Ordering the Blocks of Designs

Abstract

In Chap. 3 we discussed the various ways to order subsets, partitions and permutations. Similarly, there are various ways to order the blocks of a design. A listing of the blocks of a design such that there is a minimal change between the content of consecutive blocks is a combinatorial Gray code. Note that the minimal change definition will depend on the design parameters. A listing of points such that the blocks of a design are induced by the consecutive subwords is a more structural way of defining minimal change. Universal cycles (Ucycles) are sequences that adhere to such a definition. Another rule for ordering blocks of designs that relates both to content and structure employs configurations. A configuration describes a pattern of intersection for a set of blocks. Unlike Gray codes and Ucycles, configuration ordering is a paradigm unique to designs and can include orderings which are neither Gray codes nor Ucycles. In this chapter we formally introduce the various ways to order the blocks of a design and illustrate them with examples. We include basic results on such orderings, including bounds and fundamental enumerations. This chapter is recommended both to the researcher introducing themselves to this subject, and to the researcher familiar with some previous literature on block design orderings who is interested in generalizations of previous orderings and new ordering definitions.