On Sequences in Cyclic Groups with Distinct Partial Sums

A subset of an abelian group is sequenceable if there is an ordering (x1, . . . , xk) of its elements such that the partial sums (y0, y1, . . . , yk), given by y0 = 0 and yi =Pij=1 xj for 1 6 i 6 k, are distinct, with the possible exception that we may have yk = y0 = 0. We demonstrate the sequenceability of subsets of size k of Zn \ {0} when n = mt in many cases, including when m is either prime or has all prime factors larger than k!/2 for k 6 11 and t 6 5 and for k = 12 and t 6 4. We obtain similar, but partial, results for 13 6 k 6 15. This represents progress on a variety of questions and conjectures in the literature concerning the sequenceability of subsets of abelian groups, which we combine and summarize into the conjecture that if a subset of an abelian group does not contain 0 then it is sequenceable.