On Sequences in Cyclic Groups with Distinct Partial Sums

TL;DR

This paper investigates conditions under which subsets of cyclic groups can be ordered so that their partial sums are distinct, advancing understanding of sequenceability in abelian groups and proposing a broad conjecture.

Contribution

The authors establish new results on the sequenceability of subsets of cyclic groups, especially when the group order factors meet specific conditions, and propose a unifying conjecture.

Findings

Sequenceability holds for many subsets of inite cyclic groups under certain factorization conditions.

Progress on longstanding conjectures about the sequenceability of subsets not containing zero.

Partial results extend to subsets of size 13 to 15, indicating broader applicability.

Abstract

A subset of an abelian group is {\em sequenceable} if there is an ordering $(x_1, \ldots, x_k)$ of its elements such that the partial sums $(y_0, y_1, \ldots, y_k)$, given by $y_0 = 0$ and $y_i = \sum_{j=1}^i x_i$ for $1 \leq i \leq k$, 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 $k \leq 11$ and $t \leq 5$ and for $k=12$ and $t \leq 4$. We obtain similar, but partial, results for $13 \leq k \leq 15$. 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.