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Your data matches 34 different statistics following compositions of up to 3 maps.
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Matching statistic: St001231
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001231: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001231: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1,0]
=> [1,0]
=> 0
([],2)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
([(0,2),(1,2)],3)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
([(1,3),(2,3)],4)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 2
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(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]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
Description
The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension.
Actually the same statistics results for algebras with at most 7 simple modules when dropping the assumption that the module has projective dimension one. The author is not sure whether this holds in general.
Matching statistic: St001234
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001234: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001234: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1,0]
=> [1,0]
=> 0
([],2)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
([(0,2),(1,2)],3)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
([(1,3),(2,3)],4)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 2
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(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]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
Description
The number of indecomposable three dimensional modules with projective dimension one.
It return zero when there are no such modules.
Matching statistic: St000931
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1,0]
=> [1,0]
=> ? = 0
([],2)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
([(0,2),(1,2)],3)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
([(1,3),(2,3)],4)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 2
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(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]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,1),(0,2),(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]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3
Description
The number of occurrences of the pattern UUU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St001336
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 0
([],2)
=> ([],2)
=> ([],1)
=> 0
([(0,1)],2)
=> ([],1)
=> ([],1)
=> 0
([],3)
=> ([],3)
=> ([],1)
=> 0
([(1,2)],3)
=> ([],2)
=> ([],1)
=> 0
([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> 0
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([],4)
=> ([],4)
=> ([],1)
=> 0
([(2,3)],4)
=> ([],3)
=> ([],1)
=> 0
([(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> 0
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> ([],1)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([],5)
=> ([],5)
=> ([],1)
=> 0
([(3,4)],5)
=> ([],4)
=> ([],1)
=> 0
([(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 0
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> 0
([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ([],1)
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> ([],1)
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 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)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 0
([(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)
=> ([(0,1),(0,2),(1,2)],3)
=> 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)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 1
([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 1
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ?
=> ? = 1
([(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
=> ([(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 1
([(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,1),(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6)],7)
=> ?
=> ? = 1
([(0,1),(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
([(0,1),(0,5),(1,5),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(1,5),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
([(0,1),(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
([(0,1),(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,4),(0,6),(1,2),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,6),(1,2),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,1),(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
([(0,1),(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 1
([(0,3),(0,6),(1,3),(1,6),(2,4),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,6),(1,3),(1,6),(2,4),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
([(0,4),(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
=> ([(0,4),(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
=> ?
=> ? = 1
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,4),(0,5),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,1),(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)
=> ([(0,1),(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)
=> ?
=> ? = 3
([(0,3),(0,4),(1,2),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(1,2),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,1),(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)
=> ([(0,1),(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)
=> ?
=> ? = 3
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
Description
The minimal number of vertices in a graph whose complement is triangle-free.
Matching statistic: St000090
Mp00154: Graphs —core⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000090: Integer compositions ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000090: Integer compositions ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1] => [1] => 0
([],2)
=> ([],1)
=> [1] => [1] => 0
([(0,1)],2)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 0
([],3)
=> ([],1)
=> [1] => [1] => 0
([(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 0
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 0
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 1
([],4)
=> ([],1)
=> [1] => [1] => 0
([(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 0
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 0
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 0
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 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)
=> [3,1] => [1,3] => 2
([],5)
=> ([],1)
=> [1] => [1] => 0
([(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 0
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 0
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 0
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 0
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 0
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 0
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1,1] => [1,1] => 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,1,2] => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 1
([(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)
=> [3,1] => [1,3] => 2
([(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)
=> [3,1] => [1,3] => 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)
=> [3,1] => [1,3] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 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)
=> [2,1] => [1,2] => 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),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,3] => 2
([(3,6),(4,5)],7)
=> ?
=> ? => ? => ? = 0
([(2,3),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 0
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> ?
=> ? => ? => ? = 0
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ?
=> ? => ? => ? = 0
([(1,6),(2,5),(3,4)],7)
=> ?
=> ? => ? => ? = 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> ?
=> ? => ? => ? = 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 0
([(2,3),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ?
=> ? => ? => ? = 0
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ?
=> ? => ? => ? = 0
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ?
=> ? => ? => ? = 1
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,6),(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ?
=> ? => ? => ? = 1
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,5),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,5),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 0
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,5),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,4),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,5),(1,4),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,5),(1,2),(1,6),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(0,1),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ? => ? = 1
([(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
=> ?
=> ? => ? => ? = 1
Description
The variation of a composition.
Matching statistic: St000771
Values
([],1)
=> ([],1)
=> [1] => ([],1)
=> 1 = 0 + 1
([],2)
=> ([],1)
=> [1] => ([],1)
=> 1 = 0 + 1
([(0,1)],2)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([],3)
=> ([],1)
=> [1] => ([],1)
=> 1 = 0 + 1
([(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([],4)
=> ([],1)
=> [1] => ([],1)
=> 1 = 0 + 1
([(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 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)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([],5)
=> ([],1)
=> [1] => ([],1)
=> 1 = 0 + 1
([(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(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)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(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)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(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)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 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)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 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),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(3,6),(4,5)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(2,3),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,6),(2,5),(3,4)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,4),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,4),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,2),(1,6),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums 0, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
(4−1−2−1−14−1−2−2−14−1−1−2−14).
