Sarvate – Beam Triple Systems

A Sarvate–Beam Triple System SB(v, 3) is a set V of v elements and a collection of 3–subsets of V such that each distinct pair of elements in V occurs in i blocks, for every i in the list 1, 2, . . . , ( v 2 ) . In this paper, we completely enumerate all Sarvate-Beam Triple Systems for v = 5 and v = 6. (In the case v = 5, we extend a previous result of R. Stanton [8].) 1 Sarvate–Beam Designs The present problem under consideration has its roots in papers published since 2007 by D. Sarvate and W. Beam, R. Stanton and others. In these papers, Sarvate and Beam introduced a new type of combinatorial object called an adesign. Definition 1. An adesign AD(v, k) is a set V of v elements and a collection of k–subsets of V (called blocks) such that each distinct pair of elements in V occurs in a different number of blocks. A strict adesign SAD(v, k) is an adesign such that exactly one pair of elements occurs i times for every i in the list 1, 2, . . . , ( v 2 ) . ∗Research supported by NSERC–RGPIN 250389–06 Definition 1 was given by Sarvate and Beam [3], although the term frequency was used by Dukes [2] to refer to the number of blocks containing each distinct pair of points from V . We note that this distinct frequency condition distinguishes an adesign from a balanced incomplete block design (BIBD). The following definition was also given by Sarvate and Beam [3]: Definition 2. An aPBD(v, K) is a set V of v elements and a collection of subsets of V such that every pair of distinct elements of V occurs a distinct number of times, and the size of any block is in K. The following definition appears in [1]: Definition 3. A strict aPBD SaPBD(v, K) is a set V of v elements and a collection of subsets of V such that every pair of distinct elements of V occurs exactly once from the list 1, 2, . . . , ( v 2 ) , and the size of any block is in K. The definition of “strict adesign” was renamed by Stanton [4] as a Sarvate–Beam design (or SB design). Stanton [4] also introduced the term “SB Triple System”, referring to a SAD(v, 3). He generalized his terminology in [6] to a “SB Quad System”, which is a SAD(v, 4). Dukes [2] improved the notation, and labels it SB(v, k). Thus, we have SB(v, 3) for SB Triple Systems and SB(v, 4) for SB Quad Systems. The notation can further be used to denote a SaPBD(v, K) by SB(v, K).