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Your data matches 73 different statistics following compositions of up to 3 maps.
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Matching statistic: St001396
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(load all 4 compositions to match this statistic)
Values
([],1)
=> 0
([],2)
=> 0
([(0,1)],2)
=> 0
([],3)
=> 1
([(1,2)],3)
=> 0
([(0,1),(0,2)],3)
=> 0
([(0,2),(2,1)],3)
=> 0
([(0,2),(1,2)],3)
=> 0
([],4)
=> 4
([(2,3)],4)
=> 2
([(1,2),(1,3)],4)
=> 1
([(0,1),(0,2),(0,3)],4)
=> 1
([(0,2),(0,3),(3,1)],4)
=> 0
([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(1,2),(2,3)],4)
=> 0
([(0,3),(3,1),(3,2)],4)
=> 0
([(1,3),(2,3)],4)
=> 1
([(0,3),(1,3),(3,2)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> 0
([(0,3),(1,2),(1,3)],4)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> 0
([(0,3),(2,1),(3,2)],4)
=> 0
([(0,3),(1,2),(2,3)],4)
=> 0
([],5)
=> 10
([(3,4)],5)
=> 7
([(2,3),(2,4)],5)
=> 5
([(1,2),(1,3),(1,4)],5)
=> 4
([(0,1),(0,2),(0,3),(0,4)],5)
=> 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(4,2)],5)
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 0
([(2,3),(3,4)],5)
=> 3
([(1,4),(4,2),(4,3)],5)
=> 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> 1
([(2,4),(3,4)],5)
=> 5
([(1,4),(2,4),(4,3)],5)
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(0,4),(1,4),(2,3)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
Description
Number of triples of incomparable elements in a finite poset.
For a finite poset this is the number of 3-element sets $S \in \binom{P}{3}$ that are pairwise incomparable.
Matching statistic: St000095
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(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 0
([],2)
=> ([(0,1)],2)
=> ([],1)
=> 0
([(0,1)],2)
=> ([],2)
=> ([],2)
=> 0
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([],1)
=> 0
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([],2)
=> 0
([(0,2),(2,1)],3)
=> ([],3)
=> ([],3)
=> 0
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([],2)
=> 0
([],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
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([],2)
=> 0
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([],3)
=> 0
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> 0
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> ([],3)
=> 0
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([],3)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 0
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],4)
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([],2)
=> 0
([],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)
=> 10
([(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)
=> 7
([(2,3),(2,4)],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)
=> 5
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,1),(0,2),(0,3),(0,4)],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)
=> 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([],4)
=> 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 0
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 1
([(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)
=> 5
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 35
([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 30
([(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 26
([(3,4),(3,5),(3,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 23
([(2,3),(2,4),(2,5),(2,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 21
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6)],7)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 20
([(0,2),(0,3),(0,4),(0,5),(0,6),(6,1)],7)
=> ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 16
([(0,1),(0,2),(0,3),(0,4),(0,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 13
([(0,1),(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 11
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10
([(0,3),(0,4),(0,5),(0,6),(6,1),(6,2)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 13
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 7
([(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,1)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 9
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
([(0,1),(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
([(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,1),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 8
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 5
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
([(0,2),(0,3),(0,4),(0,5),(4,6),(5,6),(6,1)],7)
=> ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
([(0,3),(0,4),(0,5),(0,6),(5,2),(6,1)],7)
=> ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 12
([(0,2),(0,3),(0,4),(0,5),(4,6),(5,1),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10
([(0,1),(0,2),(0,3),(0,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 8
([(2,4),(2,5),(2,6),(6,3)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 17
([(0,4),(0,5),(0,6),(6,1),(6,2),(6,3)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 11
([(2,3),(2,4),(2,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 14
([(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 12
([(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,1)],7)
=> ([(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1
([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(4,6),(6,1)],7)
=> ([(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ([(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 1
([(0,3),(0,4),(0,5),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
([(0,3),(0,4),(0,5),(3,6),(4,6),(5,1),(5,2),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
([(0,3),(0,4),(0,5),(3,6),(4,6),(5,1),(6,2)],7)
=> ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2
([(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,1)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4
([(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,1),(4,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 3
([(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,1),(4,5),(4,6)],7)
=> ([(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 3
([(0,1),(0,2),(0,3),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2
([(0,2),(0,3),(0,4),(3,5),(3,6),(4,5),(4,6),(6,1)],7)
=> ([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
([(0,3),(0,4),(0,5),(4,6),(5,1),(5,6),(6,2)],7)
=> ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
([(0,4),(0,5),(0,6),(5,3),(6,1),(6,2)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 9
([(0,3),(0,4),(0,5),(4,6),(5,1),(5,2),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 8
([(0,3),(0,4),(0,5),(4,2),(4,6),(5,1),(5,6)],7)
=> ([(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7
([(0,2),(0,3),(0,4),(3,5),(3,6),(4,1),(4,5),(4,6)],7)
=> ([(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
([(0,1),(0,2),(0,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
([(0,4),(0,5),(0,6),(4,3),(5,2),(6,1)],7)
=> ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 8
([(0,3),(0,4),(0,5),(3,6),(4,2),(5,1),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 6
([(0,2),(0,3),(0,4),(2,6),(3,5),(4,1),(4,5),(4,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 5
Description
The number of triangles of a graph.
A triangle $T$ of a graph $G$ is a collection of three vertices $\{u,v,w\} \in G$ such that they form $K_3$, the complete graph on three vertices.
Matching statistic: St001651
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ? = 0
([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0
([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? = 0
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? = 0
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> 0
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ? = 4
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 2
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> 0
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> 0
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 0
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? = 0
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([],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),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,13),(1,14),(1,20),(1,28),(1,29),(1,31),(2,11),(2,12),(2,19),(2,26),(2,27),(2,31),(3,16),(3,18),(3,22),(3,27),(3,29),(3,30),(4,15),(4,17),(4,21),(4,26),(4,28),(4,30),(5,11),(5,15),(5,24),(5,29),(5,32),(5,34),(6,12),(6,16),(6,25),(6,28),(6,32),(6,35),(7,13),(7,17),(7,25),(7,27),(7,33),(7,34),(8,14),(8,18),(8,24),(8,26),(8,33),(8,35),(9,21),(9,22),(9,23),(9,31),(9,34),(9,35),(10,19),(10,20),(10,23),(10,30),(10,32),(10,33),(11,36),(11,40),(11,50),(12,36),(12,41),(12,49),(13,37),(13,42),(13,50),(14,37),(14,43),(14,49),(15,38),(15,40),(15,48),(16,39),(16,41),(16,48),(17,38),(17,42),(17,47),(18,39),(18,43),(18,47),(19,36),(19,44),(19,47),(20,37),(20,44),(20,48),(21,38),(21,45),(21,49),(22,39),(22,45),(22,50),(23,44),(23,45),(23,46),(24,40),(24,43),(24,46),(25,41),(25,42),(25,46),(26,40),(26,47),(26,49),(27,41),(27,47),(27,50),(28,42),(28,48),(28,49),(29,43),(29,48),(29,50),(30,45),(30,47),(30,48),(31,44),(31,49),(31,50),(32,36),(32,46),(32,48),(33,37),(33,46),(33,47),(34,38),(34,46),(34,50),(35,39),(35,46),(35,49),(36,51),(37,51),(38,51),(39,51),(40,51),(41,51),(42,51),(43,51),(44,51),(45,51),(46,51),(47,51),(48,51),(49,51),(50,51)],52)
=> ? = 10
([(3,4)],5)
=> ([(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),(0,5),(0,6),(0,7),(0,8),(0,9),(1,12),(1,15),(1,28),(1,31),(1,34),(2,11),(2,14),(2,28),(2,30),(2,33),(3,10),(3,13),(3,28),(3,29),(3,32),(4,10),(4,16),(4,19),(4,21),(4,30),(4,31),(5,11),(5,17),(5,20),(5,22),(5,29),(5,31),(6,12),(6,18),(6,23),(6,24),(6,29),(6,30),(7,13),(7,16),(7,20),(7,23),(7,33),(7,34),(8,14),(8,17),(8,19),(8,24),(8,32),(8,34),(9,15),(9,18),(9,21),(9,22),(9,32),(9,33),(10,25),(10,35),(10,45),(11,26),(11,36),(11,45),(12,27),(12,37),(12,45),(13,25),(13,38),(13,44),(14,26),(14,39),(14,44),(15,27),(15,40),(15,44),(16,25),(16,42),(16,43),(17,26),(17,41),(17,43),(18,27),(18,41),(18,42),(19,35),(19,39),(19,43),(20,36),(20,38),(20,43),(21,35),(21,40),(21,42),(22,36),(22,40),(22,41),(23,37),(23,38),(23,42),(24,37),(24,39),(24,41),(25,46),(26,46),(27,46),(28,44),(28,45),(29,38),(29,41),(29,45),(30,39),(30,42),(30,45),(31,40),(31,43),(31,45),(32,35),(32,41),(32,44),(33,36),(33,42),(33,44),(34,37),(34,43),(34,44),(35,46),(36,46),(37,46),(38,46),(39,46),(40,46),(41,46),(42,46),(43,46),(44,46),(45,46)],47)
=> ? = 7
([(2,3),(2,4)],5)
=> ([(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),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
=> ? = 5
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
=> ? = 4
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ? = 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
=> ? = 3
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
=> ? = 5
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
=> ? = 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ? = 4
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,28),(1,29),(2,12),(2,16),(2,20),(2,22),(2,29),(3,11),(3,15),(3,20),(3,21),(3,28),(4,13),(4,17),(4,19),(4,21),(4,29),(5,14),(5,18),(5,19),(5,22),(5,28),(6,8),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(8,27),(8,34),(8,35),(9,27),(9,30),(9,31),(10,27),(10,32),(10,33),(11,23),(11,30),(11,34),(12,24),(12,31),(12,34),(13,23),(13,31),(13,35),(14,24),(14,30),(14,35),(15,25),(15,32),(15,34),(16,26),(16,33),(16,34),(17,25),(17,33),(17,35),(18,26),(18,32),(18,35),(19,35),(19,36),(20,34),(20,36),(21,23),(21,25),(21,36),(22,24),(22,26),(22,36),(23,37),(24,37),(25,37),(26,37),(27,37),(28,30),(28,32),(28,36),(29,31),(29,33),(29,36),(30,37),(31,37),(32,37),(33,37),(34,37),(35,37),(36,37)],38)
=> ? = 2
