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Your data matches 131 different statistics following compositions of up to 3 maps.
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Matching statistic: St000527
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(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> 3
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> 3
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 2
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> 4
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> 4
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> 4
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> 4
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> 3
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> 3
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> 3
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> 3
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> 3
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 2
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> 2
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 2
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> 3
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> 3
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> 3
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 4
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> 4
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> 3
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> 4
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> 3
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 3
Description
The width of the poset.
This is the size of the poset's longest antichain, also called Dilworth number.
Matching statistic: St000010
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00193: Lattices —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 4
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> [4,1,1]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> [4,1,1]
=> 3
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [4,1,1]
=> 3
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [5,1]
=> 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> [5,1]
=> 2
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [4,1,1]
=> 3
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [5,1]
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [5,1]
=> 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> 5
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> [4,1,1,1]
=> 4
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> [4,1,1,1]
=> 4
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> [4,1,1,1]
=> 4
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> [4,1,1,1]
=> 4
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> [5,1,1]
=> 3
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> [5,1,1]
=> 3
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> [5,1,1]
=> 3
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> [5,1,1]
=> 3
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> [5,1,1]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> [5,1,1]
=> 3
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [6,1]
=> 2
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> [5,2]
=> 2
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 2
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> [5,1,1]
=> 3
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> [5,1,1]
=> 3
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> [5,1,1]
=> 3
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> [4,1,1,1]
=> 4
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> [5,1,1]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> [5,2]
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> [4,1,1,1]
=> 4
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> [5,1,1]
=> 3
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> [4,1,1,1]
=> 4
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> [4,2,1]
=> 3
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> [4,2,1]
=> 3
Description
The length of the partition.
Matching statistic: St000097
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 3
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(3,5),(4,5)],6)
=> 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 2
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(3,5),(4,5)],6)
=> 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> 5
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> 3
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 2
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> 2
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> 3
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(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)
=> 4
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(3,6),(4,5)],7)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> 3
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(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)
=> 4
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(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)
=> 3
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 3
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(3,5),(4,5)],6)
=> 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 2
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(3,5),(4,5)],6)
=> 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> 5
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> 3
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 2
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> 2
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> 3
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(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)
=> 4
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(3,6),(4,5)],7)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> 3
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(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)
=> 4
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(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)
=> 3
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St001029
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 3
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(3,5),(4,5)],6)
=> 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 2
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(3,5),(4,5)],6)
=> 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> 5
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> 3
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 2
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> 2
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> 3
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(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)
=> 4
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(3,6),(4,5)],7)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> 3
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(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)
=> 4
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(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)
=> 3
Description
The size of the core of a graph.
The core of the graph $G$ is the smallest graph $C$ such that there is a graph homomorphism from $G$ to $C$ and a graph homomorphism from $C$ to $G$.
