cp's OEIS Frontend

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

%I A082789 #14 Jan 28 2022 01:29:45
%S A082789 1,2,5,16,56,282,1865,17100,207697,3180571
%N A082789 Number of nonisomorphic configurations of n triples in Steiner triple systems.
%C A082789 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 A082789 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 A082789 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 A082789 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 A082789 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 A082789 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 A082789 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 A082789 The five configurations of 3 triples are
%e A082789 .
%e A082789      *---*---*       *---*---*
%e A082789      *---*---*       *---*---*
%e A082789      *---*---*        \
%e A082789                        *
%e A082789                         \
%e A082789           *       *      *
%e A082789          / \     /
%e A082789         *   *   *                *
%e A082789        /     \ /                /|
%e A082789       *       *                * |
%e A082789                               /  |
%e A082789      *---*---*---*---*       *   *
%e A082789               \               \  |
%e A082789                *               * |
%e A082789                 \               \|
%e A082789                  *               *
%Y A082789 Cf. A082790, A050913, A058790.
%K A082789 nonn,nice,more
%O A082789 1,2
%A A082789 T. Forbes (anthony.d.forbes(AT)googlemail.com), May 24 2003