([(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),(0,5),(0,6),(1,16),(1,17),(1,18),(1,25),(2,13),(2,14),(2,15),(2,25),(3,10),(3,11),(3,12),(3,25),(4,8),(4,9),(4,12),(4,15),(4,18),(5,7),(5,9),(5,11),(5,14),(5,17),(6,7),(6,8),(6,10),(6,13),(6,16),(7,21),(7,24),(7,29),(8,19),(8,22),(8,29),(9,20),(9,23),(9,29),(10,26),(10,29),(11,27),(11,29),(12,28),(12,29),(13,19),(13,21),(13,26),(14,20),(14,21),(14,27),(15,19),(15,20),(15,28),(16,22),(16,24),(16,26),(17,23),(17,24),(17,27),(18,22),(18,23),(18,28),(19,30),(20,30),(21,30),(22,30),(23,30),(24,30),(25,26),(25,27),(25,28),(26,30),(27,30),(28,30),(29,30)],31)
=> ? = 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 0
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 2
([(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),(0,5),(0,6),(0,7),(0,8),(1,16),(1,20),(1,24),(1,30),(1,32),(2,15),(2,19),(2,23),(2,30),(2,31),(3,17),(3,21),(3,23),(3,29),(3,32),(4,18),(4,22),(4,24),(4,29),(4,31),(5,10),(5,13),(5,14),(5,17),(5,18),(5,30),(6,10),(6,11),(6,12),(6,15),(6,16),(6,29),(7,9),(7,11),(7,13),(7,19),(7,22),(7,32),(8,9),(8,12),(8,14),(8,20),(8,21),(8,31),(9,35),(9,36),(9,41),(10,33),(10,34),(10,41),(11,25),(11,38),(11,41),(12,26),(12,37),(12,41),(13,28),(13,39),(13,41),(14,27),(14,40),(14,41),(15,25),(15,33),(15,37),(16,26),(16,33),(16,38),(17,27),(17,34),(17,39),(18,28),(18,34),(18,40),(19,25),(19,35),(19,39),(20,26),(20,36),(20,40),(21,27),(21,36),(21,37),(22,28),(22,35),(22,38),(23,37),(23,39),(24,38),(24,40),(25,42),(26,42),(27,42),(28,42),(29,34),(29,37),(29,38),(30,33),(30,39),(30,40),(31,35),(31,37),(31,40),(32,36),(32,38),(32,39),(33,42),(34,42),(35,42),(36,42),(37,42),(38,42),(39,42),(40,42),(41,42)],43)
=> ? = 4
([(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ? = 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
([(0,4),(1,2),(1,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,28),(1,29),(2,12),(2,16),(2,20),(2,22),(2,29),(3,11),(3,15),(3,20),(3,21),(3,28),(4,13),(4,17),(4,19),(4,21),(4,29),(5,14),(5,18),(5,19),(5,22),(5,28),(6,8),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(8,27),(8,34),(8,35),(9,27),(9,30),(9,31),(10,27),(10,32),(10,33),(11,23),(11,30),(11,34),(12,24),(12,31),(12,34),(13,23),(13,31),(13,35),(14,24),(14,30),(14,35),(15,25),(15,32),(15,34),(16,26),(16,33),(16,34),(17,25),(17,33),(17,35),(18,26),(18,32),(18,35),(19,35),(19,36),(20,34),(20,36),(21,23),(21,25),(21,36),(22,24),(22,26),(22,36),(23,37),(24,37),(25,37),(26,37),(27,37),(28,30),(28,32),(28,36),(29,31),(29,33),(29,36),(30,37),(31,37),(32,37),(33,37),(34,37),(35,37),(36,37)],38)
=> ? = 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,16),(1,17),(1,18),(1,25),(2,13),(2,14),(2,15),(2,25),(3,10),(3,11),(3,12),(3,25),(4,8),(4,9),(4,12),(4,15),(4,18),(5,7),(5,9),(5,11),(5,14),(5,17),(6,7),(6,8),(6,10),(6,13),(6,16),(7,21),(7,24),(7,29),(8,19),(8,22),(8,29),(9,20),(9,23),(9,29),(10,26),(10,29),(11,27),(11,29),(12,28),(12,29),(13,19),(13,21),(13,26),(14,20),(14,21),(14,27),(15,19),(15,20),(15,28),(16,22),(16,24),(16,26),(17,23),(17,24),(17,27),(18,22),(18,23),(18,28),(19,30),(20,30),(21,30),(22,30),(23,30),(24,30),(25,26),(25,27),(25,28),(26,30),(27,30),(28,30),(29,30)],31)
=> ? = 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 0
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(1,4),(3,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 0
([(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,15),(1,20),(1,21),(2,9),(2,10),(2,14),(2,18),(2,19),(3,8),(3,13),(3,17),(3,19),(3,21),(4,8),(4,12),(4,16),(4,18),(4,20),(5,7),(5,14),(5,15),(5,16),(5,17),(6,7),(6,10),(6,11),(6,12),(6,13),(7,31),(7,32),(8,30),(8,32),(9,30),(9,31),(10,22),(10,23),(10,31),(11,24),(11,25),(11,31),(12,22),(12,24),(12,32),(13,23),(13,25),(13,32),(14,26),(14,27),(14,31),(15,28),(15,29),(15,31),(16,26),(16,28),(16,32),(17,27),(17,29),(17,32),(18,22),(18,26),(18,30),(19,23),(19,27),(19,30),(20,24),(20,28),(20,30),(21,25),(21,29),(21,30),(22,33),(23,33),(24,33),(25,33),(26,33),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 0
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> ? = 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 0
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 0
([],6)
=> ([(0,1),(0,2),(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(1,40),(1,41),(1,42),(1,43),(1,44),(1,45),(1,82),(1,83),(1,84),(1,85),(2,18),(2,19),(2,25),(2,30),(2,31),(2,37),(2,78),(2,79),(2,81),(2,83),(3,16),(3,17),(3,24),(3,28),(3,29),(3,36),(3,76),(3,77),(3,81),(3,82),(4,21),(4,23),(4,27),(4,33),(4,35),(4,39),(4,77),(4,79),(4,80),(4,85),(5,20),(5,22),(5,26),(5,32),(5,34),(5,38),(5,76),(5,78),(5,80),(5,84),(6,22),(6,23),(6,24),(6,46),(6,47),(6,54),(6,83),(6,86),(6,87),(6,91),(7,20),(7,21),(7,25),(7,48),(7,49),(7,55),(7,82),(7,88),(7,89),(7,91),(8,17),(8,19),(8,26),(8,50),(8,52),(8,56),(8,85),(8,86),(8,88),(8,90),(9,16),(9,18),(9,27),(9,51),(9,53),(9,57),(9,84),(9,87),(9,89),(9,90),(10,28),(10,32),(10,43),(10,48),(10,51),(10,59),(10,79),(10,86),(10,92),(10,94),(11,29),(11,33),(11,42),(11,49),(11,50),(11,60),(11,78),(11,87),(11,92),(11,95),(12,30),(12,34),(12,41),(12,46),(12,53),(12,60),(12,77),(12,88),(12,93),(12,94),(13,31),(13,35),(13,40),(13,47),(13,52),(13,59),(13,76),(13,89),(13,93),(13,95),(14,38),(14,39),(14,45),(14,56),(14,57),(14,58),(14,81),(14,91),(14,94),(14,95),(15,36),(15,37),(15,44),(15,54),(15,55),(15,58),(15,80),(15,90),(15,92),(15,93),(16,61),(16,107),(16,112),(16,126),(16,134),(16,170),(17,62),(17,106),(17,113),(17,127),(17,134),(17,171),(18,63),(18,109),(18,112),(18,129),(18,135),(18,172),(19,64),(19,108),(19,113),(19,128),(19,135),(19,173),(20,65),(20,104),(20,110),(20,132),(20,136),(20,170),(21,66),(21,105),(21,110),(21,133),(21,137),(21,171),(22,67),(22,102),(22,111),(22,130),(22,136),(22,172),(23,68),(23,103),(23,111),(23,131),(23,137),(23,173),(24,69),(24,102),(24,103),(24,126),(24,127),(24,175),(25,70),(25,104),(25,105),(25,128),(25,129),(25,175),(26,71),(26,106),(26,108),(26,130),(26,132),(26,174),(27,72),(27,107),(27,109),(27,131),(27,133),(27,174),(28,61),(28,96),(28,114),(28,127),(28,138),(28,169),(29,62),(29,97),(29,115),(29,126),(29,138),(29,168),(30,63),(30,98),(30,117),(30,128),(30,139),(30,169),(31,64),(31,99),(31,116),(31,129),(31,139),(31,168),(32,65),(32,96),(32,118),(32,130),(32,140),(32,167),(33,66),(33,97),(33,119),(33,131),(33,141),(33,167),(34,67),(34,98),(34,120),(34,132),(34,140),(34,166),(35,68),(35,99),(35,121),(35,133),(35,141),(35,166),(36,69),(36,100),(36,122),(36,134),(36,138),(36,166),(37,70),(37,100),(37,123),(37,135),(37,139),(37,167),(38,71),(38,101),(38,124),(38,136),(38,140),(38,168),(39,72),(39,101),(39,125),(39,137),(39,141),(39,169),(40,73),(40,116),(40,121),(40,142),(40,144),(40,170),(41,74),(41,117),(41,120),(41,142),(41,145),(41,171),(42,74),(42,115),(42,119),(42,143),(42,144),(42,172),(43,73),(43,114),(43,118),(43,143),(43,145),(43,173),(44,75),(44,122),(44,123),(44,142),(44,143),(44,174),(45,75),(45,124),(45,125),(45,144),(45,145),(45,175),(46,67),(46,103),(46,117),(46,148),(46,152),(46,178),(47,68),(47,102),(47,116),(47,149),(47,152),(47,179),(48,65),(48,105),(48,114),(48,146),(48,153),(48,178),(49,66),(49,104),(49,115),(49,147),(49,153),(49,179),(50,62),(50,108),(50,119),(50,147),(50,154),(50,176),(51,61),(51,109),(51,118),(51,146),(51,155),(51,176),(52,64),(52,106),(52,121),(52,149),(52,154),(52,177),(53,63),(53,107),(53,120),(53,148),(53,155),(53,177),(54,69),(54,111),(54,123),(54,150),(54,152),(54,176),(55,70),(55,110),(55,122),(55,150),(55,153),(55,177),(56,71),(56,113),(56,125),(56,151),(56,154),(56,178),(57,72),(57,112),(57,124),(57,151),(57,155),(57,179),(58,75),(58,100),(58,101),(58,150),(58,151),(58,180),(59,73),(59,96),(59,99),(59,146),(59,149),(59,180),(60,74),(60,97),(60,98),(60,147),(60,148),(60,180),(61,181),(61,189),(61,190),(62,181),(62,188),(62,191),(63,182),(63,189),(63,192),(64,182),(64,188),(64,193),(65,183),(65,187),(65,190),(66,184),(66,187),(66,191),(67,183),(67,186),(67,192),(68,184),(68,186),(68,193),(69,181),(69,186),(69,194),(70,182),(70,187),(70,194),(71,183),(71,188),(71,195),(72,184),(72,189),(72,195),(73,185),(73,190),(73,193),(74,185),(74,191),(74,192),(75,185),(75,194),(75,195),(76,96),(76,102),(76,106),(76,166),(76,168),(76,170),(77,97),(77,103),(77,107),(77,166),(77,169),(77,171),(78,98),(78,104),(78,108),(78,167),(78,168),(78,172),(79,99),(79,105),(79,109),(79,167),(79,169),(79,173),(80,101),(80,110),(80,111),(80,166),(80,167),(80,174),(81,100),(81,112),(81,113),(81,168),(81,169),(81,175),(82,114),(82,115),(82,122),(82,170),(82,171),(82,175),(83,116),(83,117),(83,123),(83,172),(83,173),(83,175),(84,118),(84,120),(84,124),(84,170),(84,172),(84,174),(85,119),(85,121),(85,125),(85,171),(85,173),(85,174),(86,127),(86,130),(86,149),(86,173),(86,176),(86,178),(87,126),(87,131),(87,148),(87,172),(87,176),(87,179),(88,128),(88,132),(88,147),(88,171),(88,177),(88,178),(89,129),(89,133),(89,146),(89,170),(89,177),(89,179),(90,134),(90,135),(90,151),(90,174),(90,176),(90,177),(91,136),(91,137),(91,150),(91,175),(91,178),(91,179),(92,138),(92,143),(92,153),(92,167),(92,176),(92,180),(93,139),(93,142),(93,152),(93,166),(93,177),(93,180),(94,140),(94,145),(94,155),(94,169),(94,178),(94,180),(95,141),(95,144),(95,154),(95,168),(95,179),(95,180),(96,156),(96,190),(96,197),(97,157),(97,191),(97,197),(98,158),(98,192),(98,197),(99,159),(99,193),(99,197),(100,160),(100,194),(100,197),(101,161),(101,195),(101,197),(102,156),(102,186),(102,200),(103,157),(103,186),(103,201),(104,158),(104,187),(104,200),(105,159),(105,187),(105,201),(106,156),(106,188),(106,198),(107,157),(107,189),(107,198),(108,158),(108,188),(108,199),(109,159),(109,189),(109,199),(110,161),(110,187),(110,198),(111,161),(111,186),(111,199),(112,160),(112,189),(112,200),(113,160),(113,188),(113,201),(114,162),(114,190),(114,201),(115,162),(115,191),(115,200),(116,163),(116,193),(116,200),(117,163),(117,192),(117,201),(118,164),(118,190),(118,199),(119,165),(119,191),(119,199),(120,164),(120,192),(120,198),(121,165),(121,193),(121,198),(122,162),(122,194),(122,198),(123,163),(123,194),(123,199),(124,164),(124,195),(124,200),(125,165),(125,195),(125,201),(126,157),(126,181),(126,200),(127,156),(127,181),(127,201),(128,158),(128,182),(128,201),(129,159),(129,182),(129,200),(130,156),(130,183),(130,199),(131,157),(131,184),(131,199),(132,158),(132,183),(132,198),(133,159),(133,184),(133,198),(134,160),(134,181),(134,198),(135,160),(135,182),(135,199),(136,161),(136,183),(136,200),(137,161),(137,184),(137,201),(138,162),(138,181),(138,197),(139,163),(139,182),(139,197),(140,164),(140,183),(140,197),(141,165),(141,184),(141,197),(142,163),(142,185),(142,198),(143,162),(143,185),(143,199),(144,165),(144,185),(144,200),(145,164),(145,185),(145,201),(146,159),(146,190),(146,196),(147,158),(147,191),(147,196),(148,157),(148,192),(148,196),(149,156),(149,193),(149,196),(150,161),(150,194),(150,196),(151,160),(151,195),(151,196),(152,163),(152,186),(152,196),(153,162),(153,187),(153,196),(154,165),(154,188),(154,196),(155,164),(155,189),(155,196),(156,202),(157,202),(158,202),(159,202),(160,202),(161,202),(162,202),(163,202),(164,202),(165,202),(166,186),(166,197),(166,198),(167,187),(167,197),(167,199),(168,188),(168,197),(168,200),(169,189),(169,197),(169,201),(170,190),(170,198),(170,200),(171,191),(171,198),(171,201),(172,192),(172,199),(172,200),(173,193),(173,199),(173,201),(174,195),(174,198),(174,199),(175,194),(175,200),(175,201),(176,181),(176,196),(176,199),(177,182),(177,196),(177,198),(178,183),(178,196),(178,201),(179,184),(179,196),(179,200),(180,185),(180,196),(180,197),(181,202),(182,202),(183,202),(184,202),(185,202),(186,202),(187,202),(188,202),(189,202),(190,202),(191,202),(192,202),(193,202),(194,202),(195,202),(196,202),(197,202),(198,202),(199,202),(200,202),(201,202)],203)
=> ? = 20
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 0
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
Description
The Frankl number of a lattice.