Matching statistic: St000093
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(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)
=> 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(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)
=> 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(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)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],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)
=> 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(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),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(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),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(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),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(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)
=> 3
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(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)
=> 3
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(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)
=> 3
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(2,6),(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)
=> 3
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(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)
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(2,6),(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)
=> 3
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(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)
=> 2
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(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)
=> 2
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(2,5),(2,6),(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)
=> 3
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(2,6),(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)
=> 3
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(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)
=> 3
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(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)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(2,6),(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)
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(3,6),(4,5)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(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),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(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)
=> 3
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(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)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(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)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000147
Mp00193: Lattices —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> [1]
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [4,1,1]
=> 4
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> [4,1,1]
=> [3,1,1,1]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> [4,1,1]
=> [3,1,1,1]
=> 3
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [4,1,1]
=> [3,1,1,1]
=> 3
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [4,1,1]
=> [3,1,1,1]
=> 3
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> [3,1,1,1]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2,2,1,1]
=> 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> [2,2,1,1]
=> 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [5,1,1]
=> 5
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> [4,1,1,1]
=> [4,1,1,1]
=> 4
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> [4,1,1,1]
=> [4,1,1,1]
=> 4
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> [4,1,1,1]
=> [4,1,1,1]
=> 4
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> [4,1,1,1]
=> [4,1,1,1]
=> 4
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> [5,1,1]
=> [3,1,1,1,1]
=> 3
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> [5,1,1]
=> [3,1,1,1,1]
=> 3
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> [5,1,1]
=> [3,1,1,1,1]
=> 3
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> [5,1,1]
=> [3,1,1,1,1]
=> 3
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> [5,1,1]
=> [3,1,1,1,1]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> [5,1,1]
=> [3,1,1,1,1]
=> 3
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [6,1]
=> [2,1,1,1,1,1]
=> 2
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> [5,2]
=> [2,2,1,1,1]
=> 2
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [2,2,1,1,1]
=> 2
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> [5,1,1]
=> [3,1,1,1,1]
=> 3
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> [5,1,1]
=> [3,1,1,1,1]
=> 3
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> [5,1,1]
=> [3,1,1,1,1]
=> 3
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [4,1,1,1]
=> 4
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> [5,1,1]
=> [3,1,1,1,1]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> [5,2]
=> [2,2,1,1,1]
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> [4,1,1,1]
=> [4,1,1,1]
=> 4
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> [5,1,1]
=> [3,1,1,1,1]
=> 3
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> [4,1,1,1]
=> [4,1,1,1]
=> 4
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> [4,2,1]
=> [3,2,1,1]
=> 3
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [3,2,1,1]
=> 3
Description
The largest part of an integer partition.
Matching statistic: St000288
Mp00193: Lattices —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> 10 => 1
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 100 => 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> 1000 => 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 10000 => 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> 100010 => 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 100010 => 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 100000 => 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 100010 => 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 1001110 => 4
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> [4,1,1]
=> 1000110 => 3
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> [4,1,1]
=> 1000110 => 3
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [4,1,1]
=> 1000110 => 3
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [5,1]
=> 1000010 => 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 1000010 => 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> [5,1]
=> 1000010 => 2
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 1000010 => 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [4,1,1]
=> 1000110 => 3
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [5,1]
=> 1000010 => 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> 1000110 => 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> 100100 => 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 100100 => 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 1000000 => 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [5,1]
=> 1000010 => 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> 10011110 => 5
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> [4,1,1,1]
=> 10001110 => 4
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> [4,1,1,1]
=> 10001110 => 4
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> [4,1,1,1]
=> 10001110 => 4
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> [4,1,1,1]
=> 10001110 => 4
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> [5,1,1]
=> 10000110 => 3
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> [5,1,1]
=> 10000110 => 3
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> [5,1,1]
=> 10000110 => 3
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> [5,1,1]
=> 10000110 => 3
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> [5,1,1]
=> 10000110 => 3
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> [5,1,1]
=> 10000110 => 3
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [6,1]
=> 10000010 => 2
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> [5,2]
=> 1000100 => 2
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 1000100 => 2
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> [5,1,1]
=> 10000110 => 3
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> [5,1,1]
=> 10000110 => 3
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> [5,1,1]
