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Your data matches 21 different statistics following compositions of up to 3 maps.
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Matching statistic: St000722
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
([],1)
=> 1
([],2)
=> 1
([(0,1)],2)
=> 2
([],3)
=> 1
([(1,2)],3)
=> 3
([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,2)],3)
=> 3
([],4)
=> 1
([(2,3)],4)
=> 3
([(1,3),(2,3)],4)
=> 3
([(0,3),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2)],4)
=> 4
([(0,3),(1,2),(2,3)],4)
=> 4
([(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],5)
=> 1
([(3,4)],5)
=> 3
([(2,4),(3,4)],5)
=> 3
([(1,4),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3)],5)
=> 5
([(1,4),(2,3),(3,4)],5)
=> 5
([(0,1),(2,4),(3,4)],5)
=> 4
([(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> 5
([(0,1),(2,3),(2,4),(3,4)],5)
=> 5
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
Description
The number of different neighbourhoods in a graph.
Matching statistic: St000228
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(load all 6 compositions to match this statistic)
Mp00243: Graphs —weak duplicate order⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> 1
([],2)
=> ([],1)
=> [1]
=> 1
([(0,1)],2)
=> ([],2)
=> [1,1]
=> 2
([],3)
=> ([],1)
=> [1]
=> 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 3
([(0,2),(1,2)],3)
=> ([],2)
=> [1,1]
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> [1,1,1]
=> 3
([],4)
=> ([],1)
=> [1]
=> 1
([(2,3)],4)
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 3
([(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([],2)
=> [1,1]
=> 2
([(0,3),(1,2)],4)
=> ([],4)
=> [1,1,1,1]
=> 4
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 4
([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> 4
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,1,1]
=> 4
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],2)
=> [1,1]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],3)
=> [1,1,1]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> [1,1,1,1]
=> 4
([],5)
=> ([],1)
=> [1]
=> 1
([(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 3
([(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> 2
([(1,4),(2,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> 5
([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 5
([(0,1),(2,4),(3,4)],5)
=> ([],4)
=> [1,1,1,1]
=> 4
([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> 4
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> [2,1,1]
=> 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> [2,1,1]
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> [1,1]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],3)
=> [1,1,1]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(2,3),(2,4)],5)
=> [2,1,1,1]
=> 5
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> [1,1,1,1,1]
=> 5
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([],5)
=> [1,1,1,1,1]
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],5)
=> [1,1,1,1,1]
=> 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> 5
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,1,1,1]
=> 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(2,3)],4)
=> [2,1,1]
=> 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],3)
=> [1,1,1]
=> 3
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000293
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> 10 => 1
([],2)
=> ([],1)
=> [1]
=> 10 => 1
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 100 => 2
([],3)
=> ([],1)
=> [1]
=> 10 => 1
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 1010 => 3
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [2]
=> 100 => 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1000 => 3
([],4)
=> ([],1)
=> [1]
=> 10 => 1
([(2,3)],4)
=> ([(1,2)],3)
=> [2,1]
=> 1010 => 3
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> [2,1]
=> 1010 => 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> 100 => 2
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 1100 => 4
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 10000 => 4
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 4
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 10000 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [2]
=> 100 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1000 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 10000 => 4
([],5)
=> ([],1)
=> [1]
=> 10 => 1
([(3,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> 1010 => 3
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> 1010 => 3
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> 1010 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> 100 => 2
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> 11010 => 5
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 100010 => 5
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 1100 => 4
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 4
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 10000 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 10000 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> 1010 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 10000 => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 10000 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [2]
=> 100 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1000 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 100000 => 5
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> 10100 => 5
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 100000 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 100000 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 100000 => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 5
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 10000 => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1000 => 3
Description
The number of inversions of a binary word.
Matching statistic: St001034
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> [1,0]
=> 1
([],2)
=> ([],1)
=> [1]
=> [1,0]
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
([],3)
=> ([],1)
=> [1]
=> [1,0]
=> 1
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([],4)
=> ([],1)
=> [1]
=> [1,0]
=> 1
([(2,3)],4)
=> ([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> 4
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
([],5)
=> ([],1)
=> [1]
=> [1,0]
=> 1
([(3,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 5
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> 4
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
Description
The area of the parallelogram polyomino associated with the Dyck path.
The (bivariate) generating function is given in [1].
Matching statistic: St001746
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1)],2)
=> ([],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
([],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 3
([(0,2),(1,2)],3)
=> ([],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 3
([(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
([(0,3),(1,2)],4)
=> ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 3
([(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 5
([(0,1),(2,4),(3,4)],5)
=> ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
Description
The coalition number of a graph.
This is the maximal cardinality of a set partition such that each block is either a dominating set of cardinality one, or is not a dominating set but can be joined with a second block to form a dominating set.
