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Your data matches 23 different statistics following compositions of up to 3 maps.
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Matching statistic: St001176
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1]
=> 0
([],2)
=> [2] => [2]
=> 0
([(0,1)],2)
=> [1,1] => [1,1]
=> 1
([],3)
=> [3] => [3]
=> 0
([(1,2)],3)
=> [2,1] => [2,1]
=> 1
([(0,2),(1,2)],3)
=> [2,1] => [2,1]
=> 1
([],4)
=> [4] => [4]
=> 0
([(2,3)],4)
=> [3,1] => [3,1]
=> 1
([(1,3),(2,3)],4)
=> [3,1] => [3,1]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [3,1]
=> 1
([(0,3),(1,2)],4)
=> [2,2] => [2,2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [2,2]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [2,2]
=> 2
([],5)
=> [5] => [5]
=> 0
([(3,4)],5)
=> [4,1] => [4,1]
=> 1
([(2,4),(3,4)],5)
=> [4,1] => [4,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [4,1]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [4,1]
=> 1
([(1,4),(2,3)],5)
=> [3,2] => [3,2]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [3,2]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [3,2]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [3,2]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [3,2]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [3,2]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [3,2]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [3,2]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,2,1]
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [2,2,1]
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 3
([],6)
=> [6] => [6]
=> 0
([(4,5)],6)
=> [5,1] => [5,1]
=> 1
([(3,5),(4,5)],6)
=> [5,1] => [5,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [5,1] => [5,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => [5,1]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => [5,1]
=> 1
([(2,5),(3,4)],6)
=> [4,2] => [4,2]
=> 2
([(2,5),(3,4),(4,5)],6)
=> [4,2] => [4,2]
=> 2
([(1,2),(3,5),(4,5)],6)
=> [4,2] => [4,2]
=> 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => [4,2]
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2] => [4,2]
=> 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => [4,2]
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => [4,2]
=> 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> [4,2] => [4,2]
=> 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,2] => [4,2]
=> 2
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St001232
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1,0]
=> 0
([],2)
=> [2] => [1,1,0,0]
=> 0
([(0,1)],2)
=> [1,1] => [1,0,1,0]
=> 1
([],3)
=> [3] => [1,1,1,0,0,0]
=> 0
([(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> 1
([(0,2),(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> 1
([],4)
=> [4] => [1,1,1,1,0,0,0,0]
=> 0
([(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
([(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
([(0,3),(1,2)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([],5)
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
([(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
([(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
([(1,4),(2,3)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([],6)
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(4,5)],6)
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(3,5),(4,5)],6)
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(2,5),(3,4)],6)
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
([(2,5),(3,4),(4,5)],6)
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
([(1,2),(3,5),(4,5)],6)
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000171
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1] => ([],1)
=> 0
([],2)
=> [2] => [2] => ([],2)
=> 0
([(0,1)],2)
=> [1,1] => [1,1] => ([(0,1)],2)
=> 1
([],3)
=> [3] => [3] => ([],3)
=> 0
([(1,2)],3)
=> [2,1] => [1,2] => ([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> [2,1] => [1,2] => ([(1,2)],3)
=> 1
([],4)
=> [4] => [4] => ([],4)
=> 0
([(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([],5)
=> [5] => [5] => ([],5)
=> 0
([(3,4)],5)
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
([(1,4),(2,3)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([],6)
=> [6] => [6] => ([],6)
=> 0
([(4,5)],6)
=> [5,1] => [1,5] => ([(4,5)],6)
=> 1
([(3,5),(4,5)],6)
=> [5,1] => [1,5] => ([(4,5)],6)
=> 1
([(2,5),(3,5),(4,5)],6)
=> [5,1] => [1,5] => ([(4,5)],6)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => [1,5] => ([(4,5)],6)
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => [1,5] => ([(4,5)],6)
=> 1
([(2,5),(3,4)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(2,5),(3,4),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(1,2),(3,5),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
Description
The degree of the graph.
This is the maximal vertex degree of a graph.
