Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St000422
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => 2
[1,1]
 => [1,0,1,1,0,0]
 => [1,2] => ([(1,2)],3)
 => 2
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 4
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => ([(2,3)],4)
 => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => ([(2,3)],4)
 => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => ([(3,4)],5)
 => 2
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 4
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => 2
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,5] => ([(4,5)],6)
 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => 2
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => 2
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,5] => ([(4,5)],6)
 => 2
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,6] => ([(5,6)],7)
 => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => 2
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,5] => ([(4,5)],6)
 => 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 4
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,5] => ([(4,5)],6)
 => 2
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,6] => ([(5,6)],7)
 => 2
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,5] => ([(4,5)],6)
 => 2
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 4
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,5] => ([(4,5)],6)
 => 2
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,6] => ([(5,6)],7)
 => 2
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,5] => ([(4,5)],6)
 => 2
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,6] => ([(5,6)],7)
 => 2
[4,4,1]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,5] => ([(4,5)],6)
 => 2
[4,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,6] => ([(5,6)],7)
 => 2
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,5] => ([(4,5)],6)
 => 2
[3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,5] => ([(4,5)],6)
 => 2
[3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,6] => ([(5,6)],7)
 => 2
[2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 4
[2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,6] => ([(5,6)],7)
 => 2
[5,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,6] => ([(5,6)],7)
 => 2
[4,4,2]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 8
[4,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,5] => ([(4,5)],6)
 => 2
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,5] => ([(4,5)],6)
 => 2
[4,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,6] => ([(5,6)],7)
 => 2
[3,3,2,1,1]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,5] => ([(4,5)],6)
 => 2
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.