Regular Perfect Systems of Sets of Iterated Differences

✍ Scribed by G.M. Hamilton; I.T. Roberts; D.G. Rogers

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 192 KB
• Volume: 19
• Category: Article
• ISSN: 0195-6698
• DOI: 10.1006/eujc.1997.0144

✦ Synopsis

For s ≥ 2, a set {a(i, j) : 1 ≤ j ≤ s + 1i ≤ s} where a(1, j), 1 ≤ j ≤ s, are some prescribed integers and a(i + 1, j) = |a(i, j)a(i, j + 1)|, for 1 ≤ i < s and 1 ≤ j ≤ si, is called a set of iterated differences. Such a set has size s and is full if it contains s(s + 1)/2 distinct integers. Kreweras and Loeb suggested the problem of partitioning a run of ms(s + 1)/2 integers starting with c into m full sets of iterated differences of size s. We show that necessary conditions for this are that 2 ≤ s ≤ 9, and that m be sufficiently large in comparison with c. In particular, a single set of iterated differences of size s contains the integers 1 to s(s + 1)/2 (inclusive) iff 2 ≤ s ≤ 5. We also discuss connections between this problem and the theory of perfect systems of difference sets.