Counting Configurations in Designs

✍ Scribed by Robert A. Beezer

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 211 KB
• Volume: 96
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.2001.3185

✦ Synopsis

Given a t-(v, k, *) design, form all of the subsets of the set of blocks. Partition this collection of configurations according to isomorphism and consider the cardinalities of the resulting isomorphism classes. Generalizing previous results for regular graphs and Steiner triple systems, we give linear equations relating these cardinalities. For any fixed choice of t and k, the coefficients in these equations can be expressed as functions of v and * and so depend only on the design's parameters, and not its structure. This provides a characterization of the elements of a generating set for m-line configurations of an arbitrary design.

2001 Academic Press

Definition 1.1. The pair (V, B) is a t-(v, k, *) design if V is a set of v elements called points (or vertices) and B is a set of k element subsets of V called blocks (or lines) with the property that every t-element subset of V is a subset of exactly * blocks from B.