Matching statistic: St000228
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1]
=> []
=> 0
([],2)
=> [2] => [2]
=> []
=> 0
([(0,1)],2)
=> [1,1] => [1,1]
=> [1]
=> 1
([],3)
=> [3] => [3]
=> []
=> 0
([(1,2)],3)
=> [2,1] => [2,1]
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [2,1] => [2,1]
=> [1]
=> 1
([],4)
=> [4] => [4]
=> []
=> 0
([(2,3)],4)
=> [3,1] => [3,1]
=> [1]
=> 1
([(1,3),(2,3)],4)
=> [3,1] => [3,1]
=> [1]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [3,1]
=> [1]
=> 1
([(0,3),(1,2)],4)
=> [2,2] => [2,2]
=> [2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [2,2]
=> [2]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [2,2]
=> [2]
=> 2
([],5)
=> [5] => [5]
=> []
=> 0
([(3,4)],5)
=> [4,1] => [4,1]
=> [1]
=> 1
([(2,4),(3,4)],5)
=> [4,1] => [4,1]
=> [1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [4,1]
=> [1]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [4,1]
=> [1]
=> 1
([(1,4),(2,3)],5)
=> [3,2] => [3,2]
=> [2]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [3,2]
=> [2]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [3,2]
=> [2]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [3,2]
=> [2]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [3,2]
=> [2]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [3,2]
=> [2]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [3,2]
=> [2]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [3,2]
=> [2]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [2,1]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [2,1]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [2,1]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,2,1]
=> [2,1]
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [2,1]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [2,1]
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [2,1]
=> 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [2,2,1]
=> [2,1]
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [2,1]
=> 3
([],6)
=> [6] => [6]
=> []
=> 0
([(4,5)],6)
=> [5,1] => [5,1]
=> [1]
=> 1
([(3,5),(4,5)],6)
=> [5,1] => [5,1]
=> [1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [5,1] => [5,1]
=> [1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => [5,1]
=> [1]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => [5,1]
=> [1]
=> 1
([(2,5),(3,4)],6)
=> [4,2] => [4,2]
=> [2]
=> 2
([(2,5),(3,4),(4,5)],6)
=> [4,2] => [4,2]
=> [2]
=> 2
([(1,2),(3,5),(4,5)],6)
=> [4,2] => [4,2]
=> [2]
=> 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => [4,2]
=> [2]
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2] => [4,2]
=> [2]
=> 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => [4,2]
=> [2]
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => [4,2]
=> [2]
=> 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> [4,2] => [4,2]
=> [2]
=> 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,2] => [4,2]
=> [2]
=> 2
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000987
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1] => ([],1)
=> 0
([],2)
=> [2] => [2] => ([],2)
=> 0
([(0,1)],2)
=> [1,1] => [1,1] => ([(0,1)],2)
=> 1
([],3)
=> [3] => [3] => ([],3)
=> 0
([(1,2)],3)
=> [2,1] => [1,2] => ([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> [2,1] => [1,2] => ([(1,2)],3)
=> 1
([],4)
=> [4] => [4] => ([],4)
=> 0
([(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([],5)
=> [5] => [5] => ([],5)
=> 0
([(3,4)],5)
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
([(1,4),(2,3)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([],6)
=> [6] => [6] => ([],6)
=> 0
([(4,5)],6)
=> [5,1] => [1,5] => ([(4,5)],6)
=> 1
([(3,5),(4,5)],6)
=> [5,1] => [1,5] => ([(4,5)],6)
=> 1
([(2,5),(3,5),(4,5)],6)
=> [5,1] => [1,5] => ([(4,5)],6)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => [1,5] => ([(4,5)],6)
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => [1,5] => ([(4,5)],6)
=> 1
([(2,5),(3,4)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(2,5),(3,4),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(1,2),(3,5),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
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.
Matching statistic: St001721
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St001721: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St001721: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1] => 1 => 0
([],2)
=> [2] => [1,1] => 11 => 0
([(0,1)],2)
=> [1,1] => [2] => 10 => 1
([],3)
=> [3] => [1,1,1] => 111 => 0
([(1,2)],3)
=> [2,1] => [1,2] => 110 => 1
([(0,2),(1,2)],3)
=> [2,1] => [1,2] => 110 => 1
([],4)
=> [4] => [1,1,1,1] => 1111 => 0
([(2,3)],4)
=> [3,1] => [1,1,2] => 1110 => 1
([(1,3),(2,3)],4)
=> [3,1] => [1,1,2] => 1110 => 1
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [1,1,2] => 1110 => 1
([(0,3),(1,2)],4)
=> [2,2] => [1,2,1] => 1101 => 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [1,2,1] => 1101 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [1,2,1] => 1101 => 2
([],5)
=> [5] => [1,1,1,1,1] => 11111 => 0