For a lattice $L$ on at least two elements, this is
$$
\max_x(|L|-2|[x, 1]|),
$$
where we maximize over all join irreducible elements and $[x, 1]$ denotes the interval from $x$ to the top element. Frankl's conjecture asserts that this number is non-negative, and zero if and only if $L$ is a Boolean lattice.
Matching statistic: St001845
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 0
([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0
([(0,1)],2)
=> ([],2)
=> ([],1)
=> 0
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 0
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> 0
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ? = 4
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 2
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> 0
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> 0
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 0
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([],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),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,13),(1,14),(1,20),(1,28),(1,29),(1,31),(2,11),(2,12),(2,19),(2,26),(2,27),(2,31),(3,16),(3,18),(3,22),(3,27),(3,29),(3,30),(4,15),(4,17),(4,21),(4,26),(4,28),(4,30),(5,11),(5,15),(5,24),(5,29),(5,32),(5,34),(6,12),(6,16),(6,25),(6,28),(6,32),(6,35),(7,13),(7,17),(7,25),(7,27),(7,33),(7,34),(8,14),(8,18),(8,24),(8,26),(8,33),(8,35),(9,21),(9,22),(9,23),(9,31),(9,34),(9,35),(10,19),(10,20),(10,23),(10,30),(10,32),(10,33),(11,36),(11,40),(11,50),(12,36),(12,41),(12,49),(13,37),(13,42),(13,50),(14,37),(14,43),(14,49),(15,38),(15,40),(15,48),(16,39),(16,41),(16,48),(17,38),(17,42),(17,47),(18,39),(18,43),(18,47),(19,36),(19,44),(19,47),(20,37),(20,44),(20,48),(21,38),(21,45),(21,49),(22,39),(22,45),(22,50),(23,44),(23,45),(23,46),(24,40),(24,43),(24,46),(25,41),(25,42),(25,46),(26,40),(26,47),(26,49),(27,41),(27,47),(27,50),(28,42),(28,48),(28,49),(29,43),(29,48),(29,50),(30,45),(30,47),(30,48),(31,44),(31,49),(31,50),(32,36),(32,46),(32,48),(33,37),(33,46),(33,47),(34,38),(34,46),(34,50),(35,39),(35,46),(35,49),(36,51),(37,51),(38,51),(39,51),(40,51),(41,51),(42,51),(43,51),(44,51),(45,51),(46,51),(47,51),(48,51),(49,51),(50,51)],52)
=> ? = 10
([(3,4)],5)
=> ([(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),(0,5),(0,6),(0,7),(0,8),(0,9),(1,12),(1,15),(1,28),(1,31),(1,34),(2,11),(2,14),(2,28),(2,30),(2,33),(3,10),(3,13),(3,28),(3,29),(3,32),(4,10),(4,16),(4,19),(4,21),(4,30),(4,31),(5,11),(5,17),(5,20),(5,22),(5,29),(5,31),(6,12),(6,18),(6,23),(6,24),(6,29),(6,30),(7,13),(7,16),(7,20),(7,23),(7,33),(7,34),(8,14),(8,17),(8,19),(8,24),(8,32),(8,34),(9,15),(9,18),(9,21),(9,22),(9,32),(9,33),(10,25),(10,35),(10,45),(11,26),(11,36),(11,45),(12,27),(12,37),(12,45),(13,25),(13,38),(13,44),(14,26),(14,39),(14,44),(15,27),(15,40),(15,44),(16,25),(16,42),(16,43),(17,26),(17,41),(17,43),(18,27),(18,41),(18,42),(19,35),(19,39),(19,43),(20,36),(20,38),(20,43),(21,35),(21,40),(21,42),(22,36),(22,40),(22,41),(23,37),(23,38),(23,42),(24,37),(24,39),(24,41),(25,46),(26,46),(27,46),(28,44),(28,45),(29,38),(29,41),(29,45),(30,39),(30,42),(30,45),(31,40),(31,43),(31,45),(32,35),(32,41),(32,44),(33,36),(33,42),(33,44),(34,37),(34,43),(34,44),(35,46),(36,46),(37,46),(38,46),(39,46),(40,46),(41,46),(42,46),(43,46),(44,46),(45,46)],47)
=> ? = 7
([(2,3),(2,4)],5)
=> ([(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),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
=> ? = 5
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
=> ? = 4
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ? = 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
=> ? = 3
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
=> ? = 5
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
=> ? = 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ? = 4
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,28),(1,29),(2,12),(2,16),(2,20),(2,22),(2,29),(3,11),(3,15),(3,20),(3,21),(3,28),(4,13),(4,17),(4,19),(4,21),(4,29),(5,14),(5,18),(5,19),(5,22),(5,28),(6,8),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(8,27),(8,34),(8,35),(9,27),(9,30),(9,31),(10,27),(10,32),(10,33),(11,23),(11,30),(11,34),(12,24),(12,31),(12,34),(13,23),(13,31),(13,35),(14,24),(14,30),(14,35),(15,25),(15,32),(15,34),(16,26),(16,33),(16,34),(17,25),(17,33),(17,35),(18,26),(18,32),(18,35),(19,35),(19,36),(20,34),(20,36),(21,23),(21,25),(21,36),(22,24),(22,26),(22,36),(23,37),(24,37),(25,37),(26,37),(27,37),(28,30),(28,32),(28,36),(29,31),(29,33),(29,36),(30,37),(31,37),(32,37),(33,37),(34,37),(35,37),(36,37)],38)
=> ? = 2
([(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),(0,5),(0,6),(1,16),(1,17),(1,18),(1,25),(2,13),(2,14),(2,15),(2,25),(3,10),(3,11),(3,12),(3,25),(4,8),(4,9),(4,12),(4,15),(4,18),(5,7),(5,9),(5,11),(5,14),(5,17),(6,7),(6,8),(6,10),(6,13),(6,16),(7,21),(7,24),(7,29),(8,19),(8,22),(8,29),(9,20),(9,23),(9,29),(10,26),(10,29),(11,27),(11,29),(12,28),(12,29),(13,19),(13,21),(13,26),(14,20),(14,21),(14,27),(15,19),(15,20),(15,28),(16,22),(16,24),(16,26),(17,23),(17,24),(17,27),(18,22),(18,23),(18,28),(19,30),(20,30),(21,30),(22,30),(23,30),(24,30),(25,26),(25,27),(25,28),(26,30),(27,30),(28,30),(29,30)],31)
=> ? = 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 0
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 2
([(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),(0,5),(0,6),(0,7),(0,8),(1,16),(1,20),(1,24),(1,30),(1,32),(2,15),(2,19),(2,23),(2,30),(2,31),(3,17),(3,21),(3,23),(3,29),(3,32),(4,18),(4,22),(4,24),(4,29),(4,31),(5,10),(5,13),(5,14),(5,17),(5,18),(5,30),(6,10),(6,11),(6,12),(6,15),(6,16),(6,29),(7,9),(7,11),(7,13),(7,19),(7,22),(7,32),(8,9),(8,12),(8,14),(8,20),(8,21),(8,31),(9,35),(9,36),(9,41),(10,33),(10,34),(10,41),(11,25),(11,38),(11,41),(12,26),(12,37),(12,41),(13,28),(13,39),(13,41),(14,27),(14,40),(14,41),(15,25),(15,33),(15,37),(16,26),(16,33),(16,38),(17,27),(17,34),(17,39),(18,28),(18,34),(18,40),(19,25),(19,35),(19,39),(20,26),(20,36),(20,40),(21,27),(21,36),(21,37),(22,28),(22,35),(22,38),(23,37),(23,39),(24,38),(24,40),(25,42),(26,42),(27,42),(28,42),(29,34),(29,37),(29,38),(30,33),(30,39),(30,40),(31,35),(31,37),(31,40),(32,36),(32,38),(32,39),(33,42),(34,42),(35,42),(36,42),(37,42),(38,42),(39,42),(40,42),(41,42)],43)
=> ? = 4
([(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ? = 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 0
([(0,4),(1,2),(1,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,28),(1,29),(2,12),(2,16),(2,20),(2,22),(2,29),(3,11),(3,15),(3,20),(3,21),(3,28),(4,13),(4,17),(4,19),(4,21),(4,29),(5,14),(5,18),(5,19),(5,22),(5,28),(6,8),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(8,27),(8,34),(8,35),(9,27),(9,30),(9,31),(10,27),(10,32),(10,33),(11,23),(11,30),(11,34),(12,24),(12,31),(12,34),(13,23),(13,31),(13,35),(14,24),(14,30),(14,35),(15,25),(15,32),(15,34),(16,26),(16,33),(16,34),(17,25),(17,33),(17,35),(18,26),(18,32),(18,35),(19,35),(19,36),(20,34),(20,36),(21,23),(21,25),(21,36),(22,24),(22,26),(22,36),(23,37),(24,37),(25,37),(26,37),(27,37),(28,30),(28,32),(28,36),(29,31),(29,33),(29,36),(30,37),(31,37),(32,37),(33,37),(34,37),(35,37),(36,37)],38)
=> ? = 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,16),(1,17),(1,18),(1,25),(2,13),(2,14),(2,15),(2,25),(3,10),(3,11),(3,12),(3,25),(4,8),(4,9),(4,12),(4,15),(4,18),(5,7),(5,9),(5,11),(5,14),(5,17),(6,7),(6,8),(6,10),(6,13),(6,16),(7,21),(7,24),(7,29),(8,19),(8,22),(8,29),(9,20),(9,23),(9,29),(10,26),(10,29),(11,27),(11,29),(12,28),(12,29),(13,19),(13,21),(13,26),(14,20),(14,21),(14,27),(15,19),(15,20),(15,28),(16,22),(16,24),(16,26),(17,23),(17,24),(17,27),(18,22),(18,23),(18,28),(19,30),(20,30),(21,30),(22,30),(23,30),(24,30),(25,26),(25,27),(25,28),(26,30),(27,30),(28,30),(29,30)],31)
=> ? = 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 0
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 0
