On Sequences in Cyclic Groups with Distinct Partial Sums - Costa, Simone; Della Fiore, Stefano; Ollis, M. A.; Rovner-Frydman, Sarah Z. (The Electronic Journal of Combinatorics, 2022)
A subset of an abelian group is sequenceable if there is an ordering (x_1, …, x_k) of its elements such that the partial sums (y_0, y_1, …, y_k), given by y_0 = 0 and y_i = _j=1^i x_j for 1 ≤ i ≤ k, are distinct, with the possible exception that we may have y_k = y_0 = 0. We demonstrate the sequenceability of subsets of size k of Z_n { 0 } when n = mt in many cases, including when m is either prime or has all prime factors larger than k! /2 for k ≤ 11 and t ≤ 5 and for k=12 and t ≤ 4. We obtain similar, but partial, results for 13 ≤ k ≤ 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.