Identifier
Values
[[1]] => [1] => ([],1) => 0
[[1,2]] => [2] => ([],2) => 0
[[1],[2]] => [2] => ([],2) => 0
[[1,2,3]] => [3] => ([],3) => 0
[[1,2],[3]] => [3] => ([],3) => 0
[[1],[2],[3]] => [3] => ([],3) => 0
[[1,2,3,4]] => [4] => ([],4) => 0
[[1,2,3],[4]] => [4] => ([],4) => 0
[[1,2],[3],[4]] => [4] => ([],4) => 0
[[1],[2],[3],[4]] => [4] => ([],4) => 0
[[1,2,3,4,5]] => [5] => ([],5) => 0
[[1,2,3,5],[4]] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
[[1,2,3,4],[5]] => [5] => ([],5) => 0
[[1,2,3],[4,5]] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
[[1,2,5],[3],[4]] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
[[1,2,3],[4],[5]] => [5] => ([],5) => 0
[[1,2],[3,5],[4]] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
[[1,5],[2],[3],[4]] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
[[1,2],[3],[4],[5]] => [5] => ([],5) => 0
[[1],[2],[3],[4],[5]] => [5] => ([],5) => 0
[[1,2,3,4,5,6]] => [6] => ([],6) => 0
[[1,2,3,5,6],[4]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,2,3,4,5],[6]] => [6] => ([],6) => 0
[[1,2,3,6],[4,5]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,2,3,5],[4,6]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,2,5,6],[3],[4]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,2,3,5],[4],[6]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,2,3,4],[5],[6]] => [6] => ([],6) => 0
[[1,2,3],[4,5,6]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,2,6],[3,5],[4]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,2,5],[3,6],[4]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,2,3],[4,5],[6]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,5,6],[2],[3],[4]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,2,5],[3],[4],[6]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,2,3],[4],[5],[6]] => [6] => ([],6) => 0
[[1,2],[3,5],[4,6]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,5],[2,6],[3],[4]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,2],[3,5],[4],[6]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,5],[2],[3],[4],[6]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,2],[3],[4],[5],[6]] => [6] => ([],6) => 0
[[1],[2],[3],[4],[5],[6]] => [6] => ([],6) => 0
[[1,2,3,4,5,6,7]] => [7] => ([],7) => 0
[[1,2,3,5,6,7],[4]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,2,3,4,5,6],[7]] => [7] => ([],7) => 0
[[1,2,3,6,7],[4,5]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,2,3,5,6],[4,7]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,2,5,6,7],[3],[4]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,2,3,5,6],[4],[7]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,2,3,4,5],[6],[7]] => [7] => ([],7) => 0
[[1,2,3,7],[4,5,6]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,2,3,6],[4,5,7]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,2,6,7],[3,5],[4]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,2,5,6],[3,7],[4]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,2,3,6],[4,5],[7]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,2,3,5],[4,6],[7]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,5,6,7],[2],[3],[4]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,2,5,6],[3],[4],[7]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,2,3,5],[4],[6],[7]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,2,3,4],[5],[6],[7]] => [7] => ([],7) => 0
[[1,2,6],[3,5,7],[4]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,2,3],[4,5,6],[7]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,2,6],[3,5],[4,7]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,2,5],[3,6],[4,7]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,5,6],[2,7],[3],[4]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,2,6],[3,5],[4],[7]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,2,5],[3,6],[4],[7]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,2,3],[4,5],[6],[7]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,5,6],[2],[3],[4],[7]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,2,5],[3],[4],[6],[7]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,2,3],[4],[5],[6],[7]] => [7] => ([],7) => 0
[[1,5],[2,6],[3,7],[4]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,2],[3,5],[4,6],[7]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,5],[2,6],[3],[4],[7]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,2],[3,5],[4],[6],[7]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,5],[2],[3],[4],[6],[7]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[[1,2],[3],[4],[5],[6],[7]] => [7] => ([],7) => 0
[[1],[2],[3],[4],[5],[6],[7]] => [7] => ([],7) => 0
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 threshold graph
Description
The threshold graph corresponding to the composition.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.
Map
valley composition
Description
The composition corresponding to the valley set of a standard tableau.
Let $T$ be a standard tableau of size $n$.
An entry $i$ of $T$ is a descent if $i+1$ is in a lower row (in English notation), otherwise $i$ is an ascent.
An entry $2 \leq i \leq n-1$ is a valley if $i-1$ is a descent and $i$ is an ascent.
This map returns the composition $c_1,\dots,c_k$ of $n$ such that $\{c_1, c_1+c_2,\dots, c_1+\dots+c_k\}$ is the valley set of $T$.