On Sequences in Cyclic Groups with Distinct Partial Sums

Abstract

A subset of an abelian group is 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 $y_i = \sum_{j=1}^i x_j$ 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.

Weak sequenceability in cyclic groups

“…Since a t-weakly sequenceable set A is also t ( − 1)-weakly sequenceable, we can suppose that t = 7. By [10] we know that each subset ⊆ ⧹ A {0} p of cardinality ≤ k 12 is sequenceable, therefore we can assume that ≥ k 13 and thus p > 13. We divide the proof considering two different ranges of k.…”

“…Other known results on this conjecture and related problems can be found in [7,10,12]. Inspired by the graph-theoretical interpretation, we propose the following weakening of the concept of sequenceability.…”

“…One of the most important tools in the construction of combinatorial designs of various kinds is given by the difference method and its variations. The research of new and efficient ways to exploit these methods leads to several interesting conjectures proposed by many authors, such as Archdeacon and Graham (and many others, see [7, 8, 10, 11, 15, 17]). Results of these conjectures have several implications in the constructions of combinatorial designs and graph decompositions and have applications to other combinatorial structures, such as the Heffter arrays (see, e.g., [6]).…”

A survey of Heffter arrays

“…the latter was obtained by computer verification). Recently, several other papers on this topic have been published, see [3,27,29,38,50,67].…”

“…We should note that most of the recent progress on this problem has been made by use of the polynomial method. This method relies on the non-vanishing corollary to the Combinatorial Nullstellensatz (see [2,27,29,50,67]). (2) k ≤ 12 when v = pq where p is prime and q ≤ 4 [29, 50];…”