A082789 Number of nonisomorphic configurations of n triples in Steiner triple systems.

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

Offset: 1

Author

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

Keywords

• nonn
• nice
• more

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.

Examples

			The five configurations of 3 triples are
.
     *---*---*       *---*---*
     *---*---*       *---*---*
     *---*---*        \
                       *
                        \
          *       *      *
         / \     /
        *   *   *                *
       /     \ /                /|
      *       *                * |
                              /  |
     *---*---*---*---*       *   *
              \               \  |
               *               * |
                \               \|
                 *               *
		

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.

Crossrefs

Cf. A082790, A050913, A058790.