Identifier
Values
[1] => [1,0,1,0] => [.,[.,.]] => ([(0,1)],2) => 2
[1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [.,[[[.,.],[.,.]],[.,.]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[6,2] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0] => [[[[.,.],.],[.,[.,.]]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[6,3] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0] => [[[.,[.,.]],[.,[.,.]]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0] => [.,[[.,.],[[.,.],[.,.]]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[6,2,2,2] => [1,1,1,0,0,1,1,1,0,0,0,0,1,0] => [[.,.],[[[.,.],.],[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[6,3,2,2] => [1,1,1,0,0,1,1,0,1,0,0,0,1,0] => [[.,.],[[.,[.,.]],[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[6,2,2,2,1] => [1,1,0,1,0,1,1,1,0,0,0,0,1,0] => [[[[[.,.],.],[.,[.,.]]],.],.] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[5,2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,1,0,0] => [[.,[[[.,.],.],[[.,.],.]]],.] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[5,2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,1,0,0] => [.,[[[[.,.],.],[[.,.],.]],.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[6,3,2,2,1] => [1,1,0,1,0,1,1,0,1,0,0,0,1,0] => [[[[.,[.,.]],[.,[.,.]]],.],.] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[6,2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [[.,[[[.,.],.],[.,[.,.]]]],.] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[6,2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,0,1,0] => [.,[[[[.,.],.],[.,[.,.]]],.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[5,3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,0,1,0,0] => [[.,[[.,[.,.]],[[.,.],.]]],.] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[5,3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,1,0,0] => [.,[[[.,[.,.]],[[.,.],.]],.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[5,2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [.,[.,[[[.,.],.],[[.,.],.]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[6,3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,0,0,1,0] => [[.,[[.,[.,.]],[.,[.,.]]]],.] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[6,3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,0,1,0] => [.,[[[.,[.,.]],[.,[.,.]]],.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[6,2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [.,[.,[[[.,.],.],[.,[.,.]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[5,3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [.,[.,[[.,[.,.]],[[.,.],.]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[6,3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0] => [.,[.,[[.,[.,.]],[.,[.,.]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
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.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
to graph
Description
Return the undirected graph obtained from the tree nodes and edges, with leaves being ignored.
Map
logarithmic height to pruning number
Description
Francon's map from Dyck paths to binary trees.
This bijection sends the logarithmic height of the Dyck path, St000920The logarithmic height of a Dyck path., to the pruning number of the binary tree, St000396The register function (or Horton-Strahler number) of a binary tree.. The implementation is a literal translation of Knuth's [2].