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Your data matches 191 different statistics following compositions of up to 3 maps.
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(click to perform a complete search on your data)
Matching statistic: St000447
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0
([],2)
=> ([],2)
=> ([],2)
=> ([(0,1)],2)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([],3)
=> ([],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
([],4)
=> ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0
([(0,3),(1,2),(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)
=> ([(2,3)],4)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0
([],5)
=> ([],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(2,4),(3,4)],5)
=> ([(2,3),(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
([(1,4),(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)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 0
([(1,4),(2,3)],5)
=> ([(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
([(1,4),(2,3),(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)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([(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
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(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
([(0,4),(1,4),(2,3),(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)
=> ([(2,4),(3,4)],5)
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,3),(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)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> 0
([(0,4),(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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 0
([(0,3),(0,4),(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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 0
([(0,4),(1,3),(2,3),(2,4)],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)
=> ([(1,4),(2,3),(3,4)],5)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 0
([(0,3),(1,2),(1,4),(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)
=> ([(2,4),(3,4)],5)
=> 0
([(0,3),(0,4),(1,2),(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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 0
([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 0
([(0,4),(1,2),(1,3),(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)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 0
([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 0
([(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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 0
Description
The number of pairs of vertices of a graph with distance 3.
This is the coefficient of the cubic term of the Wiener polynomial, also called Wiener polarity index.
Matching statistic: St000379
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ? = 0
([],2)
=> ([],2)
=> ([],2)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> ([],1)
=> ? = 0
([],3)
=> ([],3)
=> ([],3)
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> ([],2)
=> 0
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ? = 0
([],4)
=> ([],4)
=> ([],4)
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> ([],3)
=> 0
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ? = 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],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)
=> ([],5)
=> ([],5)
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> ([],4)
=> 0
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 0
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> 0
([(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)
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(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)
=> ([(0,2),(1,2)],3)
=> 0
([(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)
=> ([],1)
=> ? = 0
([(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)
=> ([],1)
=> ? = 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ? = 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 0
([(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
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 0
([(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)
=> ([(0,2),(1,2)],3)
=> 0
([(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)
=> ([],1)
=> ? = 0
([(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
([(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)
=> ([],1)
=> ? = 0
([(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)
=> ([],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
([],6)
=> ([],6)
=> ([],6)
=> 0
([(4,5)],6)
=> ([(4,5)],6)
=> ([],5)
=> 0
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> 0
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ([],4)
=> 0
([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> 0
([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> 0
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> ? = 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ? = 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> ? = 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 2
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> ? = 0
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> ? = 0
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([],1)
=> ? = 0
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6)
=> ([],1)
=> ? = 0
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(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
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(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
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([],1)
=> ? = 0
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],1)
=> ? = 0
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
([(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)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
Description
The number of Hamiltonian cycles in a graph.
A Hamiltonian cycle in a graph $G$ is a subgraph (this is, a subset of the edges) that is a cycle which contains every vertex of $G$.
Since it is unclear whether the graph on one vertex is Hamiltonian, the statistic is undefined for this graph.
Matching statistic: St000781
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 62%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 62%●distinct values known / distinct values provided: 25%
Values
([],1)
=> [1]
=> []
=> ? = 0 + 1
([],2)
=> [1,1]
=> [1]
=> 1 = 0 + 1
([(0,1)],2)
=> [2]
=> []
=> ? = 0 + 1
([],3)
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,2)],3)
=> [2,1]
=> [1]
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> [3]
=> []
=> ? = 0 + 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? = 0 + 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 0 + 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? = 0 + 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 0 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 0 + 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? = 1 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1 = 0 + 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 0 + 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1 = 0 + 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1 = 0 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 1 = 0 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ? = 1 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? = 1 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 1 = 0 + 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 1 = 0 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 1 + 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1 = 0 + 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 1 + 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
Matching statistic: St001901
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 62%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 62%●distinct values known / distinct values provided: 25%
Values
([],1)
=> [1]
=> []
=> ? = 0 + 1
([],2)
=> [1,1]
=> [1]
=> 1 = 0 + 1
([(0,1)],2)
=> [2]
=> []
=> ? = 0 + 1
([],3)
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,2)],3)
=> [2,1]
=> [1]
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> [3]
=> []
=> ? = 0 + 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? = 0 + 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 0 + 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? = 0 + 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 0 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 0 + 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? = 1 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 0 + 1
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1 = 0 + 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 0 + 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1 = 0 + 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1 = 0 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 1 = 0 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ? = 1 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? = 1 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 1 = 0 + 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 1 = 0 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 1 + 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1 = 0 + 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 1 + 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 0 + 1
Description
The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition.
