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Your data matches 246 different statistics following compositions of up to 3 maps.
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(click to perform a complete search on your data)
Matching statistic: St001638
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(load all 10 compositions to match this statistic)
Values
([],1)
=> 0
([],2)
=> 0
([(0,1)],2)
=> 0
([],3)
=> 0
([(1,2)],3)
=> 0
([(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(1,2)],3)
=> 1
([],4)
=> 0
([(2,3)],4)
=> 0
([(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> 0
([(0,3),(1,2),(2,3)],4)
=> 0
([(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([],5)
=> 0
([(3,4)],5)
=> 0
([(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3)],5)
=> 0
([(1,4),(2,3),(3,4)],5)
=> 0
([(0,1),(2,4),(3,4)],5)
=> 0
([(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
Description
The book thickness of a graph.
The book thickness (or pagenumber, or stacknumber) of a graph is the minimal number of pages required for a book embedding of a graph.
Matching statistic: St001689
Values
([],1)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0
([],2)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(1,2)],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
([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(1,2)],4)
=> ([(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)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(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
([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(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
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(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
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(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
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(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
([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 0
([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(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)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(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
([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(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),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(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),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
Description
The number of celebrities in a graph.
A celebrity is a vertex such that more than half of its neighbours have smaller degree.
Matching statistic: St000985
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0
([],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ([],2)
=> 0
([],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> ([],2)
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],3)
=> 0
([],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0
([(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ([],2)
=> 0
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ([],2)
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],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)
=> ([],4)
=> ([],4)
=> 0
([],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0
([(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ([],2)
=> 0
([(1,4),(2,3)],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),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3),(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),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
([(1,4),(2,3),(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)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 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)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
([(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)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ([],2)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],3)
=> 0
([(0,4),(1,3),(2,3),(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),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(2,3),(2,4),(3,4)],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),(3,4)],5)
=> 1
([(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),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(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)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 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)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(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)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
([(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),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
([(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)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(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)
=> ([(1,4),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
([(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)
=> ([],3)
=> 0
Description
The number of positive eigenvalues of the adjacency matrix of the graph.
Matching statistic: St001181
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001181: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 98%●distinct values known / distinct values provided: 75%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001181: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 98%●distinct values known / distinct values provided: 75%
Values
([],1)
=> [1]
=> [1,0]
=> [1,0]
=> 0
([],2)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
([(0,2),(1,2)],3)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
([(1,3),(2,3)],4)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
Description
Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra.
Matching statistic: St000741
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 0
([],2)
=> ([],1)
=> ([],1)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> ([],1)
=> 0
([],3)
=> ([],1)
=> ([],1)
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> ([],2)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> 0
([],4)
=> ([],1)
=> ([],1)
=> 0
([(2,3)],4)
=> ([(1,2)],3)
=> ([],2)
=> 1
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([],2)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(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)
=> ([],2)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> 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)
=> ([],1)
=> 0
([],5)
=> ([],1)
=> ([],1)
=> 0
([(3,4)],5)
=> ([(1,2)],3)
=> ([],2)
=> 1
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([],2)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([],2)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> 1
([(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)
=> ([(1,2)],3)
=> ? ∊ {2,2,3}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ([],2)
=> 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)
=> ([],2)
=> 1
([(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
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> 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)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
([(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)],2)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ? ∊ {2,2,3}
([(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)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ? ∊ {2,2,3}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(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)
=> ([(0,2),(1,2)],3)
=> 2
([(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)
=> ([],2)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> 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)
=> ([],1)
=> 0
([(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)
=> ([],1)
=> 0
Description
The Colin de Verdière graph invariant.
Matching statistic: St001330
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
([],2)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
([],3)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 0 + 1
([],4)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
([(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1 = 0 + 1
([],5)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
([(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,3} + 1
([(1,4),(2,3),(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)
=> ? ∊ {2,2,2,2,2,2,3} + 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
([(1,4),(2,3),(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)
=> ? ∊ {2,2,2,2,2,2,3} + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 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)
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(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)
=> ? ∊ {2,2,2,2,2,2,3} + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(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)
=> ? ∊ {2,2,2,2,2,2,3} + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {2,2,2,2,2,2,3} + 1
([(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),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,3} + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 1 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(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),(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(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
([(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)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
([(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)
=> 1 = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 1 = 0 + 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000259
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 0
([],2)
=> ([],2)
=> ([],1)
=> 0
([(0,1)],2)
=> ([],1)
=> ([],1)
=> 0
([],3)
=> ([],3)
=> ([],1)
=> 0
([(1,2)],3)
=> ([],2)
=> ([],1)
=> 0
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> 0
([],4)
=> ([],4)
=> ([],1)
=> 0
([(2,3)],4)
=> ([],3)
=> ([],1)
=> 0
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> 0
([],5)
=> ([],5)
=> ([],1)
=> 0
([(3,4)],5)
=> ([],4)
=> ([],1)
=> 0
([(2,4),(3,4)],5)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,2,2,2,2}
