|
|
A263293
|
|
Triangle read by rows: T(n,k) is the number of graphs with n vertices and maximum vertex degree k, (0 <= k < n).
|
|
8
|
|
|
1, 1, 1, 1, 1, 2, 1, 2, 4, 4, 1, 2, 8, 12, 11, 1, 3, 15, 43, 60, 34, 1, 3, 25, 121, 360, 378, 156, 1, 4, 41, 378, 2166, 4869, 3843, 1044, 1, 4, 65, 1095, 14306, 68774, 113622, 64455, 12346, 1, 5, 100, 3441, 104829, 1141597, 3953162, 4605833, 1921532, 274668
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,6
|
|
COMMENTS
|
Terms may be computed without generating each graph by enumerating the number of graphs by degree sequence. A PARI program showing this technique for graphs with labeled vertices is given in A327366. Burnside's lemma can be used to extend this method to the unlabeled case. - Andrew Howroyd, Mar 10 2020
|
|
LINKS
|
|
|
FORMULA
|
G.f. for column k=0: A(x)=1/(1-x).
G.f. for column k=1: B(x)=x^2/((1-x^2)(1-x)).
G.f. for column k=2: 1/((1-x)(1-x^2))*Product_{i>=3} 1/(1-x^i)^2 - B(x) - A(x).
(End)
T(n, 0) = 1.
|
|
EXAMPLE
|
Triangle begins:
1,
1, 1,
1, 1, 2,
1, 2, 4, 4,
1, 2, 8, 12, 11,
1, 3, 15, 43, 60, 34,
1, 3, 25, 121, 360, 378, 156,
1, 4, 41, 378, 2166, 4869, 3843, 1044,
...
|
|
CROSSREFS
|
Row sums are A000088 (simple graphs on n nodes).
Cf. A294217 (triangle of n-node minimum vertex degree counts).
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|