4)
=> ([(1,2)],3)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([],1)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],1)
=> 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],1)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0
([(3,4)],5)
=> ([(1,2)],3)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([],1)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([],1)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],2)
=> 0
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0
([(4,5)],6)
=> ([(1,2)],3)
=> ([],1)
=> 1
([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([],1)
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([],1)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([],1)
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],1)
=> 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ([],2)
=> 0
([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
Description
The size of the preimage of the map 'to graph' from Ordered trees to Graphs.
Matching statistic: St000264
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 61%●distinct values known / distinct values provided: 17%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 61%●distinct values known / distinct values provided: 17%
Values
([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> ? = 1 + 3
([(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> ? = 1 + 3
([(0,2),(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> ? = 1 + 3
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 3
([(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 3
([(0,3),(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 3
([(0,3),(1,2)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 3
([(0,3),(1,2),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 3
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 3
([(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 3
([(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 3
([(1,4),(2,3)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 3
([(1,4),(2,3),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 3
([(0,1),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 3
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(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),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3 + 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
([(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 3
([(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 3
([(2,5),(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 3
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 3
([(2,5),(3,4)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 3
([(2,5),(3,4),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 3
([(1,2),(3,5),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 3
([(3,4),(3,5),(4,5)],6)
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 3
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 + 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 3
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 + 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(0,5),(1,4),(2,3)],6)
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 0 + 3
([(1,5),(2,4),(3,4),(3,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3 + 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 0 + 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 0 + 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 0 + 3
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 4 + 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 0 + 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3 + 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001570
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1
([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(2,3)],4)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 0
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 0
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(4,5)],6)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1
([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1
([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 0
([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 0
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> ? = 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 0
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St001060
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 + 3
([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 3
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 + 3
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(2,3)],4)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 3
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 + 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 0 + 3
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 3
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 + 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 3
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 3
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 + 3
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 0 + 3
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 3
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 0 + 3
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 + 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 3 + 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 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)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 3 = 0 + 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(4,5)],6)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 3
([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 3
([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 3
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 + 3
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 0 + 3
([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 3
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 0 + 3
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 0 + 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 0 + 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 3
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 3 + 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 3
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 3 + 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 + 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> ? = 0 + 3
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 3 + 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 3 = 0 + 3
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 3
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 3 = 0 + 3
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 4 + 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 0 + 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 3 + 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
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: St000567
Mp00154: Graphs —core⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000567: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 20%●distinct values known / distinct values provided: 17%
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000567: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 20%●distinct values known / distinct values provided: 17%
Values
([(0,1)],2)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,2)],3)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 0
([(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 0
([(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 0
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
Description
The sum of the products of all pairs of parts.
This is the evaluation of the second elementary symmetric polynomial which is equal to
$$e_2(\lambda) = \binom{n+1}{2} - \sum_{i=1}^\ell\binom{\lambda_i+1}{2}$$
for a partition $\lambda = (\lambda_1,\dots,\lambda_\ell) \vdash n$, see [1].
This is the maximal number of inversions a permutation with the given shape can have, see [2, cor.2.4].
Matching statistic: St000929
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00154: Graphs —core⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 20%●distinct values known / distinct values provided: 17%
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 20%●distinct values known / distinct values provided: 17%
Values
([(0,1)],2)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,2)],3)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 0
([(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 0
([(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 0
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St000936
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00154: Graphs —core⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000936: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 20%●distinct values known / distinct values provided: 17%
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000936: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 20%●distinct values known / distinct values provided: 17%
Values
([(0,1)],2)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,2)],3)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 0
([(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 0
([(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 0
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
Description
The number of even values of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugace class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $4$.
It is shown in [1] that the sum of the values of the statistic over all partitions of a given size is even.
Matching statistic: St000938
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00154: Graphs —core⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 20%●distinct values known / distinct values provided: 17%
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 20%●distinct values known / distinct values provided: 17%
Values
([(0,1)],2)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,2)],3)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 0
([(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 0
([(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 0
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
Description
The number of zeros of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
Matching statistic: St001097
Mp00154: Graphs —core⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St001097: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 20%●distinct values known / distinct values provided: 17%
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St001097: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 20%●distinct values known / distinct values provided: 17%
Values
([(0,1)],2)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,2)],3)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 0
([(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 0
([(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 0
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
Description
The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders.
For a generating function $f$ the associated formal group law is the symmetric function $f(f^{(-1)}(x_1) + f^{(-1)}(x_2), \dots)$, see [1].
This statistic records the coefficient of the monomial symmetric function $m_\lambda$ in the formal group law for linear orders, with generating function $f(x) = x/(1-x)$, see [1, sec. 3.4].
This statistic gives the number of Smirnov arrangements of a set of letters with $\lambda_i$ of the $i$th letter, where a Smirnov word is a word with no repeated adjacent letters. e.g., [3,2,1] = > 10 since there are 10 Smirnov rearrangements of the word 'aaabbc': 'ababac', 'ababca', 'abacab', 'abacba', 'abcaba', 'acabab', 'acbaba', 'babaca', 'bacaba', 'cababa'.
Matching statistic: St001098
Mp00154: Graphs —core⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St001098: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 20%●distinct values known / distinct values provided: 17%
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St001098: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 20%●distinct values known / distinct values provided: 17%
Values
([(0,1)],2)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,2)],3)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 0
([(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 0
([(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 0
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? = 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
Description
The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees.
For a generating function $f$ the associated formal group law is the symmetric function $f(f^{(-1)}(x_1) + f^{(-1)}(x_2), \dots)$, see [1].
This statistic records the coefficient of the monomial symmetric function $m_\lambda$ times the product of the factorials of the parts of $\lambda$ in the formal group law for vertex labelled trees, whose reversal of the generating function $f^{(-1)}(x) = x\exp(-x)$, see [1, sec. 3.3]
Fix a set of distinguishable vertices and a coloring of the vertices so that $\lambda_i$ are colored $i$. Then this statistic gives the number of ways of putting a rooted tree on this set of colored vertices so that no leaf is the same color as its parent.
The following 69 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000934The 2-degree of an integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000260The radius of a connected graph. St000259The diameter of a connected graph. 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. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001432The order dimension of the partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000944The 3-degree of an integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. 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 $eA$. 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 $eA$. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001624The breadth of a lattice.
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