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Your data matches 17 different statistics following compositions of up to 3 maps.
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Matching statistic: St000905
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
St000905: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 1
[2] => 1
[1,1,1] => 1
[1,2] => 1
[2,1] => 1
[3] => 1
[1,1,1,1] => 1
[1,1,2] => 2
[1,2,1] => 2
[1,3] => 1
[2,1,1] => 2
[2,2] => 1
[3,1] => 1
[4] => 1
[1,1,1,1,1] => 1
[1,1,1,2] => 2
[1,1,2,1] => 2
[1,1,3] => 2
[1,2,1,1] => 2
[1,2,2] => 2
[1,3,1] => 2
[1,4] => 1
[2,1,1,1] => 2
[2,1,2] => 2
[2,2,1] => 2
[2,3] => 1
[3,1,1] => 2
[3,2] => 1
[4,1] => 1
[5] => 1
[1,1,1,1,1,1] => 1
[1,1,1,1,2] => 2
[1,1,1,2,1] => 2
[1,1,1,3] => 2
[1,1,2,1,1] => 2
[1,1,2,2] => 1
[1,1,3,1] => 2
[1,1,4] => 2
[1,2,1,1,1] => 2
[1,2,1,2] => 1
[1,2,2,1] => 1
[1,2,3] => 1
[1,3,1,1] => 2
[1,3,2] => 1
[1,4,1] => 2
[1,5] => 1
[2,1,1,1,1] => 2
[2,1,1,2] => 1
[2,1,2,1] => 1
Description
The number of different multiplicities of parts of an integer composition.
Matching statistic: St000897
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000897: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000897: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 1
[1,1] => [1,1]
=> 1
[2] => [2]
=> 1
[1,1,1] => [1,1,1]
=> 1
[1,2] => [2,1]
=> 1
[2,1] => [2,1]
=> 1
[3] => [3]
=> 1
[1,1,1,1] => [1,1,1,1]
=> 1
[1,1,2] => [2,1,1]
=> 2
[1,2,1] => [2,1,1]
=> 2
[1,3] => [3,1]
=> 1
[2,1,1] => [2,1,1]
=> 2
[2,2] => [2,2]
=> 1
[3,1] => [3,1]
=> 1
[4] => [4]
=> 1
[1,1,1,1,1] => [1,1,1,1,1]
=> 1
[1,1,1,2] => [2,1,1,1]
=> 2
[1,1,2,1] => [2,1,1,1]
=> 2
[1,1,3] => [3,1,1]
=> 2
[1,2,1,1] => [2,1,1,1]
=> 2
[1,2,2] => [2,2,1]
=> 2
[1,3,1] => [3,1,1]
=> 2
[1,4] => [4,1]
=> 1
[2,1,1,1] => [2,1,1,1]
=> 2
[2,1,2] => [2,2,1]
=> 2
[2,2,1] => [2,2,1]
=> 2
[2,3] => [3,2]
=> 1
[3,1,1] => [3,1,1]
=> 2
[3,2] => [3,2]
=> 1
[4,1] => [4,1]
=> 1
[5] => [5]
=> 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 1
[1,1,1,1,2] => [2,1,1,1,1]
=> 2
[1,1,1,2,1] => [2,1,1,1,1]
=> 2
[1,1,1,3] => [3,1,1,1]
=> 2
[1,1,2,1,1] => [2,1,1,1,1]
=> 2
[1,1,2,2] => [2,2,1,1]
=> 1
[1,1,3,1] => [3,1,1,1]
=> 2
[1,1,4] => [4,1,1]
=> 2
[1,2,1,1,1] => [2,1,1,1,1]
=> 2
[1,2,1,2] => [2,2,1,1]
=> 1
[1,2,2,1] => [2,2,1,1]
=> 1
[1,2,3] => [3,2,1]
=> 1
[1,3,1,1] => [3,1,1,1]
=> 2
[1,3,2] => [3,2,1]
=> 1
[1,4,1] => [4,1,1]
=> 2
[1,5] => [5,1]
=> 1
[2,1,1,1,1] => [2,1,1,1,1]
=> 2
[2,1,1,2] => [2,2,1,1]
=> 1
[2,1,2,1] => [2,2,1,1]
=> 1
Description
The number of different multiplicities of parts of an integer partition.