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(1,4),(3,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 0
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 0
([(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,15),(1,20),(1,21),(2,9),(2,10),(2,14),(2,18),(2,19),(3,8),(3,13),(3,17),(3,19),(3,21),(4,8),(4,12),(4,16),(4,18),(4,20),(5,7),(5,14),(5,15),(5,16),(5,17),(6,7),(6,10),(6,11),(6,12),(6,13),(7,31),(7,32),(8,30),(8,32),(9,30),(9,31),(10,22),(10,23),(10,31),(11,24),(11,25),(11,31),(12,22),(12,24),(12,32),(13,23),(13,25),(13,32),(14,26),(14,27),(14,31),(15,28),(15,29),(15,31),(16,26),(16,28),(16,32),(17,27),(17,29),(17,32),(18,22),(18,26),(18,30),(19,23),(19,27),(19,30),(20,24),(20,28),(20,30),(21,25),(21,29),(21,30),(22,33),(23,33),(24,33),(25,33),(26,33),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 0
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 0
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 0
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 0
([],6)
=> ([(0,1),(0,2),(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(1,40),(1,41),(1,42),(1,43),(1,44),(1,45),(1,82),(1,83),(1,84),(1,85),(2,18),(2,19),(2,25),(2,30),(2,31),(2,37),(2,78),(2,79),(2,81),(2,83),(3,16),(3,17),(3,24),(3,28),(3,29),(3,36),(3,76),(3,77),(3,81),(3,82),(4,21),(4,23),(4,27),(4,33),(4,35),(4,39),(4,77),(4,79),(4,80),(4,85),(5,20),(5,22),(5,26),(5,32),(5,34),(5,38),(5,76),(5,78),(5,80),(5,84),(6,22),(6,23),(6,24),(6,46),(6,47),(6,54),(6,83),(6,86),(6,87),(6,91),(7,20),(7,21),(7,25),(7,48),(7,49),(7,55),(7,82),(7,88),(7,89),(7,91),(8,17),(8,19),(8,26),(8,50),(8,52),(8,56),(8,85),(8,86),(8,88),(8,90),(9,16),(9,18),(9,27),(9,51),(9,53),(9,57),(9,84),(9,87),(9,89),(9,90),(10,28),(10,32),(10,43),(10,48),(10,51),(10,59),(10,79),(10,86),(10,92),(10,94),(11,29),(11,33),(11,42),(11,49),(11,50),(11,60),(11,78),(11,87),(11,92),(11,95),(12,30),(12,34),(12,41),(12,46),(12,53),(12,60),(12,77),(12,88),(12,93),(12,94),(13,31),(13,35),(13,40),(13,47),(13,52),(13,59),(13,76),(13,89),(13,93),(13,95),(14,38),(14,39),(14,45),(14,56),(14,57),(14,58),(14,81),(14,91),(14,94),(14,95),(15,36),(15,37),(15,44),(15,54),(15,55),(15,58),(15,80),(15,90),(15,92),(15,93),(16,61),(16,107),(16,112),(16,126),(16,134),(16,170),(17,62),(17,106),(17,113),(17,127),(17,134),(17,171),(18,63),(18,109),(18,112),(18,129),(18,135),(18,172),(19,64),(19,108),(19,113),(19,128),(19,135),(19,173),(20,65),(20,104),(20,110),(20,132),(20,136),(20,170),(21,66),(21,105),(21,110),(21,133),(21,137),(21,171),(22,67),(22,102),(22,111),(22,130),(22,136),(22,172),(23,68),(23,103),(23,111),(23,131),(23,137),(23,173),(24,69),(24,102),(24,103),(24,126),(24,127),(24,175),(25,70),(25,104),(25,105),(25,128),(25,129),(25,175),(26,71),(26,106),(26,108),(26,130),(26,132),(26,174),(27,72),(27,107),(27,109),(27,131),(27,133),(27,174),(28,61),(28,96),(28,114),(28,127),(28,138),(28,169),(29,62),(29,97),(29,115),(29,126),(29,138),(29,168),(30,63),(30,98),(30,117),(30,128),(30,139),(30,169),(31,64),(31,99),(31,116),(31,129),(31,139),(31,168),(32,65),(32,96),(32,118),(32,130),(32,140),(32,167),(33,66),(33,97),(33,119),(33,131),(33,141),(33,167),(34,67),(34,98),(34,120),(34,132),(34,140),(34,166),(35,68),(35,99),(35,121),(35,133),(35,141),(35,166),(36,69),(36,100),(36,122),(36,134),(36,138),(36,166),(37,70),(37,100),(37,123),(37,135),(37,139),(37,167),(38,71),(38,101),(38,124),(38,136),(38,140),(38,168),(39,72),(39,101),(39,125),(39,137),(39,141),(39,169),(40,73),(40,116),(40,121),(40,142),(40,144),(40,170),(41,74),(41,117),(41,120),(41,142),(41,145),(41,171),(42,74),(42,115),(42,119),(42,143),(42,144),(42,172),(43,73),(43,114),(43,118),(43,143),(43,145),(43,173),(44,75),(44,122),(44,123),(44,142),(44,143),(44,174),(45,75),(45,124),(45,125),(45,144),(45,145),(45,175),(46,67),(46,103),(46,117),(46,148),(46,152),(46,178),(47,68),(47,102),(47,116),(47,149),(47,152),(47,179),(48,65),(48,105),(48,114),(48,146),(48,153),(48,178),(49,66),(49,104),(49,115),(49,147),(49,153),(49,179),(50,62),(50,108),(50,119),(50,147),(50,154),(50,176),(51,61),(51,109),(51,118),(51,146),(51,155),(51,176),(52,64),(52,106),(52,121),(52,149),(52,154),(52,177),(53,63),(53,107),(53,120),(53,148),(53,155),(53,177),(54,69),(54,111),(54,123),(54,150),(54,152),(54,176),(55,70),(55,110),(55,122),(55,150),(55,153),(55,177),(56,71),(56,113),(56,125),(56,151),(56,154),(56,178),(57,72),(57,112),(57,124),(57,151),(57,155),(57,179),(58,75),(58,100),(58,101),(58,150),(58,151),(58,180),(59,73),(59,96),(59,99),(59,146),(59,149),(59,180),(60,74),(60,97),(60,98),(60,147),(60,148),(60,180),(61,181),(61,189),(61,190),(62,181),(62,188),(62,191),(63,182),(63,189),(63,192),(64,182),(64,188),(64,193),(65,183),(65,187),(65,190),(66,184),(66,187),(66,191),(67,183),(67,186),(67,192),(68,184),(68,186),(68,193),(69,181),(69,186),(69,194),(70,182),(70,187),(70,194),(71,183),(71,188),(71,195),(72,184),(72,189),(72,195),(73,185),(73,190),(73,193),(74,185),(74,191),(74,192),(75,185),(75,194),(75,195),(76,96),(76,102),(76,106),(76,166),(76,168),(76,170),(77,97),(77,103),(77,107),(77,166),(77,169),(77,171),(78,98),(78,104),(78,108),(78,167),(78,168),(78,172),(79,99),(79,105),(79,109),(79,167),(79,169),(79,173),(80,101),(80,110),(80,111),(80,166),(80,167),(80,174),(81,100),(81,112),(81,113),(81,168),(81,169),(81,175),(82,114),(82,115),(82,122),(82,170),(82,171),(82,175),(83,116),(83,117),(83,123),(83,172),(83,173),(83,175),(84,118),(84,120),(84,124),(84,170),(84,172),(84,174),(85,119),(85,121),(85,125),(85,171),(85,173),(85,174),(86,127),(86,130),(86,149),(86,173),(86,176),(86,178),(87,126),(87,131),(87,148),(87,172),(87,176),(87,179),(88,128),(88,132),(88,147),(88,171),(88,177),(88,178),(89,129),(89,133),(89,146),(89,170),(89,177),(89,179),(90,134),(90,135),(90,151),(90,174),(90,176),(90,177),(91,136),(91,137),(91,150),(91,175),(91,178),(91,179),(92,138),(92,143),(92,153),(92,167),(92,176),(92,180),(93,139),(93,142),(93,152),(93,166),(93,177),(93,180),(94,140),(94,145),(94,155),(94,169),(94,178),(94,180),(95,141),(95,144),(95,154),(95,168),(95,179),(95,180),(96,156),(96,190),(96,197),(97,157),(97,191),(97,197),(98,158),(98,192),(98,197),(99,159),(99,193),(99,197),(100,160),(100,194),(100,197),(101,161),(101,195),(101,197),(102,156),(102,186),(102,200),(103,157),(103,186),(103,201),(104,158),(104,187),(104,200),(105,159),(105,187),(105,201),(106,156),(106,188),(106,198),(107,157),(107,189),(107,198),(108,158),(108,188),(108,199),(109,159),(109,189),(109,199),(110,161),(110,187),(110,198),(111,161),(111,186),(111,199),(112,160),(112,189),(112,200),(113,160),(113,188),(113,201),(114,162),(114,190),(114,201),(115,162),(115,191),(115,200),(116,163),(116,193),(116,200),(117,163),(117,192),(117,201),(118,164),(118,190),(118,199),(119,165),(119,191),(119,199),(120,164),(120,192),(120,198),(121,165),(121,193),(121,198),(122,162),(122,194),(122,198),(123,163),(123,194),(123,199),(124,164),(124,195),(124,200),(125,165),(125,195),(125,201),(126,157),(126,181),(126,200),(127,156),(127,181),(127,201),(128,158),(128,182),(128,201),(129,159),(129,182),(129,200),(130,156),(130,183),(130,199),(131,157),(131,184),(131,199),(132,158),(132,183),(132,198),(133,159),(133,184),(133,198),(134,160),(134,181),(134,198),(135,160),(135,182),(135,199),(136,161),(136,183),(136,200),(137,161),(137,184),(137,201),(138,162),(138,181),(138,197),(139,163),(139,182),(139,197),(140,164),(140,183),(140,197),(141,165),(141,184),(141,197),(142,163),(142,185),(142,198),(143,162),(143,185),(143,199),(144,165),(144,185),(144,200),(145,164),(145,185),(145,201),(146,159),(146,190),(146,196),(147,158),(147,191),(147,196),(148,157),(148,192),(148,196),(149,156),(149,193),(149,196),(150,161),(150,194),(150,196),(151,160),(151,195),(151,196),(152,163),(152,186),(152,196),(153,162),(153,187),(153,196),(154,165),(154,188),(154,196),(155,164),(155,189),(155,196),(156,202),(157,202),(158,202),(159,202),(160,202),(161,202),(162,202),(163,202),(164,202),(165,202),(166,186),(166,197),(166,198),(167,187),(167,197),(167,199),(168,188),(168,197),(168,200),(169,189),(169,197),(169,201),(170,190),(170,198),(170,200),(171,191),(171,198),(171,201),(172,192),(172,199),(172,200),(173,193),(173,199),(173,201),(174,195),(174,198),(174,199),(175,194),(175,200),(175,201),(176,181),(176,196),(176,199),(177,182),(177,196),(177,198),(178,183),(178,196),(178,201),(179,184),(179,196),(179,200),(180,185),(180,196),(180,197),(181,202),(182,202),(183,202),(184,202),(185,202),(186,202),(187,202),(188,202),(189,202),(190,202),(191,202),(192,202),(193,202),(194,202),(195,202),(196,202),(197,202),(198,202),(199,202),(200,202),(201,202)],203)
