Identifier
Values
{{1}} => [1] => [.,.] => ([],1) => 0
{{1,2}} => [2,1] => [[.,.],.] => ([(0,1)],2) => 2
{{1},{2}} => [1,2] => [.,[.,.]] => ([(0,1)],2) => 2
{{1,3,5,6},{2},{4}} => [3,2,5,4,6,1] => [[[.,.],[[.,.],[.,.]]],.] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
{{1,3,6},{2},{4},{5}} => [3,2,6,4,5,1] => [[[.,.],[[.,.],[.,.]]],.] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
{{1,5,6},{2,3,4}} => [5,3,4,2,6,1] => [[[[.,.],[.,.]],[.,.]],.] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
{{1,6},{2,3,4},{5}} => [6,3,4,2,5,1] => [[[[.,.],[.,.]],[.,.]],.] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
{{1},{2,3},{4,5},{6}} => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
{{1},{2,4,5},{3},{6}} => [1,4,3,5,2,6] => [.,[[[.,.],[.,.]],[.,.]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
{{1,6},{2,4},{3,5}} => [6,4,5,2,3,1] => [[[[.,.],[.,.]],[.,.]],.] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
{{1,5,6},{2},{3,4}} => [5,2,4,3,6,1] => [[[.,.],[[.,.],[.,.]]],.] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
{{1},{2,5},{3,4,6}} => [1,5,4,6,2,3] => [.,[[[.,.],[.,.]],[.,.]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
{{1,6},{2},{3,4},{5}} => [6,2,4,3,5,1] => [[[.,.],[[.,.],[.,.]]],.] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
{{1},{2,5},{3},{4,6}} => [1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
{{1},{2,5},{3},{4},{6}} => [1,5,3,4,2,6] => [.,[[[.,.],[.,.]],[.,.]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
{{1,2,5,6,7},{3,4}} => [2,5,4,3,6,7,1] => [[.,[[[.,.],.],[.,[.,.]]]],.] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,2,5,7},{3,4},{6}} => [2,5,4,3,7,6,1] => [[.,[[[.,.],.],[[.,.],.]]],.] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,2},{3,4,5},{6,7}} => [2,1,4,5,3,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,2},{3,4,5},{6},{7}} => [2,1,4,5,3,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,2,6,7},{3,4},{5}} => [2,6,4,3,5,7,1] => [[.,[[[.,.],.],[.,[.,.]]]],.] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,2,7},{3,4},{5,6}} => [2,7,4,3,6,5,1] => [[.,[[[.,.],.],[[.,.],.]]],.] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,2,7},{3,4},{5},{6}} => [2,7,4,3,5,6,1] => [[.,[[[.,.],.],[.,[.,.]]]],.] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,2},{3,5},{4,6,7}} => [2,1,5,6,3,7,4] => [[.,.],[[.,[.,.]],[[.,.],.]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,2},{3,5},{4,6},{7}} => [2,1,5,6,3,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,2},{3,5},{4,7},{6}} => [2,1,5,7,3,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,2},{3,5},{4},{6,7}} => [2,1,5,4,3,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,2},{3,5},{4},{6},{7}} => [2,1,5,4,3,6,7] => [[.,.],[[[.,.],.],[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,4,5,6},{2,3},{7}} => [4,3,2,5,6,1,7] => [[[[.,.],.],[.,[.,.]]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,4,6},{2,3},{5,7}} => [4,3,2,6,7,1,5] => [[[[.,.],.],[.,[.,.]]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,4,6},{2,3},{5},{7}} => [4,3,2,6,5,1,7] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,5,6},{2,3},{4,7}} => [5,3,2,7,6,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,5,6},{2,3},{4},{7}} => [5,3,2,4,6,1,7] => [[[[.,.],.],[.,[.,.]]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,6},{2,3},{4,5,7}} => [6,3,2,5,7,1,4] => [[[[.,.],.],[.,[.,.]]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,6},{2,3},{4,5},{7}} => [6,3,2,5,4,1,7] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,6},{2,3},{4,7},{5}} => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,6},{2,3},{4},{5,7}} => [6,3,2,4,7,1,5] => [[[[.,.],.],[.,[.,.]]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,6},{2,3},{4},{5},{7}} => [6,3,2,4,5,1,7] => [[[[.,.],.],[.,[.,.]]