([(3,4)],5)
=> [4,1] => [1,1,1,2] => 11110 => 1
([(2,4),(3,4)],5)
=> [4,1] => [1,1,1,2] => 11110 => 1
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,2] => 11110 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,2] => 11110 => 1
([(1,4),(2,3)],5)
=> [3,2] => [1,1,2,1] => 11101 => 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,2,1] => 11101 => 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [1,1,2,1] => 11101 => 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,2,1] => 11101 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,2,1] => 11101 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,2,1] => 11101 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,2,1] => 11101 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [1,1,2,1] => 11101 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 11010 => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 11010 => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 11010 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => 11010 => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 11010 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 11010 => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 11010 => 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [1,2,2] => 11010 => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 11010 => 3
([],6)
=> [6] => [1,1,1,1,1,1] => 111111 => 0
([(4,5)],6)
=> [5,1] => [1,1,1,1,2] => 111110 => 1
([(3,5),(4,5)],6)
=> [5,1] => [1,1,1,1,2] => 111110 => 1
([(2,5),(3,5),(4,5)],6)
=> [5,1] => [1,1,1,1,2] => 111110 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => [1,1,1,1,2] => 111110 => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => [1,1,1,1,2] => 111110 => 1
([(2,5),(3,4)],6)
=> [4,2] => [1,1,1,2,1] => 111101 => 2
([(2,5),(3,4),(4,5)],6)
=> [4,2] => [1,1,1,2,1] => 111101 => 2
([(1,2),(3,5),(4,5)],6)
=> [4,2] => [1,1,1,2,1] => 111101 => 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => [1,1,1,2,1] => 111101 => 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2] => [1,1,1,2,1] => 111101 => 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => [1,1,1,2,1] => 111101 => 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => [1,1,1,2,1] => 111101 => 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> [4,2] => [1,1,1,2,1] => 111101 => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,2] => [1,1,1,2,1] => 111101 => 2
Description
The degree of a binary word.
A valley in a binary word is a letter $0$ which is not immediately followed by a $1$. A peak is a letter $1$ which is not immediately followed by a $0$.
Let $f$ be the map that replaces every valley with a peak. The degree of a binary word $w$ is the number of times $f$ has to be applied to obtain a binary word without zeros.
Matching statistic: St000738
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1]
=> [[1]]
=> 1 = 0 + 1
([],2)
=> [2] => [2]
=> [[1,2]]
=> 1 = 0 + 1
([(0,1)],2)
=> [1,1] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
([],3)
=> [3] => [3]
=> [[1,2,3]]
=> 1 = 0 + 1
([(1,2)],3)
=> [2,1] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> [2,1] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
([],4)
=> [4] => [4]
=> [[1,2,3,4]]
=> 1 = 0 + 1
([(2,3)],4)
=> [3,1] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> [3,1] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> [2,2] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
([],5)
=> [5] => [5]
=> [[1,2,3,4,5]]
=> 1 = 0 + 1
([(3,4)],5)
=> [4,1] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> [4,1] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
([(1,4),(2,3)],5)
=> [3,2] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
([],6)
=> [6] => [6]
=> [[1,2,3,4,5,6]]
=> 1 = 0 + 1
([(4,5)],6)
=> [5,1] => [5,1]
=> [[1,3,4,5,6],[2]]
=> 2 = 1 + 1
([(3,5),(4,5)],6)
=> [5,1] => [5,1]
=> [[1,3,4,5,6],[2]]
=> 2 = 1 + 1
([(2,5),(3,5),(4,5)],6)
=> [5,1] => [5,1]
=> [[1,3,4,5,6],[2]]
=> 2 = 1 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => [5,1]
=> [[1,3,4,5,6],[2]]
=> 2 = 1 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => [5,1]
=> [[1,3,4,5,6],[2]]
=> 2 = 1 + 1
([(2,5),(3,4)],6)
=> [4,2] => [4,2]
=> [[1,2,5,6],[3,4]]
=> 3 = 2 + 1
([(2,5),(3,4),(4,5)],6)
=> [4,2] => [4,2]
=> [[1,2,5,6],[3,4]]
=> 3 = 2 + 1
([(1,2),(3,5),(4,5)],6)
=> [4,2] => [4,2]
=> [[1,2,5,6],[3,4]]
=> 3 = 2 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => [4,2]
=> [[1,2,5,6],[3,4]]
=> 3 = 2 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2] => [4,2]
=> [[1,2,5,6],[3,4]]
=> 3 = 2 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => [4,2]
=> [[1,2,5,6],[3,4]]
=> 3 = 2 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => [4,2]
=> [[1,2,5,6],[3,4]]
=> 3 = 2 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [4,2] => [4,2]
=> [[1,2,5,6],[3,4]]
=> 3 = 2 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,2] => [4,2]
=> [[1,2,5,6],[3,4]]
=> 3 = 2 + 1
Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see [[St000734]].
