Processing math: 100%

Identifier
Values
[1] => [1,0] => [2,1] => ([(0,1)],2) => -1
[2] => [1,0,1,0] => [3,1,2] => ([(0,2),(1,2)],3) => 0
[1,1] => [1,1,0,0] => [2,3,1] => ([(0,2),(1,2)],3) => 0
[3] => [1,0,1,0,1,0] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4) => 0
[1,1,1] => [1,1,0,1,0,0] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
[4] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[2,2] => [1,1,1,0,0,0] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4) => 0
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[5] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 1
[3,3] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 0
[2,2,2] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 1
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 0
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [6,7,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 0
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [7,6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [7,3,4,5,6,1,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 0
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],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 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
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
graph of inversions
Description
The graph of inversions of a permutation.
For a permutation of {1,…,n}, this is the graph with vertices {1,…,n}, where (i,j) is an edge if and only if it is an inversion of the permutation.