Matching statistic: St001645
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1)],2)
=> ([],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
([],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 3
([(0,2),(1,2)],3)
=> ([],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 3
([(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 3
([(0,3),(1,3),(2,3)],4)
=> ([],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
([(0,3),(1,2)],4)
=> ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 3
([(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 5
([(0,1),(2,4),(3,4)],5)
=> ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 3
([(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 3
([(2,5),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 3
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
([(2,5),(3,4)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 5
([(1,2),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
([(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 5
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 5
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 5
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 5
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 5
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 5
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 5
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,5),(1,4),(2,3)],6)
=> ([],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],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),(4,5)],6)
=> 6
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 6
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(2,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],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),(4,5)],6)
=> 6
Description
The pebbling number of a connected graph.
Matching statistic: St000189
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([],2)
=> 2
([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 3
([(0,2),(1,2)],3)
=> ([],2)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 3
([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 3
([(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([],2)
=> 2
([(0,3),(1,2)],4)
=> ([],4)
=> 4
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2)],4)
=> 4
([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 4
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 4
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],2)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],3)
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 4
([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 3
([(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],2)
=> 2
([(1,4),(2,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5
([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 5
([(0,1),(2,4),(3,4)],5)
=> ([],4)
=> 4
([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 4
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(1,2)],3)
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],3)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(2,3),(2,4)],5)
=> 5
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 5
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([],5)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],5)
=> 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3)],5)
=> 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(2,3)],4)
=> 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],3)
=> 3
([(1,6),(2,5),(3,4)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 7
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 7
([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> ? = 7
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ? = 7
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(6,5)],7)
=> ? = 7
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ? = 7
([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ? = 7
([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ? = 7
([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> ? = 7
([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> ? = 7
([(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(1,6),(2,3),(3,6),(4,5),(6,5)],7)
=> ? = 7
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> ? = 7
([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 7
([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(3,6)],7)
=> ? = 7
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? = 7
([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 7
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 7
([(0,5),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(2,6),(3,6),(4,5),(4,6)],7)
=> ? = 7
([(0,6),(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5)],7)
=> ? = 7
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ? = 7
([(0,6),(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 7
([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 7
([(0,6),(1,4),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ([(2,6),(3,4),(4,5),(4,6)],7)
=> ? = 7
([(0,6),(1,5),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,6),(1,4),(1,5),(2,3),(3,5),(3,6)],7)
=> ? = 7
([(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> ? = 7
([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(2,3),(3,5),(4,6)],7)
=> ? = 7
([(0,6),(1,5),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(4,6)],7)
=> ? = 7
([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> ? = 7
([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
=> ? = 7
([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
=> ? = 7
([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,6),(3,6),(6,4),(6,5)],7)
=> ? = 7
([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 7
([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 7
([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 7
([(0,6),(1,2),(1,3),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(4,6),(5,6)],7)
=> ? = 7
([(1,6),(2,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 7
([(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 7
([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,5),(4,5)],7)
=> ? = 7
([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 7
([(1,6),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 7
([(0,4),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,6),(2,6),(3,4),(3,5)],7)
=> ? = 7
([(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,2),(0,3),(1,4),(1,5),(4,6),(5,6)],7)
=> ? = 7
([(1,2),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> ? = 7
([(0,6),(1,2),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(5,6)],7)
=> ? = 7
([(0,6),(1,4),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,3),(1,5),(2,4),(2,5),(4,6)],7)
=> ? = 7
([(0,6),(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 7
([(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 7
([(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? = 7
([(0,6),(1,4),(2,3),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> ? = 7
([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5)],7)
=> ? = 7
Description
The number of elements in the poset.