=> ? = 20
([(4,5)],6)
=> ([(0,2),(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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(1,16),(1,21),(1,22),(1,28),(1,47),(1,52),(1,53),(1,88),(1,89),(1,91),(2,15),(2,19),(2,20),(2,27),(2,46),(2,50),(2,51),(2,86),(2,87),(2,91),(3,18),(3,24),(3,26),(3,30),(3,49),(3,55),(3,57),(3,87),(3,89),(3,90),(4,17),(4,23),(4,25),(4,29),(4,48),(4,54),(4,56),(4,86),(4,88),(4,90),(5,15),(5,31),(5,32),(5,39),(5,47),(5,54),(5,55),(5,92),(5,93),(5,97),(6,16),(6,33),(6,34),(6,40),(6,46),(6,56),(6,57),(6,94),(6,95),(6,97),(7,17),(7,35),(7,37),(7,41),(7,49),(7,50),(7,52),(7,92),(7,94),(7,96),(8,18),(8,36),(8,38),(8,42),(8,48),(8,51),(8,53),(8,93),(8,95),(8,96),(9,19),(9,23),(9,31),(9,35),(9,44),(9,82),(9,84),(9,89),(9,95),(10,20),(10,24),(10,32),(10,36),(10,45),(10,82),(10,85),(10,88),(10,94),(11,21),(11,25),(11,33),(11,37),(11,45),(11,83),(11,84),(11,87),(11,93),(12,22),(12,26),(12,34),(12,38),(12,44),(12,83),(12,85),(12,86),(12,92),(13,29),(13,30),(13,41),(13,42),(13,43),(13,84),(13,85),(13,91),(13,97),(14,27),(14,28),(14,39),(14,40),(14,43),(14,82),(14,83),(14,90),(14,96),(15,70),(15,71),(15,78),(15,152),(15,153),(15,157),(16,72),(16,73),(16,79),(16,154),(16,155),(16,157),(17,74),(17,76),(17,80),(17,152),(17,154),(17,156),(18,75),(18,77),(18,81),(18,153),(18,155),(18,156),(19,58),(19,70),(19,99),(19,122),(19,134),(19,168),(20,59),(20,71),(20,98),(20,123),(20,134),(20,167),(21,60),(21,72),(21,101),(21,124),(21,135),(21,168),(22,61),(22,73),(22,100),(22,125),(22,135),(22,167),(23,62),(23,74),(23,104),(23,122),(23,136),(23,166),(24,63),(24,75),(24,105),(24,123),(24,137),(24,166),(25,64),(25,76),(25,102),(25,124),(25,136),(25,165),(26,65),(26,77),(26,103),(26,125),(26,137),(26,165),(27,66),(27,78),(27,106),(27,126),(27,134),(27,165),(28,67),(28,79),(28,107),(28,126),(28,135),(28,166),(29,68),(29,80),(29,108),(29,127),(29,136),(29,167),(30,69),(30,81),(30,109),(30,127),(30,137),(30,168),(31,62),(31,70),(31,113),(31,128),(31,138),(31,172),(32,63),(32,71),(32,112),(32,129),(32,138),(32,171),(33,64),(33,72),(33,111),(33,130),(33,139),(33,172),(34,65),(34,73),(34,110),(34,131),(34,139),(34,171),(35,58),(35,74),(35,117),(35,128),(35,140),(35,170),(36,59),(36,75),(36,116),(36,129),(36,141),(36,170),(37,60),(37,76),(37,115),(37,130),(37,140),(37,169),(38,61),(38,77),(38,114),(38,131),(38,141),(38,169),(39,67),(39,78),(39,118),(39,132),(39,138),(39,169),(40,66),(40,79),(40,119),(40,132),(40,139),(40,170),(41,69),(41,80),(41,120),(41,133),(41,140),(41,171),(42,68),(42,81),(42,121),(42,133),(42,141),(42,172),(43,126),(43,127),(43,132),(43,133),(43,158),(44,122),(44,125),(44,128),(44,131),(44,158),(45,123),(45,124),(45,129),(45,130),(45,158),(46,66),(46,98),(46,99),(46,110),(46,111),(46,157),(47,67),(47,100),(47,101),(47,112),(47,113),(47,157),(48,68),(48,102),(48,104),(48,114),(48,116),(48,156),(49,69),(49,103),(49,105),(49,115),(49,117),(49,156),(50,58),(50,98),(50,106),(50,115),(50,120),(50,152),(51,59),(51,99),(51,106),(51,114),(51,121),(51,153),(52,60),(52,100),(52,107),(52,117),(52,120),(52,154),(53,61),(53,101),(53,107),(53,116),(53,121),(53,155),(54,62),(54,102),(54,108),(54,112),(54,118),(54,152),(55,63),(55,103),(55,109),(55,113),(55,118),(55,153),(56,64),(56,104),(56,108),(56,110),(56,119),(56,154),(57,65),(57,105),(57,109),(57,111),(57,119),(57,155),(58,159),(58,173),(58,180),(59,160),(59,173),(59,179),(60,161),(60,174),(60,180),(61,162),(61,174),(61,179),(62,159),(62,175),(62,178),(63,160),(63,176),(63,178),(64,161),(64,175),(64,177),(65,162),(65,176),(65,177),(66,163),(66,173),(66,177),(67,163),(67,174),(67,178),(68,164),(68,175),(68,179),(69,164),(69,176),(69,180),(70,142),(70,159),(70,184),(71,142),(71,160),(71,183),(72,143),(72,161),(72,184),(73,143),(73,162),(73,183),(74,144),(74,159),(74,182),(75,145),(75,160),(75,182),(76,144),(76,161),(76,181),(77,145),(77,162),(77,181),(78,142),(78,163),(78,181),(79,143),(79,163),(79,182),(80,144),(80,164),(80,183),(81,145),(81,164),(81,184),(82,134),(82,138),(82,158),(82,166),(82,170),(83,135),(83,139),(83,158),(83,165),(83,169),(84,136),(84,140),(84,158),(84,168),(84,172),(85,137),(85,141),(85,158),(85,167),(85,171),(86,110),(86,114),(86,122),(86,152),(86,165),(86,167),(87,111),(87,115),(87,123),(87,153),(87,165),(87,168),(88,112),(88,116),(88,124),(88,154),(88,166),(88,167),(89,113),(89,117),(89,125),(89,155),(89,166),(89,168),(90,118),(90,119),(90,127),(90,156),(90,165),(90,166),(91,120),(91,121),(91,126),(91,157),(91,167),(91,168),(92,100),(92,103),(92,128),(92,152),(92,169),(92,171),(93,101),(93,102),(93,129),(93,153),(93,169),(93,172),(94,98),(94,105),(94,130),(94,154),(94,170),(94,171),(95,99),(95,104),(95,131),(95,155),(95,170),(95,172),(96,106),(96,107),(96,133),(96,156),(96,169),(96,170),(97,108),(97,109),(97,132),(97,157),(97,171),(97,172),(98,147),(98,173),(98,183),(99,146),(99,173),(99,184),(100,149),(100,174),(100,183),(101,148),(101,174),(101,184),(102,148),(102,175),(102,181),(103,149),(103,176),(103,181),(104,146),(104,175),(104,182),(105,147),(105,176),(105,182),(106,150),(106,173),(106,181),(107,150),(107,174),(107,182),(108,151),(108,175),(108,183),(109,151),(109,176),(109,184),(110,146),(110,177),(110,183),(111,147),(111,177),(111,184),(112,148),(112,178),(112,183),(113,149),(113,178),(113,184),(114,146),(114,179),(114,181),(115,147),(115,180),(115,181),(116,148),(116,179),(116,182),(117,149),(117,180),(117,182),(118,151),(118,178),(118,181),(119,151),(119,177),(119,182),(120,150),(120,180),(120,183),(121,150),(121,179),(121,184),(122,146),(122,159),(122,186),(123,147),(123,160),(123,186),(124,148),(124,161),(124,186),(125,149),(125,162),(125,186),(126,150),(126,163),(126,186),(127,151),(127,164),(127,186),(128,149),(128,159),(128,185),(129,148),(129,160),(129,185),(130,147),(130,161),(130,185),(131,146),(131,162),(131,185),(132,151),(132,163),(132,185),(133,150),(133,164),(133,185),(134,142),(134,173),(134,186),(135,143),(135,174),(135,186),(136,144),(136,175),(136,186),(137,145),(137,176),(137,186),(138,142),(138,178),(138,185),(139,143),(139,177),(139,185),(140,144),(140,180),(140,185),(141,145),(141,179),(141,185),(142,187),(143,187),(144,187),(145,187),(146,187),(147,187),(148,187),(149,187),(150,187),(151,187),(152,159),(152,181),(152,183),(153,160),(153,181),(153,184),(154,161),(154,182),(154,183),(155,162),(155,182),(155,184),(156,164),(156,181),(156,182),(157,163),(157,183),(157,184),(158,185),(158,186),(159,187),(160,187),(161,187),(162,187),(163,187),(164,187),(165,177),(165,181),(165,186),(166,178),(166,182),(166,186),(167,179),(167,183),(167,186),(168,180),(168,184),(168,186),(169,174),(169,181),(169,185),(170,173),(170,182),(170,185),(171,176),(171,183),(171,185),(172,175),(172,184),(172,185),(173,187),(174,187),(175,187),(176,187),(177,187),(178,187),(179,187),(180,187),(181,187),(182,187),(183,187),(184,187),(185,187),(186,187)],188)
=> ? = 16
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 0
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
Description
The number of join irreducibles minus the rank of a lattice.
A lattice is join-extremal, if this statistic is $0$.
