Identifier
Values
[1] => [.,.] => [[]] => ([(0,1)],2) => 2
[1,4,3,2] => [.,[[[.,.],.],.]] => [[[],[],[]]] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
[4,3,2,1] => [[[[.,.],.],.],.] => [[],[],[],[]] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]] => [[[],[[],[]]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]] => [[[],[[],[]]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]] => [[[],[[],[]]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[1,5,2,4,3] => [.,[[.,[[.,.],.]],.]] => [[[[],[]],[]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]] => [[],[],[[],[]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[3,2,5,1,4] => [[[.,.],.],[[.,.],.]] => [[],[],[[],[]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[3,2,5,4,1] => [[[.,.],.],[[.,.],.]] => [[],[],[[],[]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[3,5,2,1,4] => [[[.,.],.],[[.,.],.]] => [[],[],[[],[]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[3,5,2,4,1] => [[[.,.],.],[[.,.],.]] => [[],[],[[],[]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[3,5,4,2,1] => [[[.,.],.],[[.,.],.]] => [[],[],[[],[]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[5,2,1,4,3] => [[[.,.],[[.,.],.]],.] => [[],[[],[]],[]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[5,2,4,1,3] => [[[.,.],[[.,.],.]],.] => [[],[[],[]],[]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[5,2,4,3,1] => [[[.,.],[[.,.],.]],.] => [[],[[],[]],[]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[5,4,1,3,2] => [[[.,[[.,.],.]],.],.] => [[[],[]],[],[]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[1,2,5,3,4,6] => [.,[.,[[.,[.,.]],[.,.]]]] => [[[[[]],[[]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[1,2,5,3,6,4] => [.,[.,[[.,[.,.]],[.,.]]]] => [[[[[]],[[]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[1,2,5,6,3,4] => [.,[.,[[.,[.,.]],[.,.]]]] => [[[[[]],[[]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[2,1,5,3,4,6] => [[.,.],[[.,[.,.]],[.,.]]] => [[],[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[2,1,5,3,6,4] => [[.,.],[[.,[.,.]],[.,.]]] => [[],[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[2,1,5,6,3,4] => [[.,.],[[.,[.,.]],[.,.]]] => [[],[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[2,5,1,3,4,6] => [[.,.],[[.,[.,.]],[.,.]]] => [[],[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[2,5,1,3,6,4] => [[.,.],[[.,[.,.]],[.,.]]] => [[],[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[2,5,1,6,3,4] => [[.,.],[[.,[.,.]],[.,.]]] => [[],[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[2,5,3,1,4,6] => [[.,.],[[.,[.,.]],[.,.]]] => [[],[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[2,5,3,1,6,4] => [[.,.],[[.,[.,.]],[.,.]]] => [[],[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[2,5,3,4,1,6] => [[.,.],[[.,[.,.]],[.,.]]] => [[],[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[2,5,3,4,6,1] => [[.,.],[[.,[.,.]],[.,.]]] => [[],[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[2,5,3,6,1,4] => [[.,.],[[.,[.,.]],[.,.]]] => [[],[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[2,5,3,6,4,1] => [[.,.],[[.,[.,.]],[.,.]]] => [[],[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[2,5,6,1,3,4] => [[.,.],[[.,[.,.]],[.,.]]] => [[],[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[2,5,6,3,1,4] => [[.,.],[[.,[.,.]],[.,.]]] => [[],[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[2,5,6,3,4,1] => [[.,.],[[.,[.,.]],[.,.]]] => [[],[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[5,3,1,2,4,6] => [[[.,[.,.]],[.,.]],[.,.]] => [[[]],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[5,3,1,2,6,4] => [[[.,[.,.]],[.,.]],[.,.]] => [[[]],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[5,3,1,4,2,6] => [[[.,[.,.]],[.,.]],[.,.]] => [[[]],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[5,3,1,4,6,2] => [[[.,[.,.]],[.,.]],[.,.]] => [[[]],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[5,3,1,6,2,4] => [[[.,[.,.]],[.,.]],[.,.]] => [[[]],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[5,3,1,6,4,2] => [[[.,[.,.]],[.,.]],[.,.]] => [[[]],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[5,3,4,1,2,6] => [[[.,[.,.]],[.,.]],[.,.]] => [[[]],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[5,3,4,1,6,2] => [[[.,[.,.]],[.,.]],[.,.]] => [[[]],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[5,3,4,6,1,2] => [[[.,[.,.]],[.,.]],[.,.]] => [[[]],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[5,3,6,1,2,4] => [[[.,[.,.]],[.,.]],[.,.]] => [[[]],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[5,3,6,1,4,2] => [[[.,[.,.]],[.,.]],[.,.]] => [[[]],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[5,3,6,4,1,2] => [[[.,[.,.]],[.,.]],[.,.]] => [[[]],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[5,6,3,1,2,4] => [[[.,[.,.]],[.,.]],[.,.]] => [[[]],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[5,6,3,1,4,2] => [[[.,[.,.]],[.,.]],[.,.]] => [[[]],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[5,6,3,4,1,2] => [[[.,[.,.]],[.,.]],[.,.]] => [[[]],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[6,1,4,2,3,5] => [[.,[[.,[.,.]],[.,.]]],.] => [[[[]],[[]]],[]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[6,1,4,2,5,3] => [[.,[[.,[.,.]],[.,.]]],.] => [[[[]],[[]]],[]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[6,1,4,5,2,3] => [[.,[[.,[.,.]],[.,.]]],.] => [[[[]],[[]]],[]] => ([(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 ordered tree: left child = left brother
Description
Return an ordered tree of size $n+1$ by the following recursive rule:
  • if $x$ is the left child of $y$, $x$ becomes the left brother of $y$,
  • if $x$ is the right child of $y$, $x$ becomes the last child of $y$.
Map
binary search tree: left to right
Description
Return the shape of the binary search tree of the permutation as a non labelled binary tree.
Map
to graph
Description
Return the undirected graph obtained from the tree nodes and edges.