Matching statistic: St001342
Values
([],1)
=> ([],1)
=> ([],2)
=> 2 = 1 + 1
([],2)
=> ([],1)
=> ([],2)
=> 2 = 1 + 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([(1,2)],3)
=> 3 = 2 + 1
([],3)
=> ([],1)
=> ([],2)
=> 2 = 1 + 1
([(1,2)],3)
=> ([(1,2)],3)
=> ([(2,3)],4)
=> 4 = 3 + 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(1,2)],3)
=> 3 = 2 + 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
([],4)
=> ([],1)
=> ([],2)
=> 2 = 1 + 1
([(2,3)],4)
=> ([(1,2)],3)
=> ([(2,3)],4)
=> 4 = 3 + 1
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(2,3)],4)
=> 4 = 3 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(1,2)],3)
=> 3 = 2 + 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(1,4),(2,3)],5)
=> 5 = 4 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 5 = 4 + 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(1,2)],3)
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
([],5)
=> ([],1)
=> ([],2)
=> 2 = 1 + 1
([(3,4)],5)
=> ([(1,2)],3)
=> ([(2,3)],4)
=> 4 = 3 + 1
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(2,3)],4)
=> 4 = 3 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(2,3)],4)
=> 4 = 3 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(1,2)],3)
=> 3 = 2 + 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(2,5),(3,4)],6)
=> 6 = 5 + 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(2,5),(3,4),(4,5)],6)
=> 6 = 5 + 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(1,4),(2,3)],5)
=> 5 = 4 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 5 = 4 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ([(2,3)],4)
=> 4 = 3 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 5 = 4 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(1,2)],3)
=> 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 6 = 5 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 6 = 5 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 6 = 5 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 6 = 5 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 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)
=> ([(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)
=> ([(1,2),(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)
=> ? = 6 + 1
([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> ([(2,7),(3,6),(4,5)],8)
=> ? = 7 + 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(2,3),(4,7),(5,6),(6,7)],8)
=> ? = 7 + 1
([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 7 + 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 + 1
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 7 + 1
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,7),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 7 + 1
([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 7 + 1
([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(2,7),(3,4),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 7 + 1
([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> ([(2,7),(3,6),(4,5),(5,6),(5,7),(6,7)],8)
=> ? = 7 + 1
([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,7),(3,4),(3,7),(4,6),(5,6),(5,7),(6,7)],8)
=> ? = 7 + 1
([(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 + 1
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 + 1
([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7)],8)
=> ? = 7 + 1
([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 7 + 1
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(2,6),(3,4),(3,7),(4,7),(5,6),(5,7)],8)
=> ? = 7 + 1
([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,7),(3,5),(4,6),(4,7),(5,6),(6,7)],8)
=> ? = 7 + 1
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(3,7),(4,6),(5,6),(5,7),(6,7)],8)
=> ? = 7 + 1
([(0,5),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(1,6),(2,7),(3,4),(3,7),(4,7),(5,6),(5,7)],8)
=> ? = 7 + 1
([(0,6),(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,4),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 + 1
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 + 1
([(0,6),(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,4),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 + 1
([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
=> ([(1,7),(2,6),(3,4),(3,6),(4,7),(5,6),(5,7)],8)
=> ? = 7 + 1
([(0,6),(1,4),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ([(1,7),(2,5),(3,5),(3,7),(4,6),(4,7),(5,6),(6,7)],8)
=> ? = 7 + 1
([(0,6),(1,5),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 + 1
([(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,7),(3,6),(3,7),(4,5),(4,6),(5,6),(5,7),(6,7)],8)
=> ? = 7 + 1
([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,4),(3,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 + 1
([(0,6),(1,5),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 + 1
([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 7 + 1
([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,7),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 + 1
([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 + 1
([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,7),(2,6),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 7 + 1
([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 + 1
([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ([(2,6),(2,7),(3,4),(3,5),(4,7),(5,6)],8)
=> ? = 7 + 1
([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(2,5),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 7 + 1
([(0,6),(1,2),(1,3),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,3),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,7),(2,3),(2,4),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 7 + 1
([(1,6),(2,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> ([(2,7),(3,4),(3,6),(4,5),(5,6),(5,7),(6,7)],8)
=> ? = 7 + 1
([(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,7),(4,6),(5,6),(5,7),(6,7)],8)
=> ? = 7 + 1
([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 7 + 1
([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,7),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 7 + 1
([(1,6),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,6),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,7),(3,5),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 7 + 1
([(0,4),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,5),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 7 + 1
([(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 7 + 1
([(1,2),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,7),(3,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 + 1
([(0,6),(1,2),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,3),(3,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 + 1
([(0,6),(1,4),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,5),(3,4),(3,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 + 1
([(0,6),(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,3),(2,4),(3,7),(4,6),(5,6),(5,7),(6,7)],8)
=> ? = 7 + 1
([(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 7 + 1
([(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 7 + 1
([(0,6),(1,4),(2,3),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 + 1
Description
The number of vertices in the center of a graph.
The center of a graph is the set of vertices whose maximal distance to any other vertex is minimal. In particular, if the graph is disconnected, all vertices are in the certer.
Matching statistic: St001622
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([(0,1)],2)
=> 1
([],2)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(0,1)],2)
=> ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([],3)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3
([(0,2),(1,2)],3)
=> ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([(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)
=> 3
([],4)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3
([(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 4
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 4
([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 4
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> 4
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],3)
=> ([(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)
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 4
([],5)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3
([(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,4),(2,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> 5
([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> 5
([(0,1),(2,4),(3,4)],5)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 4
([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 4
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,11),(2,10),(3,8),(3,9),(4,7),(4,8),(5,7),(5,9),(7,12),(8,2),(8,12),(9,1),(9,12),(10,6),(11,6),(12,10),(12,11)],13)
=> 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([(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)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(1,11),(1,13),(2,11),(2,12),(3,4),(3,5),(3,12),(3,13),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,15),(6,17),(7,15),(7,18),(8,16),(8,17),(9,16),(9,18),(10,15),(10,16),(11,14),(12,6),(12,7),(12,14),(13,8),(13,9),(13,14),(14,17),(14,18),(15,19),(16,19),(17,19),(18,19)],20)
=> 5
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ?
