Identifier
Values
[1] => [1,0] => [[]] => ([(0,1)],2) => 2
[1,2,3,4] => [1,0,1,0,1,0,1,0] => [[],[],[],[]] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
[2,3,4,1] => [1,1,0,1,0,1,0,0] => [[[],[],[]]] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => [[],[],[[],[]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => [[],[[],[]],[]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => [[[],[]],[],[]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0] => [[[],[[],[]]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0] => [[[],[[],[]]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0] => [[[[],[]],[]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0] => [[[[],[]],[]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[1,4,2,6,3,5] => [1,0,1,1,1,0,0,1,1,0,0,0] => [[],[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[1,4,2,6,5,3] => [1,0,1,1,1,0,0,1,1,0,0,0] => [[],[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[1,4,3,6,2,5] => [1,0,1,1,1,0,0,1,1,0,0,0] => [[],[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[1,4,3,6,5,2] => [1,0,1,1,1,0,0,1,1,0,0,0] => [[],[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0] => [[[]],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[3,1,5,2,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0] => [[[[]],[[]]],[]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[3,1,5,4,2,6] => [1,1,1,0,0,1,1,0,0,0,1,0] => [[[[]],[[]]],[]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[3,2,5,1,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0] => [[[[]],[[]]],[]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[3,2,5,4,1,6] => [1,1,1,0,0,1,1,0,0,0,1,0] => [[[[]],[[]]],[]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[4,1,6,2,3,5] => [1,1,1,1,0,0,1,1,0,0,0,0] => [[[[[]],[[]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[4,1,6,2,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0] => [[[[[]],[[]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[4,1,6,3,2,5] => [1,1,1,1,0,0,1,1,0,0,0,0] => [[[[[]],[[]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[4,1,6,3,5,2] => [1,1,1,1,0,0,1,1,0,0,0,0] => [[[[[]],[[]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[4,1,6,5,2,3] => [1,1,1,1,0,0,1,1,0,0,0,0] => [[[[[]],[[]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[4,1,6,5,3,2] => [1,1,1,1,0,0,1,1,0,0,0,0] => [[[[[]],[[]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[4,2,6,1,3,5] => [1,1,1,1,0,0,1,1,0,0,0,0] => [[[[[]],[[]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0] => [[[[[]],[[]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[4,2,6,3,1,5] => [1,1,1,1,0,0,1,1,0,0,0,0] => [[[[[]],[[]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[4,2,6,3,5,1] => [1,1,1,1,0,0,1,1,0,0,0,0] => [[[[[]],[[]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[4,2,6,5,1,3] => [1,1,1,1,0,0,1,1,0,0,0,0] => [[[[[]],[[]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[4,2,6,5,3,1] => [1,1,1,1,0,0,1,1,0,0,0,0] => [[[[[]],[[]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[4,3,6,1,2,5] => [1,1,1,1,0,0,1,1,0,0,0,0] => [[[[[]],[[]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[4,3,6,1,5,2] => [1,1,1,1,0,0,1,1,0,0,0,0] => [[[[[]],[[]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[4,3,6,2,1,5] => [1,1,1,1,0,0,1,1,0,0,0,0] => [[[[[]],[[]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[4,3,6,2,5,1] => [1,1,1,1,0,0,1,1,0,0,0,0] => [[[[[]],[[]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[4,3,6,5,1,2] => [1,1,1,1,0,0,1,1,0,0,0,0] => [[[[[]],[[]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[4,3,6,5,2,1] => [1,1,1,1,0,0,1,1,0,0,0,0] => [[[[[]],[[]]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[] => [] => [] => ([],1) => 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
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
Map
to graph
Description
Return the undirected graph obtained from the tree nodes and edges.
Map
to ordered tree
Description
Sends a Dyck path to the ordered tree encoding the heights of the path.
This map is recursively defined as follows: A Dyck path $D$ of semilength $n$ may be decomposed, according to its returns (St000011The number of touch points (or returns) of a Dyck path.), into smaller paths $D_1,\dots,D_k$ of respective semilengths $n_1,\dots,n_k$ (so one has $n = n_1 + \dots n_k$) each of which has no returns.
Denote by $\tilde D_i$ the path of semilength $n_i-1$ obtained from $D_i$ by removing the initial up- and the final down-step.
This map then sends $D$ to the tree $T$ having a root note with ordered children $T_1,\dots,T_k$ which are again ordered trees computed from $D_1,\dots,D_k$ respectively.
The unique path of semilength $1$ is sent to the tree consisting of a single node.