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Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St000259

Mapping

Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ 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)
 => 0
([],3)
 => [3] => [1] => ([],1)
 => 0
([(1,2)],3)
 => [1,2] => [1,1] => ([(0,1)],2)
 => 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1
([],4)
 => [4] => [1] => ([],1)
 => 0
([(2,3)],4)
 => [1,3] => [1,1] => ([(0,1)],2)
 => 1
([(1,3),(2,3)],4)
 => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1] => ([(0,1)],2)
 => 1
([],5)
 => [5] => [1] => ([],1)
 => 0
([(3,4)],5)
 => [1,4] => [1,1] => ([(0,1)],2)
 => 1
([(2,4),(3,4)],5)
 => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
([(1,4),(2,3)],5)
 => [2,3] => [1,1] => ([(0,1)],2)
 => 1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1] => ([(0,1)],2)
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [1,1] => ([(0,1)],2)
 => 1
([],6)
 => [6] => [1] => ([],1)
 => 0
([(4,5)],6)
 => [1,5] => [1,1] => ([(0,1)],2)
 => 1
([(3,5),(4,5)],6)
 => [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
 => 2
([(2,5),(3,5),(4,5)],6)
 => [1,2,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
([(2,5),(3,4)],6)
 => [2,4] => [1,1] => ([(0,1)],2)
 => 1
([(2,5),(3,4),(4,5)],6)
 => [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
([(1,2),(3,5),(4,5)],6)
 => [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [1,1] => ([(0,1)],2)
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1

Description

The diameter of a connected graph. This is the greatest distance between any pair of vertices.

Matching statistic: St001093

Mapping

Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001093: 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] => [1] => ([],1)
 => 1 = 0 + 1
([],3)
 => [3] => [1] => ([],1)
 => 1 = 0 + 1
([(1,2)],3)
 => [1,2] => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
([],4)
 => [4] => [1] => ([],1)
 => 1 = 0 + 1
([(2,3)],4)
 => [1,3] => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
([(1,3),(2,3)],4)
 => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [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)
 => [3,1] => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
([],5)
 => [5] => [1] => ([],1)
 => 1 = 0 + 1
([(3,4)],5)
 => [1,4] => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
([(2,4),(3,4)],5)
 => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
 => 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
([(1,4),(2,3)],5)
 => [2,3] => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [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)],5)
 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3 = 2 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3 = 2 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3 = 2 + 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
([],6)
 => [6] => [1] => ([],1)
 => 1 = 0 + 1
([(4,5)],6)
 => [1,5] => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
([(3,5),(4,5)],6)
 => [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
 => 3 = 2 + 1
([(2,5),(3,5),(4,5)],6)
 => [1,2,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
([(2,5),(3,4)],6)
 => [2,4] => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
([(2,5),(3,4),(4,5)],6)
 => [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(1,2),(3,5),(4,5)],6)
 => [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1

Description

The detour number of a graph. This is the number of vertices in a longest induced path in a graph. Note that [1] defines the detour number as the number of edges in a longest induced path, which is unsuitable for the empty graph.

Matching statistic: St001418

Mapping

Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001418: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 99%●distinct values known / distinct values provided: 67%

Values

([],1)
 => [1] => [1] => [1,0]
 => ? = 0 - 1
([],2)
 => [2] => [1] => [1,0]
 => ? = 0 - 1
([],3)
 => [3] => [1] => [1,0]
 => ? = 0 - 1
([(1,2)],3)
 => [1,2] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
([],4)
 => [4] => [1] => [1,0]
 => ? = 0 - 1
([(2,3)],4)
 => [1,3] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
([(1,3),(2,3)],4)
 => [1,1,2] => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
([],5)
 => [5] => [1] => [1,0]
 => ? = 0 - 1
([(3,4)],5)
 => [1,4] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => [1,1,3] => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => [2,3] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
([],6)
 => [6] => [1] => [1,0]
 => ? = 0 - 1
([(4,5)],6)
 => [1,5] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
([(3,5),(4,5)],6)
 => [1,1,4] => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(2,5),(3,5),(4,5)],6)
 => [1,2,3] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,3,2] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,4,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => [2,4] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
 => [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(1,2),(3,5),(4,5)],6)
 => [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,3] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => [1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
([],7)
 => [7] => [1] => [1,0]
 => ? = 0 - 1
([],8)
 => [8] => [1] => [1,0]
 => ? = 0 - 1

Description

Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. The stable Auslander algebra is by definition the stable endomorphism ring of the direct sum of all indecomposable modules.

Matching statistic: St000455

Mapping

Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 79%●distinct values known / distinct values provided: 67%

