|
|
A106238
|
|
Triangle read by rows: T(n,m) is the number of semi-strong digraphs on n unlabeled nodes with m connected components.
|
|
4
|
|
|
1, 1, 1, 5, 1, 1, 83, 6, 1, 1, 5048, 88, 6, 1, 1, 1047008, 5146, 89, 6, 1, 1, 705422362, 1052471, 5151, 89, 6, 1, 1, 1580348371788, 706498096, 1052569, 5152, 89, 6, 1, 1, 12139024825260556, 1581059448174, 706503594, 1052574, 5152, 89, 6, 1, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,4
|
|
COMMENTS
|
The formula T(n,m) is the sum over the partitions of n with m parts 1K1 + 2K2 + ... + nKn, of Product_{i=1..n} binomial(f(i) + Ki - 1, Ki) can be used to count unlabeled graphs of order n with m components if f(i) is the number of non-isomorphic connected components of order i. (In general, f denotes a sequence that counts unlabeled connected combinatorial objects.)
A digraph is semi-strong if all its weakly connected components are strongly connected. - Andrew Howroyd, Jan 14 2022
|
|
LINKS
|
|
|
FORMULA
|
Triangle read by rows: T(n, m) is the sum over the partitions of n with m parts 1K1 + 2K2 + ... + nKn, of Product_{i=1..n} binomial(A035512(i) + Ki - 1, Ki).
|
|
EXAMPLE
|
Triangle begins:
1;
1, 1;
5, 1, 1;
83, 6, 1, 1;
5048, 88, 6, 1, 1;
1047008, 5146, 89, 6, 1, 1;
705422362, 1052471, 5151, 89, 6, 1, 1;
...
T(4,2) = 6 because there are 6 digraphs of order 4 with 2 strongly connected components.
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|