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Your data matches 194 different statistics following compositions of up to 3 maps.
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Matching statistic: St000068
Values
([],1)
=> 1
([],2)
=> 2
([(0,1)],2)
=> 1
([],3)
=> 3
([(1,2)],3)
=> 2
([(0,1),(0,2)],3)
=> 1
([(0,2),(2,1)],3)
=> 1
([(0,2),(1,2)],3)
=> 2
([],4)
=> 4
([(2,3)],4)
=> 3
([(1,2),(1,3)],4)
=> 2
([(0,1),(0,2),(0,3)],4)
=> 1
([(0,2),(0,3),(3,1)],4)
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(1,2),(2,3)],4)
=> 2
([(0,3),(3,1),(3,2)],4)
=> 1
([(1,3),(2,3)],4)
=> 3
([(0,3),(1,3),(3,2)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2)],4)
=> 2
([(0,3),(1,2),(1,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,3),(2,1),(3,2)],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> 2
([],5)
=> 5
([(3,4)],5)
=> 4
([(2,3),(2,4)],5)
=> 3
([(1,2),(1,3),(1,4)],5)
=> 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(4,2)],5)
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
([(2,3),(3,4)],5)
=> 3
([(1,4),(4,2),(4,3)],5)
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> 1
([(2,4),(3,4)],5)
=> 4
([(1,4),(2,4),(4,3)],5)
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> 2
([(1,4),(2,4),(3,4)],5)
=> 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(0,4),(1,4),(2,3)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
Description
The number of minimal elements in a poset.
Matching statistic: St000069
Values
([],1)
=> 1
([],2)
=> 2
([(0,1)],2)
=> 1
([],3)
=> 3
([(1,2)],3)
=> 2
([(0,1),(0,2)],3)
=> 2
([(0,2),(2,1)],3)
=> 1
([(0,2),(1,2)],3)
=> 1
([],4)
=> 4
([(2,3)],4)
=> 3
([(1,2),(1,3)],4)
=> 3
([(0,1),(0,2),(0,3)],4)
=> 3
([(0,2),(0,3),(3,1)],4)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(1,2),(2,3)],4)
=> 2
([(0,3),(3,1),(3,2)],4)
=> 2
([(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(3,2)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> 2
([(0,3),(1,2),(1,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,3),(2,1),(3,2)],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> 1
([],5)
=> 5
([(3,4)],5)
=> 4
([(2,3),(2,4)],5)
=> 4
([(1,2),(1,3),(1,4)],5)
=> 4
([(0,1),(0,2),(0,3),(0,4)],5)
=> 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(4,2)],5)
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> 3
([(1,2),(1,3),(2,4),(3,4)],5)
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(2,3),(3,4)],5)
=> 3
([(1,4),(4,2),(4,3)],5)
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> 3
([(2,4),(3,4)],5)
=> 3
([(1,4),(2,4),(4,3)],5)
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> 2
([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
Description
The number of maximal elements of a poset.
Matching statistic: St000454
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([],2)
=> ([(0,1)],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1)],2)
=> ([],2)
=> ([],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> ([],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2} - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 1 = 2 - 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2} - 1
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 1 - 1
([(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
Description
The largest eigenvalue of a graph if it is integral.
If a graph is d-regular, then its largest eigenvalue equals d. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000420
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 80%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 80%
Values
([],1)
=> [1]
=> []
=> []
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {1,2}
([(0,1)],2)
=> [2]
=> []
=> []
=> ? ∊ {1,2}
([],3)
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
([(1,2)],3)
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {1,2,2,3}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {1,2,2,3}
([(0,2),(2,1)],3)
=> [3]
=> []
=> []
=> ? ∊ {1,2,2,3}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {1,2,2,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {2,2,2,3,3,3,4}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {2,2,2,3,3,3,4}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {2,2,2,3,3,3,4}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {2,2,2,3,3,3,4}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {2,2,2,3,3,3,4}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 2
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> []
=> ? ∊ {2,2,2,3,3,3,4}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {2,2,2,3,3,3,4}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 5
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {2,2,2,2,2,3,3,4,4,4,4}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {2,2,2,2,2,3,3,4,4,4,4}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {2,2,2,2,2,3,3,4,4,4,4}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {2,2,2,2,2,3,3,4,4,4,4}
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {2,2,2,2,2,3,3,4,4,4,4}
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {2,2,2,2,2,3,3,4,4,4,4}
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {2,2,2,2,2,3,3,4,4,4,4}
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {2,2,2,2,2,3,3,4,4,4,4}
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,3,3,4,4,4,4}
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {2,2,2,2,2,3,3,4,4,4,4}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {2,2,2,2,2,3,3,4,4,4,4}
Description
The number of Dyck paths that are weakly above a Dyck path.