Values

([],1)
 => [1] => [1] => ([],1)
 => ? = 0 - 2
([],2)
 => [2] => [1] => ([],1)
 => ? = 0 - 2
([],3)
 => [3] => [1] => ([],1)
 => ? = 0 - 2
([(1,2)],3)
 => [1,2] => [1,1] => ([(0,1)],2)
 => -1 = 1 - 2
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,1] => ([(0,1)],2)
 => -1 = 1 - 2
([],4)
 => [4] => [1] => ([],1)
 => ? = 0 - 2
([(2,3)],4)
 => [1,3] => [1,1] => ([(0,1)],2)
 => -1 = 1 - 2
([(1,3),(2,3)],4)
 => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 2 - 2
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 1 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 1 - 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1] => ([(0,1)],2)
 => -1 = 1 - 2
([],5)
 => [5] => [1] => ([],1)
 => ? = 0 - 2
([(3,4)],5)
 => [1,4] => [1,1] => ([(0,1)],2)
 => -1 = 1 - 2
([(2,4),(3,4)],5)
 => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 2 - 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 1 - 2
([(1,4),(2,3)],5)
 => [2,3] => [1,1] => ([(0,1)],2)
 => -1 = 1 - 2
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 2 - 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 2 - 2
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1] => ([(0,1)],2)
 => -1 = 1 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 2 - 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 2 - 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 1 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 2 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 2 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 1 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 2 - 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1] => ([(0,1)],2)
 => -1 = 1 - 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 2 - 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [1,1] => ([(0,1)],2)
 => -1 = 1 - 2
([],6)
 => [6] => [1] => ([],1)
 => ? = 0 - 2
([(4,5)],6)
 => [1,5] => [1,1] => ([(0,1)],2)
 => -1 = 1 - 2
([(3,5),(4,5)],6)
 => [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 2 - 2
([(2,5),(3,5),(4,5)],6)
 => [1,2,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 1 - 2
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 1 - 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 1 - 2
([(2,5),(3,4)],6)
 => [2,4] => [1,1] => ([(0,1)],2)
 => -1 = 1 - 2
([(2,5),(3,4),(4,5)],6)
 => [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 2 - 2
([(1,2),(3,5),(4,5)],6)
 => [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 2 - 2
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [1,1] => ([(0,1)],2)
 => -1 = 1 - 2
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 2 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 2 - 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 2 - 2
([(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 1 - 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 2 - 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 1 - 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 2 - 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 2 - 2
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 2 - 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 2 - 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 1 - 2
([(1,5),(2,4),(3,4),(3,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 2 - 2
([(0,1),(2,5),(3,4),(4,5)],6)
 => [1,2,1,2] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 1 - 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 1 - 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 2 - 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 2 - 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,1,2] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 1 - 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 2
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 2
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 2
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => [1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 2
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 2
([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 2
([],7)
 => [7] => [1] => ([],1)
 => ? = 0 - 2
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
 => [1,2,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => [1,2,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
 => [1,2,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [2,1,1,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 2
([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => [1,2,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [1,1,2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
 => [1,2,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => [2,1,1,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,2,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(0,6),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [2,1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [2,1,1,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(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)
 => [2,1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [2,1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => [1,2,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => [1,2,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,2,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,2,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,2,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => [1,2,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 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)],7)
 => [1,1,2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [1,2,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,2,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,2,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [2,1,1,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [3,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 2
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [3,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 2
([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [1,2,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [2,1,1,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [3,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 2
([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => [1,1,2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => [1,1,2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 2
([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [1,3,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 2
([(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [1,1,2,1,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 2
([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [3,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 2
([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [1,2,2,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 2

Description

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

Matching statistic: St001431

Mapping

Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001431: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%

Values

([],1)
 => [1] => [1] => [1,0]
 => ? = 0 - 1
([],2)
 => [2] => [1] => [1,0]
 => ? = 0 - 1
([],3)
 => [3] => [1] => [1,0]
 => ? = 0 - 1
([(1,2)],3)
 => [1,2] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
([],4)
 => [4] => [1] => [1,0]
 => ? = 0 - 1
([(2,3)],4)
 => [1,3] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
([(1,3),(2,3)],4)
 => [1,1,2] => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
([],5)
 => [5] => [1] => [1,0]
 => ? = 0 - 1
([(3,4)],5)
 => [1,4] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => [1,1,3] => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => [2,3] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
([],6)
 => [6] => [1] => [1,0]
 => ? = 0 - 1
([(4,5)],6)
 => [1,5] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
([(3,5),(4,5)],6)
 => [1,1,4] => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(2,5),(3,5),(4,5)],6)
 => [1,2,3] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,3,2] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,4,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => [2,4] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
 => [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(1,2),(3,5),(4,5)],6)
 => [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,3] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => [1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
([],7)
 => [7] => [1] => [1,0]
 => ? = 0 - 1
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2 - 1
([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2 - 1
([(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2 - 1
([(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,6),(2,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,6),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,2),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1

Description

Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. The modified algebra B is obtained from the stable Auslander algebra kQ/I by deleting all relations which contain walks of length at least three (conjectural this step of deletion is not necessary as the stable higher Auslander algebras might be quadratic) and taking as B then the algebra kQ^(op)/J when J is the quadratic perp of the ideal I. See http://www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Loewy length and Nakayama algebras associated to Dyck paths.

Matching statistic: St001330
(load all 2 compositions to match this statistic)

Mapping

Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 67%

Values

([],1)
 => [1] => [1] => ([],1)
 => 1 = 0 + 1
([],2)
 => [2] => [2] => ([],2)
 => 1 = 0 + 1
([],3)
 => [3] => [3] => ([],3)
 => 1 = 0 + 1
([(1,2)],3)
 => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,1),(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)
 => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(1,3),(2,3)],4)
 => [1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
([],5)
 => [5] => [5] => ([],5)
 => 1 = 0 + 1
([(3,4)],5)
 => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(2,4),(3,4)],5)
 => [1,1,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
([(1,4),(2,3)],5)
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
([],6)
 => [6] => [6] => ([],6)
 => 1 = 0 + 1
([(4,5)],6)
 => [1,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
([(3,5),(4,5)],6)
 => [1,1,4] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
([(2,5),(3,5),(4,5)],6)
 => [1,2,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
([(2,5),(3,4)],6)
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
([(2,5),(3,4),(4,5)],6)
 => [1,1,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
([(1,2),(3,5),(4,5)],6)
 => [1,1,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,3,1] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,2,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,3,1] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
([(1,5),(2,4),(3,4),(3,5)],6)
 => [1,1,1,1,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => [1,2,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,2,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,2] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 2 = 1 + 1
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1] => [1,5] => ([(4,5)],6)
 => 2 = 1 + 1
([(5,6)],7)
 => [1,6] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1

Description

The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of

$q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number

$HG(G)$ of a graph

$G$ is the largest integer

$q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of

$q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.