searching the database
Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000350
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
([],1)
=> 0
([],2)
=> 0
([(0,1)],2)
=> 2
([],3)
=> 0
([(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> 4
([(0,1),(0,2),(1,2)],3)
=> 6
([],4)
=> 0
([(2,3)],4)
=> 2
([(1,3),(2,3)],4)
=> 4
([(0,3),(1,3),(2,3)],4)
=> 6
([(0,3),(1,2)],4)
=> 4
([(0,3),(1,2),(2,3)],4)
=> 6
([(1,2),(1,3),(2,3)],4)
=> 6
([(0,3),(1,2),(1,3),(2,3)],4)
=> 8
([(0,2),(0,3),(1,2),(1,3)],4)
=> 8
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 10
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 12
([],5)
=> 0
([(3,4)],5)
=> 2
([(2,4),(3,4)],5)
=> 4
([(1,4),(2,4),(3,4)],5)
=> 6
([(0,4),(1,4),(2,4),(3,4)],5)
=> 8
([(1,4),(2,3)],5)
=> 4
([(1,4),(2,3),(3,4)],5)
=> 6
([(0,1),(2,4),(3,4)],5)
=> 6
([(2,3),(2,4),(3,4)],5)
=> 6
([(0,4),(1,4),(2,3),(3,4)],5)
=> 8
([(1,4),(2,3),(2,4),(3,4)],5)
=> 8
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
([(1,3),(1,4),(2,3),(2,4)],5)
=> 8
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 10
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 10
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 12
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 12
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 14
([(0,4),(1,3),(2,3),(2,4)],5)
=> 8
([(0,1),(2,3),(2,4),(3,4)],5)
=> 8
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 10
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 12
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 10
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 12
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 14
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 12
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 12
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 14
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 16
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 14
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 16
Description
The sum of the vertex degrees of a graph.
This is clearly equal to twice the number of edges, and, incidentally, also equal to the trace of the Laplacian matrix of a graph. From this it follows that it is also the sum of the squares of the eigenvalues of the adjacency matrix of the graph.
The Laplacian matrix is defined as $D-A$ where $D$ is the degree matrix (the diagonal matrix with the vertex degrees on the diagonal) and where $A$ is the adjacency matrix. See [1] for detailed definitions.
Matching statistic: St001232
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 38%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 38%
Values
([],1)
=> [1] => [1,0]
=> [1,1,0,0]
=> 0
([],2)
=> [2] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
([(0,1)],2)
=> [1,1] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
([],3)
=> [3] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(1,2)],3)
=> [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
([(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 6
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([],4)
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(2,3)],4)
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 6
([(1,3),(2,3)],4)
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ? ∊ {6,6,8,8,10,12}
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ? ∊ {6,6,8,8,10,12}
([(0,3),(1,2)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {6,6,8,8,10,12}
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {6,6,8,8,10,12}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ? ∊ {6,6,8,8,10,12}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {6,6,8,8,10,12}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
([],5)
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(3,4)],5)
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 8
([(2,4),(3,4)],5)
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 8
([(1,4),(2,3)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 6
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 6
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(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]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ? ∊ {4,6,6,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
([],6)
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
([(4,5)],6)
=> [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> 10
([(3,5),(4,5)],6)
=> [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? ∊ {6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> 10
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> 10
([(2,5),(3,4)],6)
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> 8
([(2,5),(3,4),(4,5)],6)
=> [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? ∊ {6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(1,2),(3,5),(4,5)],6)
=> [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? ∊ {6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(3,4),(3,5),(4,5)],6)
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> 8
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? ∊ {6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? ∊ {6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? ∊ {6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? ∊ {6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> ? ∊ {6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? ∊ {6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? ∊ {6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? ∊ {6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? ∊ {6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? ∊ {6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> 8
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> 6
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> 10
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> 6
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> 10
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> 4
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> 4
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> 8
([(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] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000422
Values
([],1)
=> ([],0)
=> ([],1)
=> 0
([],2)
=> ([],0)
=> ([],1)
=> 0
([(0,1)],2)
=> ([],1)
=> ([(0,1)],2)
=> 2
([],3)
=> ([],0)
=> ([],1)
=> 0
([(1,2)],3)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
([],4)
=> ([],0)
=> ([],1)
=> 0
([(2,3)],4)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> ? ∊ {4,6,8,8,10,12}
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {4,6,8,8,10,12}
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,6,8,8,10,12}
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {4,6,8,8,10,12}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,6,8,8,10,12}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? ∊ {4,6,8,8,10,12}
([],5)
=> ([],0)
=> ([],1)
=> 0
([(3,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(0,6),(1,3),(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)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(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),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(0,7),(1,2),(1,4),(1,5),(1,7),(2,3),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,3),(0,4),(0,5),(0,6),(0,8),(1,2),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,7),(2,8),(3,4),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,5),(0,6),(0,7),(1,3),(1,4),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,2),(0,3),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,8),(2,5),(2,7),(2,8),(3,4),(3,6),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(0,9),(1,2),(1,3),(1,7),(1,8),(1,9),(2,3),(2,5),(2,6),(2,8),(2,9),(3,4),(3,6),(3,7),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,8),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,10),(4,5),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,7),(6,8),(6,10),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? ∊ {4,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,14,14,14,14,16,16,18,20}
([],6)
=> ([],0)
=> ([],1)
=> 0
([(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 2
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> 10
([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> ? ∊ {4,6,6,6,8,8,8,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {4,6,6,6,8,8,8,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(1,2),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {4,6,6,6,8,8,8,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,6,6,6,8,8,8,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,6,6,6,8,8,8,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,6,6,6,8,8,8,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,6,6,6,8,8,8,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ? ∊ {4,6,6,6,8,8,8,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,6,6,6,8,8,8,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {4,6,6,6,8,8,8,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {4,6,6,6,8,8,8,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,6,6,6,8,8,8,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,6,6,6,8,8,8,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,6,6,6,8,8,8,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,6,6,6,8,8,8,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,6,6,6,8,8,8,8,8,8,8,8,10,10,10,10,10,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,24,24,24,24,24,26,26,28,30}
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!