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Number of nonisomorphic configurations of n triples in Steiner triple systems.
5

%I #14 Jan 28 2022 01:29:45

%S 1,2,5,16,56,282,1865,17100,207697,3180571

%N Number of nonisomorphic configurations of n triples in Steiner triple systems.

%C 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.

%C 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.

%C 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.

%C 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.

%D 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.

%D 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.

%H A. D. Forbes, M. J. Grannell and T. S. Griggs, <a href="http://ajc.maths.uq.edu.au/pdf/29/ajc_v29_p075.pdf">Configurations and trades in Steiner triple systems</a>, Australasian J. Combin. 29 (2004), 75-84.

%e The five configurations of 3 triples are

%e .

%e *---*---* *---*---*

%e *---*---* *---*---*

%e *---*---* \

%e *

%e \

%e * * *

%e / \ /

%e * * * *

%e / \ / /|

%e * * * |

%e / |

%e *---*---*---*---* * *

%e \ \ |

%e * * |

%e \ \|

%e * *

%Y Cf. A082790, A050913, A058790.

%K nonn,nice,more

%O 1,2

%A T. Forbes (anthony.d.forbes(AT)googlemail.com), May 24 2003