Matching statistic: St000668
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 71%●distinct values known / distinct values provided: 60%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 71%●distinct values known / distinct values provided: 60%
Values
([],1)
=> [1]
=> []
=> []
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,2}
([(0,1)],2)
=> [2]
=> []
=> []
=> ? ∊ {1,2}
([],3)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3}
([(0,2),(2,1)],3)
=> [3]
=> []
=> []
=> ? ∊ {1,1,2,3}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3,3,3,4}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3,3,3,4}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3,3,3,4}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3,3,3,4}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3,3,3,4}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3,3,3,4}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 3
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
Description
The least common multiple of the parts of the partition.
Matching statistic: St000708
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 71%●distinct values known / distinct values provided: 60%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 71%●distinct values known / distinct values provided: 60%
Values
([],1)
=> [1]
=> []
=> []
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,2}
([(0,1)],2)
=> [2]
=> []
=> []
=> ? ∊ {1,2}
([],3)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3}
([(0,2),(2,1)],3)
=> [3]
=> []
=> []
=> ? ∊ {1,1,2,3}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3,3,3,4}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3,3,3,4}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3,3,3,4}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3,3,3,4}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3,3,3,4}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3,3,3,4}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 3
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
Description
The product of the parts of an integer partition.
Matching statistic: St000933
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000933: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 71%●distinct values known / distinct values provided: 60%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000933: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 71%●distinct values known / distinct values provided: 60%
Values
([],1)
=> [1]
=> []
=> []
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,2}
([(0,1)],2)
=> [2]
=> []
=> []
=> ? ∊ {1,2}
([],3)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3}
([(0,2),(2,1)],3)
=> [3]
=> []
=> []
=> ? ∊ {1,1,2,3}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3,3,3,4}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3,3,3,4}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3,3,3,4}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3,3,3,4}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3,3,3,4}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,3,3,3,4}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 3
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,2,2,3,4,4,4,4,5}
Description
The number of multipartitions of sizes given by an integer partition.
This is, for λ=(λ1,…,λn), this is the number of n-tuples (λ(1),…,λ(n)) of partitions λ(i) such that λ(i)⊢λi.
Matching statistic: St000937
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 71%●distinct values known / distinct values provided: 60%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 71%●distinct values known / distinct values provided: 60%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,2}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {1,2}
([],3)
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
([(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,3}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,3}
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? ∊ {1,2,2,3}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {2,2,2,3,3,3,4}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {2,2,2,3,3,3,4}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {2,2,2,3,3,3,4}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {2,2,2,3,3,3,4}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {2,2,2,3,3,3,4}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [2]
=> 2
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [2]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [2]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? ∊ {2,2,2,3,3,3,4}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {2,2,2,3,3,3,4}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1,1]
=> 3
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 1
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {2,2,2,2,2,3,4,4,4,4,5}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> [2]
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> [2]
=> 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> [2]
=> 2
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> [2]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {2,2,2,2,2,3,4,4,4,4,5}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> [2]
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [2]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> [2]
=> 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> [2]
=> 2
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [2]
=> [2]
=> 2
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 3
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {2,2,2,2,2,3,4,4,4,4,5}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {2,2,2,2,2,3,4,4,4,4,5}
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {2,2,2,2,2,3,4,4,4,4,5}
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {2,2,2,2,2,3,4,4,4,4,5}
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {2,2,2,2,2,3,4,4,4,4,5}
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {2,2,2,2,2,3,4,4,4,4,5}
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,3,4,4,4,4,5}
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {2,2,2,2,2,3,4,4,4,4,5}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {2,2,2,2,2,3,4,4,4,4,5}
Description
The number of positive 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 conjugacy class (2,1,1). Therefore, the statistic on the partition (2,2) is 2.
Matching statistic: St000771
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([(0,1)],2)
=> ([],2)
=> ([],2)
=> ? = 2
([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> ([],1)
=> 1
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],3)
=> ? ∊ {2,3}
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([],2)
=> ? ∊ {2,3}
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([],1)
=> 1
([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([],1)
=> 1
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ([],4)
=> ? ∊ {1,1,2,3,3,4}
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([],3)
=> ? ∊ {1,1,2,3,3,4}
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {1,1,2,3,3,4}
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,2,3,3,4}
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {1,1,2,3,3,4}
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> 1
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> ? ∊ {1,1,2,3,3,4}
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([],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),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ([],5)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ([],4)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],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)
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
([(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,4),(1,2),(1,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(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
([(0,4),(1,2),(1,3),(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)
=> 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,4),(3,2),(4,3)],5)
=> ([(0,4),(1,4),(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)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
([(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
([(0,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums 0, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
(4−1−2−1−14−1−2−2−14−1−1−2−14).