Matching statistic: St000205
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 62%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 62%●distinct values known / distinct values provided: 25%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> [1]
=> 0
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? = 0
([],3)
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> 0
([(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? = 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? = 0
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? = 0
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [2]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ?
=> ? = 0
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? = 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ?
=> ? = 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? = 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? = 0
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [2]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [2]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [3]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [2]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> [2,1]
=> 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1]
=> 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [4]
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> [2]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [3]
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [2]
=> 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000206
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 62%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 62%●distinct values known / distinct values provided: 25%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> [1]
=> 0
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? = 0
([],3)
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> 0
([(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? = 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? = 0
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? = 0
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [2]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ?
=> ? = 0
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? = 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ?
=> ? = 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? = 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? = 0
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [2]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [2]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 0
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [3]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [2]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> [2,1]
=> 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1]
=> 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [4]
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> [2]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [3]
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [2]
=> 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 0
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
See also [[St000205]].
Each value in this statistic is greater than or equal to corresponding value in [[St000205]].
Matching statistic: St000290
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 25% ●values known / values provided: 62%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 25% ●values known / values provided: 62%●distinct values known / distinct values provided: 25%
Values
([],1)
=> [1]
=> []
=> ? => ? = 0
([],2)
=> [1,1]
=> [1]
=> 1 => 0
([(0,1)],2)
=> [2]
=> []
=> ? => ? = 0
([],3)
=> [1,1,1]
=> [1,1]
=> 11 => 0
([(1,2)],3)
=> [2,1]
=> [1]
=> 1 => 0
([(0,2),(1,2)],3)
=> [3]
=> []
=> ? => ? = 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? => ? = 0
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 11 => 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 => 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 0
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 0 => 0
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? => ? = 0
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? => ? = 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 0
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 01 => 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0 => 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? => ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 11111 => 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 111 => 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 011 => 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 01 => 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 111 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 1 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 00 => 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 01 => 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
Description
The major index of a binary word.
This is the sum of the positions of descents, i.e., a one followed by a zero.
For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
Matching statistic: St000291
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 25% ●values known / values provided: 62%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 25% ●values known / values provided: 62%●distinct values known / distinct values provided: 25%
Values
([],1)
=> [1]
=> []
=> ? => ? = 0
([],2)
=> [1,1]
=> [1]
=> 1 => 0
([(0,1)],2)
=> [2]
=> []
=> ? => ? = 0
([],3)
=> [1,1,1]
=> [1,1]
=> 11 => 0
([(1,2)],3)
=> [2,1]
=> [1]
=> 1 => 0
([(0,2),(1,2)],3)
=> [3]
=> []
=> ? => ? = 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? => ? = 0
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 11 => 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 => 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 0
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 0 => 0
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? => ? = 0
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? => ? = 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 0
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 01 => 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0 => 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? => ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 11111 => 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 111 => 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 011 => 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 01 => 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 111 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 1 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 00 => 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 01 => 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
Description
The number of descents of a binary word.
Matching statistic: St000293
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 25% ●values known / values provided: 62%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 25% ●values known / values provided: 62%●distinct values known / distinct values provided: 25%
Values
([],1)
=> [1]
=> []
=> ? => ? = 0
([],2)
=> [1,1]
=> [1]
=> 1 => 0
([(0,1)],2)
=> [2]
=> []
=> ? => ? = 0
([],3)
=> [1,1,1]
=> [1,1]
=> 11 => 0
([(1,2)],3)
=> [2,1]
=> [1]
=> 1 => 0
([(0,2),(1,2)],3)
=> [3]
=> []
=> ? => ? = 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? => ? = 0
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 11 => 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 => 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 0
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 0 => 0
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? => ? = 0
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? => ? = 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 0
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 01 => 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0 => 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? => ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 0
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 11111 => 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 111 => 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 011 => 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 01 => 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 111 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 1 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 00 => 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 01 => 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 0
Description
The number of inversions of a binary word.