Matching statistic: St001435
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 4% ●values known / values provided: 5%●distinct values known / distinct values provided: 4%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 4% ●values known / values provided: 5%●distinct values known / distinct values provided: 4%
Values
([],1)
=> [1]
=> [[1],[]]
=> 0
([],2)
=> [2]
=> [[2],[]]
=> 0
([(0,1)],2)
=> [1]
=> [[1],[]]
=> 0
([],3)
=> [3,3]
=> [[3,3],[]]
=> ? = 1
([(1,2)],3)
=> [3]
=> [[3],[]]
=> 0
([(0,1),(0,2)],3)
=> [2]
=> [[2],[]]
=> 0
([(0,2),(2,1)],3)
=> [1]
=> [[1],[]]
=> 0
([(0,2),(1,2)],3)
=> [2]
=> [[2],[]]
=> 0
([],4)
=> [4,4,4,4,4,4]
=> [[4,4,4,4,4,4],[]]
=> ? = 4
([(2,3)],4)
=> [4,4,4]
=> [[4,4,4],[]]
=> ? = 2
([(1,2),(1,3)],4)
=> [8]
=> [[8],[]]
=> ? = 1
([(0,1),(0,2),(0,3)],4)
=> [3,3]
=> [[3,3],[]]
=> ? = 1
([(0,2),(0,3),(3,1)],4)
=> [3]
=> [[3],[]]
=> 0
([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> [[2],[]]
=> 0
([(1,2),(2,3)],4)
=> [4]
=> [[4],[]]
=> 0
([(0,3),(3,1),(3,2)],4)
=> [2]
=> [[2],[]]
=> 0
([(1,3),(2,3)],4)
=> [8]
=> [[8],[]]
=> ? = 1
([(0,3),(1,3),(3,2)],4)
=> [2]
=> [[2],[]]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3,3]
=> [[3,3],[]]
=> ? = 1
([(0,3),(1,2)],4)
=> [4,2]
=> [[4,2],[]]
=> ? = 0
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> [[3,2],[]]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [[2,2],[]]
=> 0
([(0,3),(2,1),(3,2)],4)
=> [1]
=> [[1],[]]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [[3],[]]
=> 0
([],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
=> [[5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5],[]]
=> ? = 10
([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> [[5,5,5,5,5,5,5,5,5,5,5,5],[]]
=> ? = 7
([(2,3),(2,4)],5)
=> [10,10,10,10]
=> [[10,10,10,10],[]]
=> ? = 5
([(1,2),(1,3),(1,4)],5)
=> [15,15]
=> [[15,15],[]]
=> ? = 4
([(0,1),(0,2),(0,3),(0,4)],5)
=> [4,4,4,4,4,4]
=> [[4,4,4,4,4,4],[]]
=> ? = 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> [[4,4,4],[]]
=> ? = 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [8]
=> [[8],[]]
=> ? = 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [[3,3],[]]
=> ? = 1
([(1,3),(1,4),(4,2)],5)
=> [15]
=> [[15],[]]
=> ? = 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [8]
=> [[8],[]]
=> ? = 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [5,5]
=> [[5,5],[]]
=> ? = 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> [[2],[]]
=> 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,2]
=> [[4,2],[]]
=> ? = 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [[3,2],[]]
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2]
=> [[2,2],[]]
=> 0
([(2,3),(3,4)],5)
=> [5,5,5,5]
=> [[5,5,5,5],[]]
=> ? = 3
([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> [[5,5],[]]
=> ? = 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,3]
=> [[3,3],[]]
=> ? = 1
([(2,4),(3,4)],5)
=> [10,10,10,10]
=> [[10,10,10,10],[]]
=> ? = 5
([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> [[5,5],[]]
=> ? = 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [2,2]
=> [[2,2],[]]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [15,15]
=> [[15,15],[]]
=> ? = 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,3]
=> [[3,3],[]]
=> ? = 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,4,4,4,4,4]
=> [[4,4,4,4,4,4],[]]
=> ? = 4
([(0,4),(1,4),(2,3)],5)
=> [10,10]
=> [[10,10],[]]
=> ? = 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [12,4]
=> [[12,4],[]]
=> ? = 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [14]
=> [[14],[]]
=> ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6,6]
=> [[6,6],[]]
=> ? = 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> [[2],[]]
=> 0
([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> [[8],[]]
=> ? = 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [10,4,4]
=> [[10,4,4],[]]
=> ? = 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> [[4,4,4],[]]
=> ? = 2
([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> [[5,5,5,5,5,5],[]]
=> ? = 4
([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> [[15,5,5],[]]
=> ? = 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> [[5,4],[]]
=> ? = 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [[3,2],[]]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [5,5,5,5]
=> [[5,5,5,5],[]]
=> ? = 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> [[6],[]]
=> ? = 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [2,2]
=> [[2,2],[]]
=> 0
([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> [[7],[]]
=> ? = 0
([(0,4),(1,2),(1,3)],5)
=> [10,10]
=> [[10,10],[]]
=> ? = 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> [10,4,4]
=> [[10,4,4],[]]
=> ? = 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> [[4],[]]
=> 0
([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> [[4,4,3],[]]
=> ? = 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> [[3],[]]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> [[8],[]]
=> ? = 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> [12,4]
=> [[12,4],[]]
=> ? = 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [14]
=> [[14],[]]
=> ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [6,6]
=> [[6,6],[]]
=> ? = 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [5,3]
=> [[5,3],[]]
=> ? = 0
([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> [[5,4],[]]
=> ? = 0
([(1,4),(3,2),(4,3)],5)
=> [5]
=> [[5],[]]
=> 0
([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> [[2],[]]
=> 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [[3],[]]
=> 0
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> [[3],[]]
=> 0
([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> [[1],[]]
=> 0
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [[4],[]]
=> 0
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> [[2],[]]
=> 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [2,2]
=> [[2,2],[]]
=> 0
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [2,2]
=> [[2,2],[]]
=> 0
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [2,2]
=> [[2,2],[]]
=> 0
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> [[2],[]]
=> 0
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [2,2]
=> [[2,2],[]]
=> 0
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [3,2]
=> [[3,2],[]]
=> 0
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> [2,2]
=> [[2,2],[]]
=> 0
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [3,2]
=> [[3,2],[]]
=> 0
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [3]
=> [[3],[]]
=> 0
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> [4]
=> [[4],[]]
=> 0
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [2]
=> [[2],[]]
=> 0
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5]
=> [[5],[]]
=> 0
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> [2]
=> [[2],[]]
=> 0
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> [4]
=> [[4],[]]
=> 0
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> [[2],[]]
=> 0
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [[4],[]]
=> 0
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> [2,2]
=> [[2,2],[]]
=> 0
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> [3,2]
=> [[3,2],[]]
=> 0
Description
The number of missing boxes in the first row.
Matching statistic: St001438
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 4% ●values known / values provided: 5%●distinct values known / distinct values provided: 4%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 4% ●values known / values provided: 5%●distinct values known / distinct values provided: 4%
Values
([],1)
=> [1]
=> [[1],[]]
=> 0
([],2)
=> [2]
=> [[2],[]]
=> 0
([(0,1)],2)
=> [1]
=> [[1],[]]
=> 0
([],3)
=> [3,3]
=> [[3,3],[]]
=> ? = 1
([(1,2)],3)
=> [3]
=> [[3],[]]
=> 0
([(0,1),(0,2)],3)
=> [2]
=> [[2],[]]
=> 0
([(0,2),(2,1)],3)
=> [1]
=> [[1],[]]
=> 0
([(0,2),(1,2)],3)
=> [2]
=> [[2],[]]
=> 0
([],4)
=> [4,4,4,4,4,4]
=> [[4,4,4,4,4,4],[]]
=> ? = 4
([(2,3)],4)
=> [4,4,4]
=> [[4,4,4],[]]
=> ? = 2
([(1,2),(1,3)],4)
=> [8]
=> [[8],[]]
=> ? = 1
([(0,1),(0,2),(0,3)],4)
=> [3,3]
=> [[3,3],[]]
=> ? = 1
([(0,2),(0,3),(3,1)],4)
=> [3]
=> [[3],[]]
=> 0
([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> [[2],[]]
=> 0
([(1,2),(2,3)],4)
=> [4]
=> [[4],[]]
=> 0
([(0,3),(3,1),(3,2)],4)
=> [2]
=> [[2],[]]
=> 0
([(1,3),(2,3)],4)
=> [8]
=> [[8],[]]
=> ? = 1
([(0,3),(1,3),(3,2)],4)
=> [2]
=> [[2],[]]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3,3]
=> [[3,3],[]]
=> ? = 1
([(0,3),(1,2)],4)
=> [4,2]
=> [[4,2],[]]
=> ? = 0
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> [[3,2],[]]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [[2,2],[]]
=> 0
([(0,3),(2,1),(3,2)],4)
=> [1]
=> [[1],[]]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [[3],[]]
=> 0
([],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
=> [[5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5],[]]
=> ? = 10
([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> [[5,5,5,5,5,5,5,5,5,5,5,5],[]]
=> ? = 7
([(2,3),(2,4)],5)
=> [10,10,10,10]
=> [[10,10,10,10],[]]
=> ? = 5
([(1,2),(1,3),(1,4)],5)
=> [15,15]
=> [[15,15],[]]
=> ? = 4
([(0,1),(0,2),(0,3),(0,4)],5)
=> [4,4,4,4,4,4]
=> [[4,4,4,4,4,4],[]]
=> ? = 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> [[4,4,4],[]]
=> ? = 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [8]
=> [[8],[]]
=> ? = 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [[3,3],[]]
=> ? = 1
([(1,3),(1,4),(4,2)],5)
=> [15]
=> [[15],[]]
=> ? = 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [8]
=> [[8],[]]
=> ? = 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [5,5]
=> [[5,5],[]]
=> ? = 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> [[2],[]]
=> 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,2]
=> [[4,2],[]]
=> ? = 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [[3,2],[]]
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2]
=> [[2,2],[]]
=> 0
([(2,3),(3,4)],5)
=> [5,5,5,5]
=> [[5,5,5,5],[]]
=> ? = 3
([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> [[5,5],[]]
=> ? = 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,3]
=> [[3,3],[]]
=> ? = 1
([(2,4),(3,4)],5)
=> [10,10,10,10]
=> [[10,10,10,10],[]]
=> ? = 5
([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> [[5,5],[]]
=> ? = 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [2,2]
=> [[2,2],[]]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [15,15]
=> [[15,15],[]]
=> ? = 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,3]
=> [[3,3],[]]
=> ? = 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,4,4,4,4,4]
=> [[4,4,4,4,4,4],[]]
=> ? = 4
([(0,4),(1,4),(2,3)],5)
=> [10,10]
=> [[10,10],[]]
=> ? = 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [12,4]
=> [[12,4],[]]
=> ? = 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [14]
=> [[14],[]]
=> ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6,6]
=> [[6,6],[]]
=> ? = 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> [[2],[]]
=> 0
([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> [[8],[]]
=> ? = 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [10,4,4]
=> [[10,4,4],[]]
=> ? = 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> [[4,4,4],[]]
=> ? = 2
([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> [[5,5,5,5,5,5],[]]
=> ? = 4
([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> [[15,5,5],[]]
=> ? = 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> [[5,4],[]]
=> ? = 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [[3,2],[]]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [5,5,5,5]
=> [[5,5,5,5],[]]
=> ? = 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> [[6],[]]
=> ? = 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [2,2]
=> [[2,2],[]]
=> 0
([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> [[7],[]]
=> ? = 0
([(0,4),(1,2),(1,3)],5)
=> [10,10]
=> [[10,10],[]]
=> ? = 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> [10,4,4]
=> [[10,4,4],[]]
=> ? = 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> [[4],[]]
=> 0
([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> [[4,4,3],[]]
=> ? = 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> [[3],[]]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> [[8],[]]
=> ? = 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> [12,4]
=> [[12,4],[]]
=> ? = 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [14]
=> [[14],[]]
=> ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [6,6]
=> [[6,6],[]]
=> ? = 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [5,3]
=> [[5,3],[]]
=> ? = 0
([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> [[5,4],[]]
=> ? = 0
([(1,4),(3,2),(4,3)],5)
=> [5]
=> [[5],[]]
=> 0
([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> [[2],[]]
=> 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [[3],[]]
=> 0
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> [[3],[]]
=> 0
([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> [[1],[]]
=> 0
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [[4],[]]
=> 0
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> [[2],[]]
=> 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [2,2]
=> [[2,2],[]]
=> 0
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [2,2]
=> [[2,2],[]]
=> 0
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [2,2]
=> [[2,2],[]]
=> 0
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> [[2],[]]
=> 0
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [2,2]
=> [[2,2],[]]
=> 0
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [3,2]
=> [[3,2],[]]
=> 0
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> [2,2]
=> [[2,2],[]]
=> 0
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [3,2]
=> [[3,2],[]]
=> 0
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [3]
=> [[3],[]]
=> 0
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> [4]
=> [[4],[]]
=> 0
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [2]
=> [[2],[]]
=> 0
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5]
=> [[5],[]]
=> 0
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> [2]
=> [[2],[]]
=> 0
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> [4]
=> [[4],[]]
=> 0
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> [[2],[]]
=> 0
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [[4],[]]
=> 0
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> [2,2]
=> [[2,2],[]]
=> 0
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> [3,2]
=> [[3,2],[]]
=> 0
Description
The number of missing boxes of a skew partition.