Matching statistic: St001725
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001725: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001725: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1] => ([],1)
=> 1 = 0 + 1
([],2)
=> [2] => [2] => ([],2)
=> 1 = 0 + 1
([(0,1)],2)
=> [1,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
([],3)
=> [3] => [3] => ([],3)
=> 1 = 0 + 1
([(1,2)],3)
=> [2,1] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> [2,1] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
([],4)
=> [4] => [4] => ([],4)
=> 1 = 0 + 1
([(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
([],5)
=> [5] => [5] => ([],5)
=> 1 = 0 + 1
([(3,4)],5)
=> [4,1] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> [4,1] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
([(1,4),(2,3)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([],6)
=> [6] => [6] => ([],6)
=> 1 = 0 + 1
([(4,5)],6)
=> [5,1] => [1,5] => ([(4,5)],6)
=> 2 = 1 + 1
([(3,5),(4,5)],6)
=> [5,1] => [1,5] => ([(4,5)],6)
=> 2 = 1 + 1
([(2,5),(3,5),(4,5)],6)
=> [5,1] => [1,5] => ([(4,5)],6)
=> 2 = 1 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => [1,5] => ([(4,5)],6)
=> 2 = 1 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => [1,5] => ([(4,5)],6)
=> 2 = 1 + 1
([(2,5),(3,4)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
([(2,5),(3,4),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
([(1,2),(3,5),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
Description
The harmonious chromatic number of a graph.
A harmonious colouring is a proper vertex colouring such that any pair of colours appears at most once on adjacent vertices.
Matching statistic: St000476
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1] => [1,0]
=> ? = 0
([],2)
=> [2] => [2] => [1,1,0,0]
=> 0
([(0,1)],2)
=> [1,1] => [1,1] => [1,0,1,0]
=> 1
([],3)
=> [3] => [3] => [1,1,1,0,0,0]
=> 0
([(1,2)],3)
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
([(0,2),(1,2)],3)
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
([],4)
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
([(2,3)],4)
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
([(1,3),(2,3)],4)
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([],5)
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
([(3,4)],5)
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
([(2,4),(3,4)],5)
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
([(1,4),(2,3)],5)
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
([],6)
=> [6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(4,5)],6)
=> [5,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(3,5),(4,5)],6)
=> [5,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [5,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(2,5),(3,4)],6)
=> [4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
([(2,5),(3,4),(4,5)],6)
=> [4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
([(1,2),(3,5),(4,5)],6)
=> [4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> [4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path.
For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which
is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is
$$
\sum_v (j_v-i_v)/2.
$$
Matching statistic: St001118
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001118: Graphs ⟶ ℤResult quality: 83% ●values known / values provided: 99%●distinct values known / distinct values provided: 83%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001118: Graphs ⟶ ℤResult quality: 83% ●values known / values provided: 99%●distinct values known / distinct values provided: 83%
Values
([],1)
=> [1] => [1] => ([],1)
=> ? = 0
([],2)
=> [2] => [2] => ([],2)
=> ? = 0
([(0,1)],2)
=> [1,1] => [1,1] => ([(0,1)],2)
=> 1
([],3)
=> [3] => [3] => ([],3)
=> ? = 0
([(1,2)],3)
=> [2,1] => [1,2] => ([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> [2,1] => [1,2] => ([(1,2)],3)
=> 1
([],4)
=> [4] => [4] => ([],4)
=> ? = 0
([(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
([],5)
=> [5] => [5] => ([],5)
=> ? = 0
([(3,4)],5)
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
([(1,4),(2,3)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([],6)
=> [6] => [6] => ([],6)
=> ? = 0
([(4,5)],6)
=> [5,1] => [1,5] => ([(4,5)],6)
=> 1
([(3,5),(4,5)],6)
=> [5,1] => [1,5] => ([(4,5)],6)
=> 1
([(2,5),(3,5),(4,5)],6)
=> [5,1] => [1,5] => ([(4,5)],6)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => [1,5] => ([(4,5)],6)
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => [1,5] => ([(4,5)],6)
=> 1
([(2,5),(3,4)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(2,5),(3,4),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(1,2),(3,5),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 2
([],7)
=> [7] => [7] => ([],7)
=> ? = 0
Description
The acyclic chromatic index of a graph.
An acyclic edge coloring of a graph is a proper colouring of the edges of a graph such that the union of the edges colored with any two given colours is a forest.
The smallest number of colours such that such a colouring exists is the acyclic chromatic index.
The following 13 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000362The size of a minimal vertex cover of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000264The girth of a graph, which is not a tree. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000331The number of upper interactions of a Dyck path. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001117The game chromatic index of a graph. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000455The second largest eigenvalue of a graph if it is integral.
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