=> ? = 5
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ?
=> ? = 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([],5)
=> ?
=> ? = 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],5)
=> ?
=> ? = 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
=> 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,10),(2,7),(2,8),(3,9),(3,12),(4,9),(4,11),(5,2),(5,11),(5,12),(7,14),(8,14),(9,1),(9,13),(10,6),(11,7),(11,13),(12,8),(12,13),(13,10),(13,14),(14,6)],15)
=> 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> 5
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(2,8),(2,9),(3,9),(3,11),(3,12),(4,8),(4,10),(4,12),(5,7),(5,10),(5,11),(7,13),(7,14),(8,13),(8,15),(9,14),(9,15),(10,13),(10,16),(11,14),(11,16),(12,15),(12,16),(13,17),(14,17),(15,17),(16,1),(16,17),(17,6)],18)
=> 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
=> 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],3)
=> ([(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)
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 4
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ?
=> ? = 5
([],6)
=> ([],1)
=> ([(0,1)],2)
=> 1
([(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3
([(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3
([(0,5),(1,4),(2,3)],6)
=> ([],6)
=> ?
=> ? = 6
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ?
=> ? = 6
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4)],6)
=> ?
=> ? = 6
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 6
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4)],6)
=> ?
=> ? = 6
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ?
=> ? = 6
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ?
=> ? = 6
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4)],6)
=> ?
=> ? = 6
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 6
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 6
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 6
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ?
=> ? = 6
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(2,4),(4,5)],6)
=> ?
=> ? = 6
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(2,5),(3,4)],6)
=> ?
=> ? = 6
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([],5)
=> ?
=> ? = 5
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(4,5)],6)
=> ?
=> ? = 6
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5)],6)
=> ?
=> ? = 6
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],5)
=> ?
=> ? = 5
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(3,4)],5)
=> ?
=> ? = 5
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(3,4)],5)
=> ?
=> ? = 5
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(3,4)],5)
=> ?
=> ? = 5
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ?
=> ? = 6
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],5)
=> ?
=> ? = 5
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],5)
=> ?
=> ? = 5
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(2,4),(4,5)],6)
=> ?
=> ? = 6
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ?
=> ? = 6
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ?
=> ? = 6
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(4,5)],6)
=> ?
=> ? = 6
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([],6)
=> ?
=> ? = 6
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3)],6)
=> ?
=> ? = 6
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5)],6)
=> ?
=> ? = 6
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ?
=> ? = 6
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ?
=> ? = 6
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(3,4),(4,5)],6)
=> ?
=> ? = 6
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4)],6)
=> ?
=> ? = 6
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5)],6)
=> ?
=> ? = 6
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(4,5)],6)
=> ?
=> ? = 6
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(3,4),(3,5)],6)
=> ?
=> ? = 6
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(3,4)],5)
=> ?
=> ? = 5
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
=> ?
=> ? = 6
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(2,4),(2,5)],6)
=> ?
=> ? = 6
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ?
=> ? = 6
([(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)
=> ?
=> ? = 6
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([],6)
=> ?
=> ? = 6
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5)],6)
=> ?
=> ? = 6
Description
The number of join-irreducible elements of a lattice.
An element $j$ of a lattice $L$ is '''join irreducible''' if it is not the least element and if $j=x\vee y$, then $j\in\{x,y\}$ for all $x,y\in L$.
Matching statistic: St000987
Values
([],1)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],2)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 4
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 5
([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 4
([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 5
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1
([(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
([(3,6),(4,5)],7)
=> ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 5
([(3,6),(4,5),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 5
([(2,3),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,3),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 5
([(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 5
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,2),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 5
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 5
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 5
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 5
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 5
([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 5
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 5
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 6
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 4
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 6
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 5
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 5
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 5
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 5
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 6
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 4
([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 5
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 5
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,7),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 6
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 5
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 5
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 5
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 6
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 5
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 4
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 5
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 5
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 3
Description
The number of positive eigenvalues of the Laplacian matrix of the graph.
This is the number of vertices minus the number of connected components of the graph.
The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000718The largest Laplacian eigenvalue of a graph if it is integral. St001723The differential of a graph. St001724The 2-packing differential of a graph. St000171The degree of the graph. St001268The size of the largest ordinal summand in the poset. St001725The harmonious chromatic number of a graph. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001458The rank of the adjacency matrix of a graph. St001330The hat guessing number of a graph. St001875The number of simple modules with projective dimension at most 1.
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