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,4,5,6},{2,7},{3}} => [4,7,3,5,6,1,2] => [[[.,[.,.]],[.,[.,.]]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,4,5,7},{2,6},{3}} => [4,6,3,5,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,4,6},{2,5,7},{3}} => [4,5,3,6,7,1,2] => [[[.,[.,.]],[.,[.,.]]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,4,7},{2,5,6},{3}} => [4,5,3,7,6,2,1] => [[[[.,[.,.]],[[.,.],.]],.],.] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,4,6},{2,7},{3},{5}} => [4,7,3,6,5,1,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,4,7},{2,6},{3},{5}} => [4,6,3,7,5,2,1] => [[[[.,[.,.]],[[.,.],.]],.],.] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,5,6},{2,4,7},{3}} => [5,4,3,7,6,1,2] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,5,7},{2,4,6},{3}} => [5,4,3,6,7,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,6},{2,4,5,7},{3}} => [6,4,3,5,7,1,2] => [[[[.,.],.],[.,[.,.]]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,7},{2,4,5,6},{3}} => [7,4,3,5,6,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,6},{2,4,7},{3},{5}} => [6,4,3,7,5,1,2] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,7},{2,4,6},{3},{5}} => [7,4,3,6,5,2,1] => [[[[[.,.],.],[[.,.],.]],.],.] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1},{2,5,6,7},{3,4}} => [1,5,4,3,6,7,2] => [.,[[[[.,.],.],[.,[.,.]]],.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1},{2,5,7},{3,4},{6}} => [1,5,4,3,7,6,2] => [.,[[[[.,.],.],[[.,.],.]],.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1},{2},{3,4,5},{6,7}} => [1,2,4,5,3,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1},{2},{3,4,5},{6},{7}} => [1,2,4,5,3,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1},{2,6,7},{3,4},{5}} => [1,6,4,3,5,7,2] => [.,[[[[.,.],.],[.,[.,.]]],.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1},{2,7},{3,4},{5,6}} => [1,7,4,3,6,5,2] => [.,[[[[.,.],.],[[.,.],.]],.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1},{2,7},{3,4},{5},{6}} => [1,7,4,3,5,6,2] => [.,[[[[.,.],.],[.,[.,.]]],.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,5,6},{2,7},{3},{4}} => [5,7,3,4,6,1,2] => [[[.,[.,.]],[.,[.,.]]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,5,7},{2,6},{3},{4}} => [5,6,3,4,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,6},{2,5,7},{3},{4}} => [6,5,3,4,7,1,2] => [[[[.,.],.],[.,[.,.]]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,7},{2,5,6},{3},{4}} => [7,5,3,4,6,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1},{2},{3,5},{4,6,7}} => [1,2,5,6,3,7,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1},{2},{3,5},{4,6},{7}} => [1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1},{2},{3,5},{4,7},{6}} => [1,2,5,7,3,6,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1},{2},{3,5},{4},{6,7}} => [1,2,5,4,3,7,6] => [.,[.,[[[.,.],.],[[.,.],.]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1},{2},{3,5},{4},{6},{7}} => [1,2,5,4,3,6,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,6},{2,7},{3},{4,5}} => [6,7,3,5,4,1,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,7},{2,6},{3},{4,5}} => [7,6,3,5,4,2,1] => [[[[[.,.],.],[[.,.],.]],.],.] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,6},{2,7},{3},{4},{5}} => [6,7,3,4,5,1,2] => [[[.,[.,.]],[.,[.,.]]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
{{1,7},{2,6},{3},{4},{5}} => [7,6,3,4,5,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.] => ([(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 increasing tree
Description
Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
to graph
Description
Return the undirected graph obtained from the tree nodes and edges, with leaves being ignored.