Its eigenvalues are 0,4,4,6, so the statistic is 2.
The path on four vertices has eigenvalues 0,4.7…,6,9.2… and therefore statistic 1.
Matching statistic: St000259
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([],2)
=> ([(0,1)],2)
=> ([],2)
=> ([],2)
=> ? = 2 - 1
([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> ([],1)
=> 0 = 1 - 1
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],3)
=> ? ∊ {1,3} - 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([],2)
=> ? ∊ {1,3} - 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1 = 2 - 1
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ([],4)
=> ? ∊ {1,1,1,1,3,4} - 1
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([],3)
=> ? ∊ {1,1,1,1,3,4} - 1
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,3,4} - 1
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {1,1,1,1,3,4} - 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,3,4} - 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> ? ∊ {1,1,1,1,3,4} - 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 2 - 1
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,4,4,5} - 1
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,4,4,5} - 1
([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,4,4,5} - 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,4,4,5} - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,4,4,5} - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,4,4,5} - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,4,4,5} - 1
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,4,4,5} - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,4,4,5} - 1
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,4,4,5} - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,4,4,5} - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,4,4,5} - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 4 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,4,4,5} - 1
([(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,4,4,5} - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,4,4,5} - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2 = 3 - 1
([(0,4),(1,2),(1,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,4,4,5} - 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(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)
=> ([(0,2),(1,2)],3)
=> 2 = 3 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2 = 3 - 1
([(1,4),(3,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,4,4,5} - 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,4,4,5} - 1
([(0,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,4,4,5} - 1
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
The following 184 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000741The Colin de Verdière graph invariant. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001570The minimal number of edges to add to make a graph Hamiltonian. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000189The number of elements in the poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000260The radius of a connected graph. St000477The weight of a partition according to Alladi. St000681The Grundy value of Chomp on Ferrers diagrams. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000939The number of characters of the symmetric group whose value on the partition is positive. St001118The acyclic chromatic index of a graph. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000993The multiplicity of the largest part of an integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001568The smallest positive integer that does not appear twice in the partition. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St001060The distinguishing index of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St000120The number of left tunnels of a Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules S with grade inf at least two in the Nakayama algebra A corresponding to the Dyck path. St001192The maximal dimension of Ext_A^2(S,A) for a simple module S over the corresponding Nakayama algebra A. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between e_i J and e_j J (the radical of the indecomposable projective modules). St000015The number of peaks of a Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of projdim(S)+injdim(S) over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000100The number of linear extensions of a poset. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001632The number of indecomposable injective modules I with dim Ext^1(I,A)=1 for the incidence algebra A of a poset. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001201The grade of the simple module S_0 in the special CNakayama algebra corresponding to the Dyck path. St000331The number of upper interactions of a Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra eAe in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000455The second largest eigenvalue of a graph if it is integral. St001199The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000667The greatest common divisor of the parts of the partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. 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. St000914The sum of the values of the Möbius function of a poset. St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. 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. St001128The exponens consonantiae of a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001621The number of atoms of a lattice. St000456The monochromatic index of a connected graph. St001597The Frobenius rank of a skew partition. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000172The Grundy number of a graph. St000363The number of minimal vertex covers of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000537The cutwidth of a graph. St000553The number of blocks of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001271The competition number of a graph. St001304The number of maximally independent sets of vertices of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001725The harmonious chromatic number of a graph. St001963The tree-depth of a graph. St000171The degree of the graph. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000469The distinguishing number of a graph. St000552The number of cut vertices of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001071The beta invariant of the graph. St001305The number of induced cycles on four vertices in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001689The number of celebrities in a graph. St001691The number of kings in a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001816Eigenvalues of the top-to-random operator acting on a simple module. St001883The mutual visibility number of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St001875The number of simple modules with projective dimension at most 1. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000264The girth of a graph, which is not a tree. St001200The number of simple modules in eAe with projective dimension at most 2 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001399The distinguishing number of a poset. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000907The number of maximal antichains of minimal length in a poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001195The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000450The number of edges minus the number of vertices plus 2 of a graph. St000315The number of isolated vertices of a graph. St001117The game chromatic index of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001742The difference of the maximal and the minimal degree in a graph. St001812The biclique partition number of a graph. St001330The hat guessing number of a graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. 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. St001624The breadth of a lattice.
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