Matching statistic: St001487
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 4% ●values known / values provided: 5%●distinct values known / distinct values provided: 4%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 4% ●values known / values provided: 5%●distinct values known / distinct values provided: 4%
Values
([],1)
=> [1]
=> [[1],[]]
=> 1 = 0 + 1
([],2)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(0,1)],2)
=> [1]
=> [[1],[]]
=> 1 = 0 + 1
([],3)
=> [3,3]
=> [[3,3],[]]
=> ? = 1 + 1
([(1,2)],3)
=> [3]
=> [[3],[]]
=> 1 = 0 + 1
([(0,1),(0,2)],3)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(0,2),(2,1)],3)
=> [1]
=> [[1],[]]
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([],4)
=> [4,4,4,4,4,4]
=> [[4,4,4,4,4,4],[]]
=> ? = 4 + 1
([(2,3)],4)
=> [4,4,4]
=> [[4,4,4],[]]
=> ? = 2 + 1
([(1,2),(1,3)],4)
=> [8]
=> [[8],[]]
=> ? = 1 + 1
([(0,1),(0,2),(0,3)],4)
=> [3,3]
=> [[3,3],[]]
=> ? = 1 + 1
([(0,2),(0,3),(3,1)],4)
=> [3]
=> [[3],[]]
=> 1 = 0 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(1,2),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1 = 0 + 1
([(0,3),(3,1),(3,2)],4)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> [8]
=> [[8],[]]
=> ? = 1 + 1
([(0,3),(1,3),(3,2)],4)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> [3,3]
=> [[3,3],[]]
=> ? = 1 + 1
([(0,3),(1,2)],4)
=> [4,2]
=> [[4,2],[]]
=> ? = 0 + 1
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> [[3,2],[]]
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [[2,2],[]]
=> 1 = 0 + 1
([(0,3),(2,1),(3,2)],4)
=> [1]
=> [[1],[]]
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [[3],[]]
=> 1 = 0 + 1
([],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
=> [[5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5],[]]
=> ? = 10 + 1
([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> [[5,5,5,5,5,5,5,5,5,5,5,5],[]]
=> ? = 7 + 1
([(2,3),(2,4)],5)
=> [10,10,10,10]
=> [[10,10,10,10],[]]
=> ? = 5 + 1
([(1,2),(1,3),(1,4)],5)
=> [15,15]
=> [[15,15],[]]
=> ? = 4 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [4,4,4,4,4,4]
=> [[4,4,4,4,4,4],[]]
=> ? = 4 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> [[4,4,4],[]]
=> ? = 2 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [8]
=> [[8],[]]
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [[3,3],[]]
=> ? = 1 + 1
([(1,3),(1,4),(4,2)],5)
=> [15]
=> [[15],[]]
=> ? = 2 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [8]
=> [[8],[]]
=> ? = 1 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [5,5]
=> [[5,5],[]]
=> ? = 1 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,2]
=> [[4,2],[]]
=> ? = 0 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [[3,2],[]]
=> 1 = 0 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2]
=> [[2,2],[]]
=> 1 = 0 + 1
([(2,3),(3,4)],5)
=> [5,5,5,5]
=> [[5,5,5,5],[]]
=> ? = 3 + 1
([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> [[5,5],[]]
=> ? = 1 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,3]
=> [[3,3],[]]
=> ? = 1 + 1
([(2,4),(3,4)],5)
=> [10,10,10,10]
=> [[10,10,10,10],[]]
=> ? = 5 + 1
([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> [[5,5],[]]
=> ? = 1 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [2,2]
=> [[2,2],[]]
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> [15,15]
=> [[15,15],[]]
=> ? = 4 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,3]
=> [[3,3],[]]
=> ? = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,4,4,4,4,4]
=> [[4,4,4,4,4,4],[]]
=> ? = 4 + 1
([(0,4),(1,4),(2,3)],5)
=> [10,10]
=> [[10,10],[]]
=> ? = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [12,4]
=> [[12,4],[]]
=> ? = 1 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [14]
=> [[14],[]]
=> ? = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6,6]
=> [[6,6],[]]
=> ? = 1 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> [[8],[]]
=> ? = 1 + 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [10,4,4]
=> [[10,4,4],[]]
=> ? = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> [[4,4,4],[]]
=> ? = 2 + 1
([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> [[5,5,5,5,5,5],[]]
=> ? = 4 + 1
([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> [[15,5,5],[]]
=> ? = 3 + 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> [[5,4],[]]
=> ? = 0 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [[3,2],[]]
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [5,5,5,5]
=> [[5,5,5,5],[]]
=> ? = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> [[6],[]]
=> ? = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [2,2]
=> [[2,2],[]]
=> 1 = 0 + 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> [[7],[]]
=> ? = 0 + 1
([(0,4),(1,2),(1,3)],5)
=> [10,10]
=> [[10,10],[]]
=> ? = 2 + 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [10,4,4]
=> [[10,4,4],[]]
=> ? = 2 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> [[4],[]]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> [[4,4,3],[]]
=> ? = 1 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> [[3],[]]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> [[8],[]]
=> ? = 1 + 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> [12,4]
=> [[12,4],[]]
=> ? = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [14]
=> [[14],[]]
=> ? = 1 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [6,6]
=> [[6,6],[]]
=> ? = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [5,3]
=> [[5,3],[]]
=> ? = 0 + 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> [[5,4],[]]
=> ? = 0 + 1
([(1,4),(3,2),(4,3)],5)
=> [5]
=> [[5],[]]
=> 1 = 0 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [[3],[]]
=> 1 = 0 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> [[3],[]]
=> 1 = 0 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> [[1],[]]
=> 1 = 0 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [[4],[]]
=> 1 = 0 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [2,2]
=> [[2,2],[]]
=> 1 = 0 + 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [2,2]
=> [[2,2],[]]
=> 1 = 0 + 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [2,2]
=> [[2,2],[]]
=> 1 = 0 + 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [2,2]
=> [[2,2],[]]
=> 1 = 0 + 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [3,2]
=> [[3,2],[]]
=> 1 = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> [2,2]
=> [[2,2],[]]
=> 1 = 0 + 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [3,2]
=> [[3,2],[]]
=> 1 = 0 + 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [3]
=> [[3],[]]
=> 1 = 0 + 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> [4]
=> [[4],[]]
=> 1 = 0 + 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5]
=> [[5],[]]
=> 1 = 0 + 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> [4]
=> [[4],[]]
=> 1 = 0 + 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [[4],[]]
=> 1 = 0 + 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> [2,2]
=> [[2,2],[]]
=> 1 = 0 + 1
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> [3,2]
=> [[3,2],[]]
=> 1 = 0 + 1
Description
The number of inner corners of a skew partition.
Matching statistic: St001490
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 4% ●values known / values provided: 5%●distinct values known / distinct values provided: 4%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 4% ●values known / values provided: 5%●distinct values known / distinct values provided: 4%
Values
([],1)
=> [1]
=> [[1],[]]
=> 1 = 0 + 1
([],2)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(0,1)],2)
=> [1]
=> [[1],[]]
=> 1 = 0 + 1
([],3)
=> [3,3]
=> [[3,3],[]]
=> ? = 1 + 1
([(1,2)],3)
=> [3]
=> [[3],[]]
=> 1 = 0 + 1
([(0,1),(0,2)],3)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(0,2),(2,1)],3)
=> [1]
=> [[1],[]]
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([],4)
=> [4,4,4,4,4,4]
=> [[4,4,4,4,4,4],[]]
=> ? = 4 + 1
([(2,3)],4)
=> [4,4,4]
=> [[4,4,4],[]]
=> ? = 2 + 1
([(1,2),(1,3)],4)
=> [8]
=> [[8],[]]
=> ? = 1 + 1
([(0,1),(0,2),(0,3)],4)
=> [3,3]
=> [[3,3],[]]
=> ? = 1 + 1
([(0,2),(0,3),(3,1)],4)
=> [3]
=> [[3],[]]
=> 1 = 0 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(1,2),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1 = 0 + 1
([(0,3),(3,1),(3,2)],4)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> [8]
=> [[8],[]]
=> ? = 1 + 1
([(0,3),(1,3),(3,2)],4)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> [3,3]
=> [[3,3],[]]
=> ? = 1 + 1
([(0,3),(1,2)],4)
=> [4,2]
=> [[4,2],[]]
=> ? = 0 + 1
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> [[3,2],[]]
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [[2,2],[]]
=> 1 = 0 + 1
([(0,3),(2,1),(3,2)],4)
=> [1]
=> [[1],[]]
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [[3],[]]
=> 1 = 0 + 1
([],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
=> [[5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5],[]]
=> ? = 10 + 1
([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> [[5,5,5,5,5,5,5,5,5,5,5,5],[]]
=> ? = 7 + 1
([(2,3),(2,4)],5)
=> [10,10,10,10]
=> [[10,10,10,10],[]]
=> ? = 5 + 1
([(1,2),(1,3),(1,4)],5)
=> [15,15]
=> [[15,15],[]]
=> ? = 4 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [4,4,4,4,4,4]
=> [[4,4,4,4,4,4],[]]
=> ? = 4 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> [[4,4,4],[]]
=> ? = 2 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [8]
=> [[8],[]]
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [[3,3],[]]
=> ? = 1 + 1
([(1,3),(1,4),(4,2)],5)
=> [15]
=> [[15],[]]
=> ? = 2 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [8]
=> [[8],[]]
=> ? = 1 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [5,5]
=> [[5,5],[]]
=> ? = 1 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,2]
=> [[4,2],[]]
=> ? = 0 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [[3,2],[]]
=> 1 = 0 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2]
=> [[2,2],[]]
=> 1 = 0 + 1
([(2,3),(3,4)],5)
=> [5,5,5,5]
=> [[5,5,5,5],[]]
=> ? = 3 + 1
([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> [[5,5],[]]
=> ? = 1 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,3]
=> [[3,3],[]]
=> ? = 1 + 1
([(2,4),(3,4)],5)
=> [10,10,10,10]
=> [[10,10,10,10],[]]
=> ? = 5 + 1
([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> [[5,5],[]]
=> ? = 1 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [2,2]
=> [[2,2],[]]
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> [15,15]
=> [[15,15],[]]
=> ? = 4 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,3]
=> [[3,3],[]]
=> ? = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,4,4,4,4,4]
=> [[4,4,4,4,4,4],[]]
=> ? = 4 + 1
([(0,4),(1,4),(2,3)],5)
=> [10,10]
=> [[10,10],[]]
=> ? = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [12,4]
=> [[12,4],[]]
=> ? = 1 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [14]
=> [[14],[]]
=> ? = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6,6]
=> [[6,6],[]]
=> ? = 1 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> [[8],[]]
=> ? = 1 + 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [10,4,4]
=> [[10,4,4],[]]
=> ? = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> [[4,4,4],[]]
=> ? = 2 + 1
([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> [[5,5,5,5,5,5],[]]
=> ? = 4 + 1
([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> [[15,5,5],[]]
=> ? = 3 + 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> [[5,4],[]]
=> ? = 0 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [[3,2],[]]
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [5,5,5,5]
=> [[5,5,5,5],[]]
=> ? = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> [[6],[]]
=> ? = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [2,2]
=> [[2,2],[]]
=> 1 = 0 + 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> [[7],[]]
=> ? = 0 + 1
([(0,4),(1,2),(1,3)],5)
=> [10,10]
=> [[10,10],[]]
=> ? = 2 + 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [10,4,4]
=> [[10,4,4],[]]
=> ? = 2 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> [[4],[]]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> [[4,4,3],[]]
=> ? = 1 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> [[3],[]]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> [[8],[]]
=> ? = 1 + 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> [12,4]
=> [[12,4],[]]
=> ? = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [14]
=> [[14],[]]
=> ? = 1 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [6,6]
=> [[6,6],[]]
=> ? = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [5,3]
=> [[5,3],[]]
=> ? = 0 + 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> [[5,4],[]]
=> ? = 0 + 1
([(1,4),(3,2),(4,3)],5)
=> [5]
=> [[5],[]]
=> 1 = 0 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [[3],[]]
=> 1 = 0 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> [[3],[]]
=> 1 = 0 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> [[1],[]]
=> 1 = 0 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [[4],[]]
=> 1 = 0 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [2,2]
=> [[2,2],[]]
=> 1 = 0 + 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [2,2]
=> [[2,2],[]]
=> 1 = 0 + 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [2,2]
=> [[2,2],[]]
=> 1 = 0 + 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [2,2]
=> [[2,2],[]]
=> 1 = 0 + 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [3,2]
=> [[3,2],[]]
=> 1 = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> [2,2]
=> [[2,2],[]]
=> 1 = 0 + 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [3,2]
=> [[3,2],[]]
=> 1 = 0 + 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [3]
=> [[3],[]]
=> 1 = 0 + 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> [4]
=> [[4],[]]
=> 1 = 0 + 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5]
=> [[5],[]]
=> 1 = 0 + 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> [4]
=> [[4],[]]
=> 1 = 0 + 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [[4],[]]
=> 1 = 0 + 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> [2,2]
=> [[2,2],[]]
=> 1 = 0 + 1
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> [3,2]
=> [[3,2],[]]
=> 1 = 0 + 1
Description
The number of connected components of a skew partition.
