On Sequences in Cyclic Groups with Distinct Partial Sums

Abstract

A subset of an abelian group is {\em sequenceable} if there is an ordering $(x_1,\dots ,x_k)$ of its elements such that the partial sums $(y_0,y_1,\dots ,y_k)$, given by $y_0=0$ and $y_i={\sum }_{j=1}^i{x}_i$ for $1\le i\le k$, 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 ${\mathbb{Z}}_n\setminus \{0\}$ when $n=mt$ in many cases, including when $m$ is either prime or has all prime factors larger than $k!/2$ for $k\le 11$ and $t\le 5$ and for $k=12$ and $t\le 4$. We obtain similar, but partial, results for $13\le k\le 15$. 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.

19 pages, plus supporting tables and code