Matching statistic: St000260
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0 = 1 - 1
[1,1] => [2] => [1] => ([],1)
=> 0 = 1 - 1
[2] => [1] => [1] => ([],1)
=> 0 = 1 - 1
[1,1,1] => [3] => [1] => ([],1)
=> 0 = 1 - 1
[1,2] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1} - 1
[2,1] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1} - 1
[3] => [1] => [1] => ([],1)
=> 0 = 1 - 1
[1,1,1,1] => [4] => [1] => ([],1)
=> 0 = 1 - 1
[1,1,2] => [2,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,2,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,2} - 1
[1,3] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,2} - 1
[2,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[2,2] => [2] => [1] => ([],1)
=> 0 = 1 - 1
[3,1] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,2} - 1
[4] => [1] => [1] => ([],1)
=> 0 = 1 - 1
[1,1,1,1,1] => [5] => [1] => ([],1)
=> 0 = 1 - 1
[1,1,1,2] => [3,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,1,2,1] => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,2,2,2} - 1
[1,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,2,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,2,2] => [1,2] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,3,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,2,2,2} - 1
[1,4] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,2,2,2} - 1
[2,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[2,1,2] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,2,2,2} - 1
[2,2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[2,3] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,2,2,2} - 1
[3,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[3,2] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,2,2,2} - 1
[4,1] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,2,2,2} - 1
[5] => [1] => [1] => ([],1)
=> 0 = 1 - 1
[1,1,1,1,1,1] => [6] => [1] => ([],1)
=> 0 = 1 - 1
[1,1,1,1,2] => [4,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,1,1,2,1] => [3,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} - 1
[1,1,1,3] => [3,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,1,2,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,1,2,2] => [2,2] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} - 1
[1,1,3,1] => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} - 1
[1,1,4] => [2,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,2,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,2,1,2] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} - 1
[1,2,2,1] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,2,3] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} - 1
[1,3,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,3,2] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} - 1
[1,4,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} - 1
[1,5] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} - 1
[2,1,1,1,1] => [1,4] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[2,1,1,2] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,1,2,1] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} - 1
[2,1,3] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} - 1
[2,2,1,1] => [2,2] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} - 1
[2,2,2] => [3] => [1] => ([],1)
=> 0 = 1 - 1
[2,3,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} - 1
[2,4] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} - 1
[3,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[3,1,2] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} - 1
[3,2,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} - 1
[3,3] => [2] => [1] => ([],1)
=> 0 = 1 - 1
[4,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[4,2] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} - 1
[5,1] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} - 1
[6] => [1] => [1] => ([],1)
=> 0 = 1 - 1
[1,1,1,1,1,1,1] => [7] => [1] => ([],1)
=> 0 = 1 - 1
[1,1,1,1,1,2] => [5,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,1,1,1,2,1] => [4,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[1,1,1,1,3] => [4,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,1,1,2,2] => [3,2] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,1,1,3,1] => [3,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[1,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,1,2,1,2] => [2,1,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[1,1,2,2,1] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,1,2,3] => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[1,1,3,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,1,3,2] => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[1,1,4,1] => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[1,1,5] => [2,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,2,1,1,1,1] => [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,2,1,1,2] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,2,1,2,1] => [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[1,2,1,3] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[1,2,2,1,1] => [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[1,2,2,2] => [1,3] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3,1] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[1,2,4] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[1,3,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,3,1,2] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[1,3,2,1] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[1,3,3] => [1,2] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,4,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,4,2] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[1,5,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[1,6] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[2,1,1,2,1] => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[2,1,3,1] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[2,1,4] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[2,2,1,2] => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[2,3,2] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St001630
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
=> ([],1)
=> ([],1)
=> ? = 1
[1,1] => [[1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1}
[2] => [[2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1}
[1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1}
[1,2] => [[2,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1}
[2,1] => [[2,2],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1}
[3] => [[3],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1}
[1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,2,2,2}
[1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,2,2,2}
[1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,2,2,2}
[1,3] => [[3,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,2,2,2}
[2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,2,2,2}
[2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,2,2,2}
[3,1] => [[3,3],[2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,2,2,2}
[4] => [[4],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,2,2,2}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,4] => [[4,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[4,1] => [[4,4],[3]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[5] => [[5],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,1,1,3] => [[3,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,1,4] => [[4,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,4,1] => [[4,4,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,5] => [[5,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3] => [[4,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,4] => [[5,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,1,2] => [[4,3,3],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[4,1,1] => [[4,4,4],[3,3]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,2] => [[5,4],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[5,1] => [[5,5],[4]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[6] => [[6],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> ([(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
[1,1,2,3] => [[4,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,3,2] => [[4,3,1,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> ([(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
[1,2,1,3] => [[4,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> ([(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
[1,2,2,2] => [[4,3,2,1],[2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[1,2,3,1] => [[4,4,2,1],[3,1]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,2,4] => [[5,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,3,3] => [[5,3,1],[2]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,4,2] => [[5,4,1],[3]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[2,2,3] => [[5,3,2],[2,1]]
=> ([(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
[2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,2] => [[5,4,2],[3,1]]
=> ([(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
