Regular Perfect Systems of Sets of Iterated Differences

Fors ≥ 2, a set {a(i,j):1 ≤ j ≤ s + 1 − i ≤ s} wherea(1,j), 1 ≤ j ≤ s, are some prescribed integers anda(i + 1,j) = |a(i,j) − a(i,j + 1)|, for 1 ≤ i < sand 1 ≤ j ≤ s − i, is called a set of iterated differences. Such a set has sizesand is full if it containss(s + 1)/2 distinct integers. Kreweras and Loeb suggested the problem of partitioning a run ofms(s + 1)/2 integers starting withcintomfull sets of iterated differences of sizes. We show that necessary conditions for this are that 2 ≤ s ≤ 9, and thatmbe sufficiently large in comparison withc. In particular, a single set of iterated differences of sizescontains the integers 1 tos(s + 1)/2 (inclusive) iff 2 ≤ s ≤ 5. We also discuss connections between this problem and the theory of perfect systems of difference sets.