OFFSET
1,2
COMMENTS
A configuration is a set of triples (of points) where every pair of points occurs in at most one triple. (A Steiner triple system is a set of triples where every pair occurs exactly once; thus configurations are often called partial Steiner triple systems.) The triples are also called blocks.
A 'generator' is 'a configuration where every point occurs in at least two blocks'. The term refers to the work of Horak, Phillips, Wallis & Yucas, who show that the number of occurrences of a configuration in a Steiner triple system is expressible as a linear form in the numbers of occurrences of the generators.
If we relax the restriction on the number of times a pair of points can occur in a configuration -- so that a configuration is just any multi-set of triples - then we get A050913.
If we allow a configuration to be any *set* of triples -- i.e., configurations with multiple occurrences of blocks are not allowed, but more than one pair is allowed -- then we get A058790.
REFERENCES
Mike Grannell and Terry Griggs, 'Configurations in Steiner triple systems', in Combinatorial Designs and their Applications, Chapman & Hall, CRC Research Notes in Math. 403 (1999), 103-126.
Horak, P., Phillips, N. K. C., Wallis, W. D. and Yucas, J. L., Counting frequencies of configurations in Steiner triple systems. Ars Combin. 46 (1997), 65-75.
LINKS
A. D. Forbes, M. J. Grannell and T. S. Griggs, Configurations and trades in Steiner triple systems, Australasian J. Combin. 29 (2004), 75-84.
EXAMPLE
The five configurations of 3 triples are
.
*---*---* *---*---*
*---*---* *---*---*
*---*---* \
*
\
* * *
/ \ /
* * * *
/ \ / /|
* * * |
/ |
*---*---*---*---* * *
\ \ |
* * |
\ \|
* *
CROSSREFS
KEYWORD
nonn,nice,more
AUTHOR
T. Forbes (anthony.d.forbes(AT)googlemail.com), May 24 2003
STATUS
approved