[2,4,1] => [[5,5,2],[4,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[3,2,2] => [[5,4,3],[3,2]]
=> ([(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
[3,3,1] => [[5,5,3],[4,2]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[3,4] => [[6,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001029
Values
[1] => ([],1)
=> ([],0)
=> ([],0)
=> ? = 1
[1,1] => ([(0,1)],2)
=> ([],1)
=> ([],1)
=> 1
[2] => ([],2)
=> ([],0)
=> ([],0)
=> ? = 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2] => ([(1,2)],3)
=> ([],1)
=> ([],1)
=> 1
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 1
[3] => ([],3)
=> ([],0)
=> ([],0)
=> ? = 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 2
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3] => ([(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 2
[2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[4] => ([],4)
=> ([],0)
=> ([],0)
=> ? = 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ? ∊ {1,2,2,2}
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 2
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {1,2,2,2}
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,4] => ([(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,7),(0,8),(1,5),(1,6),(2,3),(2,4),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? ∊ {1,2,2,2}
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 2
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> 2
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1
[5] => ([],5)
=> ([],0)
=> ([],0)
=> ? ∊ {1,2,2,2}
[1,1,1,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 2
[1,1,2,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,8),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,8),(4,6),(5,6),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,9),(1,4),(1,7),(1,8),(2,3),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 2
[1,5] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 1
[2,1,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,7),(0,8),(1,5),(1,6),(2,3),(2,4),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(1,5),(1,6),(2,4),(2,6),(2,8),(3,4),(3,5),(3,7),(4,9),(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 2
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> 2
[2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 1
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> 2
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,5),(4,6),(4,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ([(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7)],9)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 1
[6] => ([],6)
=> ([],0)
=> ([],0)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,1,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,2] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,2,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,3] => ([(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),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,2,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 2
[1,1,2,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,2,1,2] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,2,2,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,2,3] => ([(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),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,3,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,6),(1,8),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,8),(1,7),(2,6),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2,1,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,1,1,2] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,1,2,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,1,3] => ([(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),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,2,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,8),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,8),(4,6),(5,6),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,7),(0,8),(1,4),(1,5),(1,6),(1,9),(2,3),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,9),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,9),(1,4),(1,7),(1,8),(2,3),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,4,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(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)
=> ([(2,5),(3,5),(4,5)],6)
=> 2
[1,5,1] => ([(0,6),(1,6),(2,6),(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)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,6] => ([(5,6)],7)
=> ([],1)
=> ([],1)
=> 1
[2,1,1,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> 2
[2,5] => ([(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([],2)
=> 1
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: St001494
Values
[1] => ([],1)
=> ([],0)
=> ([],0)
=> ? = 1
[1,1] => ([(0,1)],2)
=> ([],1)
=> ([],1)
=> 1
[2] => ([],2)
=> ([],0)
=> ([],0)
=> ? = 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2] => ([(1,2)],3)
=> ([],1)
=> ([],1)
=> 1
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 1
[3] => ([],3)
=> ([],0)
=> ([],0)
=> ? = 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 2
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3] => ([(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 2
[2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[4] => ([],4)
=> ([],0)
=> ([],0)
=> ? = 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ? ∊ {1,2,2,2}
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 2
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {1,2,2,2}
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,4] => ([(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,7),(0,8),(1,5),(1,6),(2,3),(2,4),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? ∊ {1,2,2,2}
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 2
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> 2
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1
[5] => ([],5)
=> ([],0)
=> ([],0)
=> ? ∊ {1,2,2,2}
[1,1,1,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 2
[1,1,2,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,8),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,8),(4,6),(5,6),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,9),(1,4),(1,7),(1,8),(2,3),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 2
[1,5] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 1
[2,1,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,7),(0,8),(1,5),(1,6),(2,3),(2,4),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(1,5),(1,6),(2,4),(2,6),(2,8),(3,4),(3,5),(3,7),(4,9),(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 2
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> 2
[2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 1
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> 2
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,5),(4,6),(4,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ([(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7)],9)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 1
[6] => ([],6)
=> ([],0)
=> ([],0)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,1,1,1,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,2] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,2,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,3] => ([(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),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,2,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 2
[1,1,2,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,2,1,2] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,2,2,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,2,3] => ([(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),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,3,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,6),(1,8),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,8),(1,7),(2,6),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2,1,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,1,1,2] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,1,2,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,1,3] => ([(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),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,2,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,8),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,8),(4,6),(5,6),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,7),(0,8),(1,4),(1,5),(1,6),(1,9),(2,3),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,9),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,9),(1,4),(1,7),(1,8),(2,3),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,4,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(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)
=> ([(2,5),(3,5),(4,5)],6)
=> 2
[1,5,1] => ([(0,6),(1,6),(2,6),(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)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,6] => ([(5,6)],7)
=> ([],1)
=> ([],1)
=> 1
[2,1,1,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> 2
[2,5] => ([(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([],2)
=> 1
Description
The Alon-Tarsi number of a graph.