([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,2,2,2}
([(0,4),(1,4),(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)
=> 1
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> 0
([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,2,2,2,2}
([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,2,2,2,2}
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,2,2,2,2}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,2,2,2,2}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,2,2,2,2}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(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)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 2
([(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)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1
([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 0
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000260
(load all 36 compositions to match this statistic)
(load all 36 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 0
([],2)
=> ([],2)
=> ([],1)
=> 0
([(0,1)],2)
=> ([],1)
=> ([],1)
=> 0
([],3)
=> ([],3)
=> ([],1)
=> 0
([(1,2)],3)
=> ([],2)
=> ([],1)
=> 0
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> 0
([],4)
=> ([],4)
=> ([],1)
=> 0
([(2,3)],4)
=> ([],3)
=> ([],1)
=> 0
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> 0
([],5)
=> ([],5)
=> ([],1)
=> 0
([(3,4)],5)
=> ([],4)
=> ([],1)
=> 0
([(2,4),(3,4)],5)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {1,1,2,2,2,2,3}
([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,2,2,2,2,3}
([(0,4),(1,4),(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)
=> 1
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> 0
([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {1,1,2,2,2,2,3}
([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {1,1,2,2,2,2,3}
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {1,1,2,2,2,2,3}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,2,2,2,2,3}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {1,1,2,2,2,2,3}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(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)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 2
([(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)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1
([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 0
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St001022
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001022: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 83%●distinct values known / distinct values provided: 75%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001022: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 83%●distinct values known / distinct values provided: 75%
Values
([],1)
=> []
=> []
=> []
=> ? = 0
([],2)
=> []
=> []
=> []
=> ? = 0
([(0,1)],2)
=> [1]
=> [1,0]
=> [1,0]
=> 0
([],3)
=> []
=> []
=> []
=> ? = 0
([(1,2)],3)
=> [1]
=> [1,0]
=> [1,0]
=> 0
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
([],4)
=> []
=> []
=> []
=> ? = 2
([(2,3)],4)
=> [1]
=> [1,0]
=> [1,0]
=> 0
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([],5)
=> []
=> []
=> []
=> ? ∊ {2,2,2,2,3}
([(3,4)],5)
=> [1]
=> [1,0]
=> [1,0]
=> 0
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3}
Description
Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001113
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001113: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 75%●distinct values known / distinct values provided: 50%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001113: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 75%●distinct values known / distinct values provided: 50%
Values
([],1)
=> []
=> []
=> ? = 0
([],2)
=> []
=> []
=> ? = 0
([(0,1)],2)
=> [1]
=> [1,0]
=> 0
([],3)
=> []
=> []
=> ? = 0
([(1,2)],3)
=> [1]
=> [1,0]
=> 0
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 1
([],4)
=> []
=> []
=> ? = 2
([(2,3)],4)
=> [1]
=> [1,0]
=> 0
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([],5)
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,3}
([(3,4)],5)
=> [1]
=> [1,0]
=> 0
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,3}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,3}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,3}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,3}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,3}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,3}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,3}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,3}
Description
Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra.
The following 236 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001331The size of the minimal feedback vertex set. St001572The minimal number of edges to remove to make a graph bipartite. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001871The number of triconnected components of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000671The maximin edge-connectivity for choosing a subgraph. 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 $eA$. 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. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001624The breadth of a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000928The sum of the coefficients of the character polynomial of an integer partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001570The minimal number of edges to add to make a graph Hamiltonian. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000137The Grundy value of an integer partition. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001933The largest multiplicity of a part in an integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001383The BG-rank of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. 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. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St000454The largest eigenvalue of a graph if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St001118The acyclic chromatic index of a graph. St001281The normalized isoperimetric number of a graph. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001592The maximal number of simple paths between any two different vertices of a graph. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000340The number of non-final maximal constant sub-paths of length greater than one. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000442The maximal area to the right of an up step of a Dyck path. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000547The number of even non-empty partial sums of an integer partition. St000659The number of rises of length at least 2 of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000921The number of internal inversions of a binary word. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001273The projective dimension of the first term in an injective coresolution of the regular module. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001424The number of distinct squares in a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001435The number of missing boxes in the first row. St001480The number of simple summands of the module J^2/J^3. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the 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). St001955The number of natural descents for set-valued two row standard Young tableaux. St001487The number of inner corners of a skew partition. St000456The monochromatic index of a connected graph. St000273The domination number of a graph. St001829The common independence number of a graph. St001626The number of maximal proper sublattices of a lattice. St001651The Frankl number of a lattice. St001396Number of triples of incomparable elements in a finite poset. St001964The interval resolution global dimension of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001249Sum of the odd parts of a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001561The value of the elementary symmetric function evaluated at 1. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001322The size of a minimal independent dominating set in a graph. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001712The number of natural descents of a standard Young tableau. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000929The constant term of the character polynomial of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. 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. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000284The Plancherel distribution on integer partitions. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000707The product of the factorials of the parts. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000567The sum of the products of all pairs of parts. St000658The number of rises of length 2 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000674The number of hills of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000693The modular (standard) major index of a standard tableau. St000706The product of the factorials of the multiplicities of an integer partition. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000874The position of the last double rise in a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000947The major index east count of a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000984The number of boxes below precisely one peak. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000422The energy of a graph, if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. 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. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given 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. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001360The number of covering relations in Young's lattice below a partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001722The number of minimal chains with small intervals between a binary word and the top element. St001498The normalised height of a Nakayama algebra with magnitude 1. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001645The pebbling number of a connected graph. St001060The distinguishing index of a graph.
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