Its eigenvalues are 0,4,4,6, so the statistic is 2.
The path on four vertices has eigenvalues 0,4.7…,6,9.2… and therefore statistic 1.
Matching statistic: St000772
Values
([],1)
=> ([],1)
=> [1] => ([],1)
=> 1 = 0 + 1
([],2)
=> ([],1)
=> [1] => ([],1)
=> 1 = 0 + 1
([(0,1)],2)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([],3)
=> ([],1)
=> [1] => ([],1)
=> 1 = 0 + 1
([(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([],4)
=> ([],1)
=> [1] => ([],1)
=> 1 = 0 + 1
([(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 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)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([],5)
=> ([],1)
=> [1] => ([],1)
=> 1 = 0 + 1
([(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(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)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(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)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(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)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 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)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 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),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(3,6),(4,5)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(2,3),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,6),(2,5),(3,4)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,4),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,4),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,2),(1,6),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums 0, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
(4−1−2−1−14−1−2−2−14−1−1−2−14).
Its eigenvalues are 0,4,4,6, so the statistic is 1.
The path on four vertices has eigenvalues 0,4.7…,6,9.2… and therefore also statistic 1.
The graphs with statistic n−1, n−2 and n−3 have been characterised, see [1].
Matching statistic: St000773
Values
([],1)
=> ([],1)
=> [1] => ([],1)
=> 1 = 0 + 1
([],2)
=> ([],1)
=> [1] => ([],1)
=> 1 = 0 + 1
([(0,1)],2)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([],3)
=> ([],1)
=> [1] => ([],1)
=> 1 = 0 + 1
([(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([],4)
=> ([],1)
=> [1] => ([],1)
=> 1 = 0 + 1
([(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 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)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([],5)
=> ([],1)
=> [1] => ([],1)
=> 1 = 0 + 1
([(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(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)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(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)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(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)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 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)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 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),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(3,6),(4,5)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(2,3),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,6),(2,5),(3,4)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,4),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,4),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,2),(1,6),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
Description
The multiplicity of the largest Laplacian eigenvalue in a graph.
Matching statistic: St000774
Values
([],1)
=> ([],1)
=> [1] => ([],1)
=> 1 = 0 + 1
([],2)
=> ([],1)
=> [1] => ([],1)
=> 1 = 0 + 1
([(0,1)],2)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([],3)
=> ([],1)
=> [1] => ([],1)
=> 1 = 0 + 1
([(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([],4)
=> ([],1)
=> [1] => ([],1)
=> 1 = 0 + 1
([(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 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)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([],5)
=> ([],1)
=> [1] => ([],1)
=> 1 = 0 + 1
([(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(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)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(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)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(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)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 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)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 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),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(3,6),(4,5)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(2,3),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,6),(2,5),(3,4)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,4),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,4),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,2),(1,6),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
Description
The maximal multiplicity of a Laplacian eigenvalue in a graph.
Matching statistic: St000776
Values
([],1)
=> ([],1)
=> [1] => ([],1)
=> 1 = 0 + 1
([],2)
=> ([],1)
=> [1] => ([],1)
=> 1 = 0 + 1
([(0,1)],2)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([],3)
=> ([],1)
=> [1] => ([],1)
=> 1 = 0 + 1
([(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([],4)
=> ([],1)
=> [1] => ([],1)
=> 1 = 0 + 1
([(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 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)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([],5)
=> ([],1)
=> [1] => ([],1)
=> 1 = 0 + 1
([(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(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)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(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)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(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)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 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)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 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),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(3,6),(4,5)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(2,3),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,6),(2,5),(3,4)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 0 + 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,4),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,4),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,5),(1,2),(1,6),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(0,1),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
([(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
=> ?
=> ? => ?
=> ? = 1 + 1
Description
The maximal multiplicity of an eigenvalue in a graph.
The following 24 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001917The order of toric promotion on the set of labellings of a graph. St001654The monophonic hull number of a graph. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000770The major index of an integer partition when read from bottom to top. St000993The multiplicity of the largest part of an integer partition. St001060The distinguishing index of a graph. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001845The number of join irreducibles minus the rank of a lattice. St001624The breadth of a lattice. St001651The Frankl number of a lattice. St000741The Colin de Verdière graph invariant. St001570The minimal number of edges to add to make a graph Hamiltonian. St000455The second largest eigenvalue of a graph if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St000422The energy of a graph, if it is integral. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001396Number of triples of incomparable elements in a finite poset. St001964The interval resolution global dimension of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St000544The cop number of a graph.
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