Identifier
Values
[1] => [1] => ([],1) => ([(0,1)],2) => -1
[1,2] => [1,2] => ([],2) => ([(0,2),(1,2)],3) => 0
[2,1] => [1,2] => ([],2) => ([(0,2),(1,2)],3) => 0
[1,2,3] => [1,2,3] => ([],3) => ([(0,3),(1,3),(2,3)],4) => 0
[1,3,2] => [1,2,3] => ([],3) => ([(0,3),(1,3),(2,3)],4) => 0
[2,1,3] => [1,2,3] => ([],3) => ([(0,3),(1,3),(2,3)],4) => 0
[2,3,1] => [1,2,3] => ([],3) => ([(0,3),(1,3),(2,3)],4) => 0
[1,2,3,4] => [1,2,3,4] => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[1,2,4,3] => [1,2,3,4] => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[1,3,2,4] => [1,2,3,4] => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[1,3,4,2] => [1,2,3,4] => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[2,1,3,4] => [1,2,3,4] => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[2,1,4,3] => [1,2,3,4] => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[2,3,1,4] => [1,2,3,4] => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[2,3,4,1] => [1,2,3,4] => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[1,2,3,4,5] => [1,2,3,4,5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,3,2,4,5] => [1,2,3,4,5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[2,1,3,4,5] => [1,2,3,4,5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[2,3,1,4,5] => [1,2,3,4,5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[3,5,1,2,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5) => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 1
[3,5,1,4,2] => [1,3,2,5,4] => ([(1,4),(2,3)],5) => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 1
[3,5,2,1,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5) => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 1
[3,5,2,4,1] => [1,3,2,5,4] => ([(1,4),(2,3)],5) => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,4,6,2,3,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6) => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[1,4,6,2,5,3] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6) => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[1,4,6,3,2,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6) => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[1,4,6,3,5,2] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6) => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[2,1,3,5,6,4] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[2,1,4,3,5,6] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[2,1,4,3,6,5] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[2,1,4,5,3,6] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[2,1,4,5,6,3] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[2,3,1,4,5,6] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[2,3,1,4,6,5] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[2,3,1,5,4,6] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[2,3,1,5,6,4] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[2,3,4,1,5,6] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[2,3,4,1,6,5] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[2,3,4,5,1,6] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[2,3,4,5,6,1] => [1,2,3,4,5,6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[2,4,6,1,3,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6) => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[2,4,6,1,5,3] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6) => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[2,4,6,3,1,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6) => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[2,4,6,3,5,1] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6) => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[3,1,2,6,4,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6) => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[3,1,2,6,5,4] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6) => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[3,2,1,6,4,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6) => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[3,2,1,6,5,4] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6) => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[3,4,1,6,2,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6) => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[3,4,1,6,5,2] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6) => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[3,4,2,6,1,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6) => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[3,4,2,6,5,1] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6) => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[3,5,1,2,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6) => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[3,5,1,4,2,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6) => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[3,5,1,6,2,4] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6) => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[3,5,1,6,4,2] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6) => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[3,5,2,1,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6) => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[3,5,2,4,1,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6) => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[3,5,2,6,1,4] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6) => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[3,5,2,6,4,1] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6) => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[3,6,5,1,2,4] => [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 1
[3,6,5,2,1,4] => [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 1
[3,6,5,4,1,2] => [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 1
[3,6,5,4,2,1] => [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 1
[4,1,6,3,2,5] => [1,4,3,6,5,2] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 1
[4,2,6,3,1,5] => [1,4,3,6,5,2] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 1
[4,6,1,2,5,3] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 1
[4,6,2,1,5,3] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 1
[4,6,3,1,5,2] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 1
[4,6,3,2,5,1] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 1
>>> Load all 109 entries. <<<
[4,6,5,1,2,3] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 1
[4,6,5,1,3,2] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 1
[4,6,5,2,1,3] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 1
[4,6,5,2,3,1] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 1
[6,5,1,2,4,3] => [1,6,3,2,5,4] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 1
[6,5,1,4,2,3] => [1,6,3,2,5,4] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 1
[6,5,2,1,4,3] => [1,6,3,2,5,4] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 1
[6,5,2,4,1,3] => [1,6,3,2,5,4] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 1
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 second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
Map
cone
Description
The cone of a graph.
The cone of a graph is obtained by joining a new vertex to all the vertices of the graph. The added vertex is called a universal vertex or a dominating vertex.
Map
graph of inversions
Description
The graph of inversions of a permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.