Matching statistic: St001232
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 5%●distinct values known / distinct values provided: 4%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 5%●distinct values known / distinct values provided: 4%
Values
([],1)
=> [1]
=> [1,0]
=> [1,0]
=> 0
([],2)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(0,1)],2)
=> [1]
=> [1,0]
=> [1,0]
=> 0
([],3)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
([(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(0,1),(0,2)],3)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(0,2),(2,1)],3)
=> [1]
=> [1,0]
=> [1,0]
=> 0
([(0,2),(1,2)],3)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([],4)
=> [4,4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
([(2,3)],4)
=> [4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
([(1,2),(1,3)],4)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
([(0,1),(0,2),(0,3)],4)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
([(0,2),(0,3),(3,1)],4)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(0,3),(3,1),(3,2)],4)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(1,3),(2,3)],4)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
([(0,3),(1,3),(3,2)],4)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
([(0,3),(1,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ? = 0
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? = 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0
([(0,3),(2,1),(3,2)],4)
=> [1]
=> [1,0]
=> [1,0]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
([(2,3),(2,4)],5)
=> [10,10,10,10]
=> [1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
([(1,2),(1,3),(1,4)],5)
=> [15,15]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 4
([(0,1),(0,2),(0,3),(0,4)],5)
=> [4,4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
([(1,3),(1,4),(4,2)],5)
=> [15]
=> [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]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ? = 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? = 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0
([(2,3),(3,4)],5)
=> [5,5,5,5]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3
([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
([(2,4),(3,4)],5)
=> [10,10,10,10]
=> [1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0
([(1,4),(2,4),(3,4)],5)
=> [15,15]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
([(0,4),(1,4),(2,3)],5)
=> [10,10]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [12,4]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [14]
=> [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]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [10,4,4]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? = 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [5,5,5,5]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0
([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
([(0,4),(1,2),(1,3)],5)
=> [10,10]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> [10,4,4]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(1,4),(3,2),(4,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> [1,0]
=> [1,0]
=> 0
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(1,5),(3,4),(4,2),(5,3)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
([(0,5),(1,2),(2,3),(2,5),(3,4),(5,4)],6)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001491
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 4%
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 4%
Values
([],1)
=> [1]
=> 10 => 1 = 0 + 1
([],2)
=> [2]
=> 100 => 1 = 0 + 1
([(0,1)],2)
=> [1]
=> 10 => 1 = 0 + 1
([],3)
=> [3,3]
=> 11000 => ? = 1 + 1
([(1,2)],3)
=> [3]
=> 1000 => 1 = 0 + 1
([(0,1),(0,2)],3)
=> [2]
=> 100 => 1 = 0 + 1
([(0,2),(2,1)],3)
=> [1]
=> 10 => 1 = 0 + 1
([(0,2),(1,2)],3)
=> [2]
=> 100 => 1 = 0 + 1
([],4)
=> [4,4,4,4,4,4]
=> 1111110000 => ? = 4 + 1
([(2,3)],4)
=> [4,4,4]
=> 1110000 => ? = 2 + 1
([(1,2),(1,3)],4)
=> [8]
=> 100000000 => ? = 1 + 1
([(0,1),(0,2),(0,3)],4)
=> [3,3]
=> 11000 => ? = 1 + 1
([(0,2),(0,3),(3,1)],4)
=> [3]
=> 1000 => 1 = 0 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> 100 => 1 = 0 + 1
([(1,2),(2,3)],4)
=> [4]
=> 10000 => ? = 0 + 1
([(0,3),(3,1),(3,2)],4)
=> [2]
=> 100 => 1 = 0 + 1
([(1,3),(2,3)],4)
=> [8]
=> 100000000 => ? = 1 + 1
([(0,3),(1,3),(3,2)],4)
=> [2]
=> 100 => 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> [3,3]
=> 11000 => ? = 1 + 1
([(0,3),(1,2)],4)
=> [4,2]
=> 100100 => ? = 0 + 1
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> 10100 => ? = 0 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,3),(2,1),(3,2)],4)
=> [1]
=> 10 => 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1000 => 1 = 0 + 1
([],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
=> 11111111111111111111111100000 => ? = 10 + 1
([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> 11111111111100000 => ? = 7 + 1
([(2,3),(2,4)],5)
=> [10,10,10,10]
=> 11110000000000 => ? = 5 + 1
([(1,2),(1,3),(1,4)],5)
=> [15,15]
=> 11000000000000000 => ? = 4 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [4,4,4,4,4,4]
=> 1111110000 => ? = 4 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 1110000 => ? = 2 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [8]
=> 100000000 => ? = 1 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> 11000 => ? = 1 + 1
([(1,3),(1,4),(4,2)],5)
=> [15]
=> 1000000000000000 => ? = 2 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [8]
=> 100000000 => ? = 1 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [5,5]
=> 1100000 => ? = 1 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> 100 => 1 = 0 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,2]
=> 100100 => ? = 0 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> 10100 => ? = 0 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2]
=> 1100 => 1 = 0 + 1
([(2,3),(3,4)],5)
=> [5,5,5,5]
=> 111100000 => ? = 3 + 1
([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 1100000 => ? = 1 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,3]
=> 11000 => ? = 1 + 1
([(2,4),(3,4)],5)
=> [10,10,10,10]
=> 11110000000000 => ? = 5 + 1
([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> 1100000 => ? = 1 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [2,2]
=> 1100 => 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> [15,15]
=> 11000000000000000 => ? = 4 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,3]
=> 11000 => ? = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,4,4,4,4,4]
=> 1111110000 => ? = 4 + 1
([(0,4),(1,4),(2,3)],5)
=> [10,10]
=> 110000000000 => ? = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [12,4]
=> 10000000010000 => ? = 1 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [14]
=> 100000000000000 => ? = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6,6]
=> 11000000 => ? = 1 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 100 => 1 = 0 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 100000000 => ? = 1 + 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [10,4,4]
=> 1000000110000 => ? = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 1110000 => ? = 2 + 1
([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> 11111100000 => ? = 4 + 1
([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> 100000000001100000 => ? = 3 + 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> 1010000 => ? = 0 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> 10100 => ? = 0 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [5,5,5,5]
=> 111100000 => ? = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> 1000000 => ? = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> 10000000 => ? = 0 + 1
([(0,4),(1,2),(1,3)],5)
=> [10,10]
=> 110000000000 => ? = 2 + 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [10,4,4]
=> 1000000110000 => ? = 2 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> 10000 => ? = 0 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> 1101000 => ? = 1 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> 1000 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> 100000000 => ? = 1 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> 100 => 1 = 0 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1000 => 1 = 0 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> 1000 => 1 = 0 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 10 => 1 = 0 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 100 => 1 = 0 + 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 100 => 1 = 0 + 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [3]
=> 1000 => 1 = 0 + 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [2]
=> 100 => 1 = 0 + 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> [2]
=> 100 => 1 = 0 + 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 100 => 1 = 0 + 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [3]
=> 1000 => 1 = 0 + 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 10 => 1 = 0 + 1
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 1000 => 1 = 0 + 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [2]
=> 100 => 1 = 0 + 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [3]
=> 1000 => 1 = 0 + 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,6),(1,6),(2,5),(3,5),(4,2),(4,3),(6,4)],7)
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,6),(1,6),(4,5),(5,2),(5,3),(6,4)],7)
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> [2]
=> 100 => 1 = 0 + 1
([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> [2]
=> 100 => 1 = 0 + 1
([(0,6),(1,6),(2,5),(3,5),(5,4),(6,2),(6,3)],7)
=> [2,2]
=> 1100 => 1 = 0 + 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,2),(5,4),(6,4)],7)
=> [2,2]
=> 1100 => 1 = 0 + 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
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St001875The number of simple modules with projective dimension at most 1. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001624The breadth of a lattice. St001890The maximum magnitude of the Möbius function of a poset. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000637The length of the longest cycle in a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000699The toughness times the least common multiple of 1,. St000948The chromatic discriminant of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001281The normalized isoperimetric number of a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001395The number of strictly unfriendly partitions of a graph. St001638The book thickness of a graph. St001689The number of celebrities in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001736The total number of cycles in a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001797The number of overfull subgraphs of a graph. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001272The number of graphs with the same degree sequence. St001316The domatic number of a graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St001546The number of monomials in the Tutte polynomial of a graph. St000636The hull number of a graph. St001029The size of the core of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001654The monophonic hull number of a graph.
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