Let $G$ be a graph with vertices $\{1,\dots,n\}$ and edge set $E$. Let $P_G=\prod_{i < j, (i,j)\in E} x_i-x_j$ be its graph polynomial. Then the Alon-Tarsi number is the smallest number $k$ such that $P_G$ contains a monomial with exponents strictly less than $k$.
Matching statistic: St001878
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
=> ([],1)
=> ([],1)
=> ? = 1
[1,1] => [[1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1}
[2] => [[2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1}
[1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1}
[1,2] => [[2,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1}
[2,1] => [[2,2],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1}
[3] => [[3],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1}
[1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,2,2,2}
[1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,2,2,2}
[1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,2,2,2}
[1,3] => [[3,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,2,2,2}
[2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,2,2,2}
[2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,2,2,2}
[3,1] => [[3,3],[2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,2,2,2}
[4] => [[4],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,2,2,2}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4] => [[4,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,1] => [[4,4],[3]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5] => [[5],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,3] => [[3,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,3,1] => [[3,3,1,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,4] => [[4,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],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)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,4,1] => [[4,4,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,5] => [[5,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3] => [[4,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],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)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,4] => [[5,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,1,2] => [[4,3,3],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[4,1,1] => [[4,4,4],[3,3]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,2] => [[5,4],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[5,1] => [[5,5],[4]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,2,3] => [[4,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,3,2] => [[4,3,1,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,1,3] => [[4,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,4] => [[5,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,4,2] => [[5,4,1],[3]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,4,1] => [[5,5,2],[4,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4] => [[6,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[4,2,1] => [[5,5,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[4,3] => [[6,4],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,2,2] => [[3,2,1,1,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,3,2] => [[4,3,1,1,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,1,1,1,2] => [[3,2,2,2,2,1],[1,1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,1,1,2] => [[4,3,3,3,1],[2,2,2]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,4,1,2] => [[5,4,4,1],[3,3]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[2,1,1,1,2,1] => [[3,3,2,2,2,2],[2,1,1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1,3,1] => [[4,4,2,2,2],[3,1,1,1]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[2,2,1,1,1,1] => [[3,3,3,3,3,2],[2,2,2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,4,1,1] => [[5,5,5,2],[4,4,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,1,1,2,1] => [[4,4,3,3,3],[3,2,2,2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,1,1,3] => [[5,3,3,3],[2,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[4,1,2,1] => [[5,5,4,4],[4,3,3]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001337
Values
[1] => ([],1)
=> ([],0)
=> ? = 1
[1,1] => ([(0,1)],2)
=> ([],1)
=> 1
[2] => ([],2)
=> ([],0)
=> ? = 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2] => ([(1,2)],3)
=> ([],1)
=> 1
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[3] => ([],3)
=> ([],0)
=> ? = 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3] => ([(2,3)],4)
=> ([],1)
=> 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[4] => ([],4)
=> ([],0)
=> ? = 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {1,2,2,2,2}
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,2,2,2}
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4] => ([(3,4)],5)
=> ([],1)
=> 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,2,2,2}
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,2}
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[5] => ([],5)
=> ([],0)
=> ? ∊ {1,2,2,2,2}
[1,1,1,1,1,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[1,1,2,1,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,1,1,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5] => ([(4,5)],6)
=> ([],1)
=> 1
[2,1,1,1,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[6] => ([],6)
=> ([],0)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,1,1,1,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,2] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,2,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,3] => ([(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),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,2,1,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[1,1,2,1,1,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,2,1,2] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,2,2,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,2,3] => ([(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),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,3,1,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,6),(1,8),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,1,1,1,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,1,1,2] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,1,2,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,1,3] => ([(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),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,2,1,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,7),(0,8),(1,4),(1,5),(1,6),(1,9),(2,3),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,1,1,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(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
[1,6] => ([(5,6)],7)
=> ([],1)
=> 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[2,5] => ([(4,6),(5,6)],7)
=> ([(0,1)],2)
=> 1
[3,4] => ([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
Description
The upper domination number of a graph.