=> 10000110 => 3
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> [4,1,1,1]
=> 10001110 => 4
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> [5,1,1]
=> 10000110 => 3
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> [5,2]
=> 1000100 => 2
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> [4,1,1,1]
=> 10001110 => 4
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> [5,1,1]
=> 10000110 => 3
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> [4,1,1,1]
=> 10001110 => 4
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> [4,2,1]
=> 1001010 => 3
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> [4,2,1]
=> 1001010 => 3
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000378
Mp00193: Lattices —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> [1]
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [4,1]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [3,3]
=> 4
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> [4,1,1]
=> [3,1,1,1]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> [4,1,1]
=> [3,1,1,1]
=> 3
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [4,1,1]
=> [3,1,1,1]
=> 3
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [4,1,1]
=> [3,1,1,1]
=> 3
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> [3,1,1,1]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2,2,1,1]
=> 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> [2,2,1,1]
=> 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [3,2,2]
=> 5
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> [4,1,1,1]
=> [3,2,1,1]
=> 4
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> [4,1,1,1]
=> [3,2,1,1]
=> 4
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> [4,1,1,1]
=> [3,2,1,1]
=> 4
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> [4,1,1,1]
=> [3,2,1,1]
=> 4
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> [5,1,1]
=> [3,1,1,1,1]
=> 3
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> [5,1,1]
=> [3,1,1,1,1]
=> 3
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> [5,1,1]
=> [3,1,1,1,1]
=> 3
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> [5,1,1]
=> [3,1,1,1,1]
=> 3
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> [5,1,1]
=> [3,1,1,1,1]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> [5,1,1]
=> [3,1,1,1,1]
=> 3
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [6,1]
=> [2,1,1,1,1,1]
=> 2
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> [5,2]
=> [2,2,1,1,1]
=> 2
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [2,2,1,1,1]
=> 2
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> [5,1,1]
=> [3,1,1,1,1]
=> 3
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> [5,1,1]
=> [3,1,1,1,1]
=> 3
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> [5,1,1]
=> [3,1,1,1,1]
=> 3
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [3,2,1,1]
=> 4
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> [5,1,1]
=> [3,1,1,1,1]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> [5,2]
=> [2,2,1,1,1]
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> [4,1,1,1]
=> [3,2,1,1]
=> 4
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> [5,1,1]
=> [3,1,1,1,1]
=> 3
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> [4,1,1,1]
=> [3,2,1,1]
=> 4
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> [4,2,1]
=> [5,1,1]
=> 3
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [5,1,1]
=> 3
Description
The diagonal inversion number of an integer partition.
The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000733
Mp00193: Lattices —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> [[1]]
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 4
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 3
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 3
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> 2
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 3
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> [[1,2,3,4,5,6]]
=> 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> 5
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> 4
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> 4
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> 4
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> 4
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> 3
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> 3
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> 3
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> 3
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> 3
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [6,1]
=> [[1,2,3,4,5,6],[7]]
=> 2
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 2
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 2
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> 3
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> 3
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> 3
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> 4
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> 4
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> 3
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> 4
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> 3
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> 3
Description
The row containing the largest entry of a standard tableau.
The following 121 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000820The number of compositions obtained by rotating the composition. St000822The Hadwiger number of the graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000157The number of descents of a standard tableau. St000225Difference between largest and smallest parts in a partition. St000272The treewidth of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000519The largest length of a factor maximising the subword complexity. St000536The pathwidth of a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001777The number of weak descents in an integer composition. St001962The proper pathwidth of a graph. St000632The jump number of the poset. St000172The Grundy number of a graph. St001581The achromatic number of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001644The dimension of a graph. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001280The number of parts of an integer partition that are at least two. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001323The independence gap of a graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St001330The hat guessing number of a graph. St000143The largest repeated part of a partition. St000741The Colin de Verdière graph invariant. St000454The largest eigenvalue of a graph if it is integral. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St000259The diameter of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000087The number of induced subgraphs. St000286The number of connected components of the complement of a graph. St000363The number of minimal vertex covers of a graph. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000926The clique-coclique number of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000362The size of a minimal vertex cover of a graph. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in 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. St001391The disjunction number of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001812The biclique partition number of a graph. St001949The rigidity index of a graph. St001971The number of negative eigenvalues of the adjacency matrix of the graph. St001875The number of simple modules with projective dimension at most 1. St001624The breadth of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000264The girth of a graph, which is not a tree. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000909The number of maximal chains of maximal size in a poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001577The minimal number of edges to add or remove to make a graph a cograph. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000100The number of linear extensions of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000307The number of rowmotion orbits of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001271The competition number of a graph. St001399The distinguishing number of a poset. St001779The order of promotion on the set of linear extensions of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset.
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