Matching statistic: St000296
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000296: Binary words ⟶ ℤResult quality: 25% ●values known / values provided: 62%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000296: Binary words ⟶ ℤResult quality: 25% ●values known / values provided: 62%●distinct values known / distinct values provided: 25%
Values
([],1)
=> [1]
=> []
=> => ? = 0
([],2)
=> [1,1]
=> [1]
=> 10 => 0
([(0,1)],2)
=> [2]
=> []
=> => ? = 0
([],3)
=> [1,1,1]
=> [1,1]
=> 110 => 0
([(1,2)],3)
=> [2,1]
=> [1]
=> 10 => 0
([(0,2),(1,2)],3)
=> [3]
=> []
=> => ? = 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> => ? = 0
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 110 => 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 10 => 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> => ? = 0
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 100 => 0
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> => ? = 0
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 10 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> => ? = 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> => ? = 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> => ? = 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> => ? = 0
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 110 => 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 10 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1010 => 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 10 => 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 100 => 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 110 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> => ? = 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 10 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 10 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 10 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> => ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> => ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 100 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> => ? = 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 10 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> => ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 0
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 110 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 0
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 110 => 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1010 => 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 100 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 110 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> => ? = 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 110 => 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 1000 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> => ? = 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 110 => 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> => ? = 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 10 => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> => ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> => ? = 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> => ? = 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 1100 => 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 10 => 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 100 => 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1010 => 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 100 => 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 0
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> 10 => 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 0
Description
The length of the symmetric border of a binary word.
The symmetric border of a word is the longest word which is a prefix and its reverse is a suffix.
The statistic value is equal to the length of the word if and only if the word is [[https://en.wikipedia.org/wiki/Palindrome|palindromic]].
The following 181 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000347The inversion sum of a binary word. St000629The defect of a binary word. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001175The size of a partition minus the hook length of the base cell. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000183The side length of the Durfee square of an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000847The number of standard Young tableaux whose descent set is the binary word. St000913The number of ways to refine the partition into singletons. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001722The number of minimal chains with small intervals between a binary word and the top element. St000630The length of the shortest palindromic decomposition of a binary word. St001471The magnitude of a Dyck path. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. 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 $eA$. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St000699The toughness times the least common multiple of 1,. 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. St001248Sum of the even parts of a 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. St000618The number of self-evacuating tableaux of given shape. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000264The girth of a graph, which is not a tree. St001330The hat guessing number of a graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. 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. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences 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. St001307The number of induced stars on four vertices in a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000455The second largest eigenvalue of a graph if it is integral. St000456The monochromatic index of a connected graph. St001118The acyclic chromatic index of a graph. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000879The number of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001578The minimal number of edges to add or remove to make a graph a line graph. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000069The number of maximal elements of a poset. St001846The number of elements which do not have a complement in the lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St000449The number of pairs of vertices of a graph with distance 4. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000623The number of occurrences of the pattern 52341 in a permutation. St000787The number of flips required to make a perfect matching noncrossing. 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. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001381The fertility of a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St000056The decomposition (or block) number of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001461The number of topologically connected components of the chord diagram of a permutation. St001518The number of graphs with the same ordinary spectrum as the given graph. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001845The number of join irreducibles minus the rank of a lattice. St000068The number of minimal elements in a poset. St001890The maximum magnitude of the Möbius function of a poset. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001432The order dimension of the partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000929The constant term of the character polynomial of an integer partition. St000944The 3-degree of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001651The Frankl number of a lattice. St001577The minimal number of edges to add or remove to make a graph a cograph. 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. St001703The villainy of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001877Number of indecomposable injective modules with projective dimension 2. St001301The first Betti number of the order complex associated with the poset. St000181The number of connected components of the Hasse diagram for the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001730The number of times the path corresponding to a binary word crosses the base line. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. 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. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000552The number of cut vertices of a graph. St001095The number of non-isomorphic posets with precisely one further covering relation. St001793The difference between the clique number and the chromatic number of a graph. St000553The number of blocks of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000914The sum of the values of the Möbius function of a poset. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph.
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