Identifier
Values
[3] => [1,0,1,0,1,0] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
[4] => [1,0,1,0,1,0,1,0] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3
[3,1] => [1,0,1,0,1,1,0,0] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 3
[5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 3
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 3
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 3
[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 3
[4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 3
[6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 3
[5,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 3
[4,2,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 3
[6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[5,3,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 3
[5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[4,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 3
[6,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 3
[5,3,2] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 3
[4,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 3
[5,3,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 3
search for individual values
searching the database for the individual values of this statistic
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Map
touch composition
Description
Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.
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
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.