This is the maximum cardinality of a minimal dominating set of $G$.
The smallest graph with different upper irredundance number and upper domination number has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [1].
Matching statistic: St001338
Values
[1] => ([],1)
=> ([],0)
=> ? = 1
[1,1] => ([(0,1)],2)
=> ([],1)
=> 1
[2] => ([],2)
=> ([],0)
=> ? = 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2] => ([(1,2)],3)
=> ([],1)
=> 1
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[3] => ([],3)
=> ([],0)
=> ? = 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3] => ([(2,3)],4)
=> ([],1)
=> 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[4] => ([],4)
=> ([],0)
=> ? = 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {1,2,2,2,2}
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,2,2,2}
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4] => ([(3,4)],5)
=> ([],1)
=> 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,2,2,2}
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,2}
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[5] => ([],5)
=> ([],0)
=> ? ∊ {1,2,2,2,2}
[1,1,1,1,1,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[1,1,2,1,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,1,1,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5] => ([(4,5)],6)
=> ([],1)
=> 1
[2,1,1,1,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[6] => ([],6)
=> ([],0)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,1,1,1,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,2] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,2,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,3] => ([(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),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,2,1,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[1,1,2,1,1,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,2,1,2] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,2,2,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,2,3] => ([(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),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,3,1,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,6),(1,8),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,1,1,1,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,1,1,2] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,1,2,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,1,3] => ([(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),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,2,1,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,7),(0,8),(1,4),(1,5),(1,6),(1,9),(2,3),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,1,1,1] => ([(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)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(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
[1,6] => ([(5,6)],7)
=> ([],1)
=> 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[2,5] => ([(4,6),(5,6)],7)
=> ([(0,1)],2)
=> 1
[3,4] => ([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
Description
The upper irredundance number of a graph.
A set $S$ of vertices is irredundant, if there is no vertex in $S$, whose closed neighbourhood is contained in the union of the closed neighbourhoods of the other vertices of $S$.
The upper irredundance number is the largest size of a maximal irredundant set.
The smallest graph with different upper irredundance number and upper domination number [[St001337]] has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [2].
Matching statistic: St001330
Values
[1] => ([],1)
=> ([],0)
=> ([],0)
=> ? = 1
[1,1] => ([(0,1)],2)
=> ([],1)
=> ([],1)
=> 1
[2] => ([],2)
=> ([],0)
=> ([],0)
=> ? = 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2] => ([(1,2)],3)
=> ([],1)
=> ([],1)
=> 1
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 1
[3] => ([],3)
=> ([],0)
=> ([],0)
=> ? = 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 2
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3] => ([(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 2
[2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[4] => ([],4)
=> ([],0)
=> ([],0)
=> ? = 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ? ∊ {1,2,2,2,2}
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 2
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {1,2,2,2,2}
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,4] => ([(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,7),(0,8),(1,5),(1,6),(2,3),(2,4),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? ∊ {1,2,2,2,2}
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 2
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ? ∊ {1,2,2,2,2}
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1
[5] => ([],5)
=> ([],0)
=> ([],0)
=> ? ∊ {1,2,2,2,2}
[1,1,1,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 2
[1,1,2,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,8),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,8),(4,6),(5,6),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,9),(1,4),(1,7),(1,8),(2,3),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 2
[1,5] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 1
[2,1,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,7),(0,8),(1,5),(1,6),(2,3),(2,4),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(1,5),(1,6),(2,4),(2,6),(2,8),(3,4),(3,5),(3,7),(4,9),(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 2
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 1
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,5),(4,6),(4,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ([(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7)],9)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 1
[6] => ([],6)
=> ([],0)
=> ([],0)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,1,1,1,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,2] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,2,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,3] => ([(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),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,2,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 2
[1,1,2,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,2,1,2] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,2,2,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,2,3] => ([(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),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,3,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,6),(1,8),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,8),(1,7),(2,6),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2,1,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,1,1,2] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,1,2,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,1,3] => ([(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),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,2,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,8),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,8),(4,6),(5,6),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,7),(0,8),(1,4),(1,5),(1,6),(1,9),(2,3),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,9),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,9),(1,4),(1,7),(1,8),(2,3),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(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)
=> ([(2,5),(3,5),(4,5)],6)
=> 2
[1,6] => ([(5,6)],7)
=> ([],1)
=> ([],1)
=> 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[2,5] => ([(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([],2)
=> 1
[3,4] => ([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1
[5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 1
[6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ([],6)
=> 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
The following 7 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000822The Hadwiger number of the graph. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St001520The number of strict 3-descents. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St000068The number of minimal elements in a poset.
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