Let
(
G
,
+
) be an abelian group and consider a subset
A
⊆
G with
∣
A
∣
=
k. Given an ordering
(
a
1
,
…
,
a
k
) of the elements of
A, define its
partial sums by
s
0
=
0 and
s
j
=
∑
i
=
1
j
a
i for
1
≤
j
≤
k. We consider the following conjecture of Alspach: for any cyclic group
Z
n and any subset
A
⊆
Z
n
⧹
{
0
} with
s
k
≠
0, it is possible to find an ordering of the elements of
A such that no two of its partial sums
s
i and
s
j are equal for
0
≤
i
<
j
≤
k. We show that Alspach’s Conjecture holds for prime
n when
k
≥
n
−
3 and when
k
≤
10. The former result is by direct construction, the latter is nonconstructive and uses the polynomial method. We also use the polynomial method to show that for prime
n a sequence of length
k having distinct partial sums exists in any subset of
Z
n
⧹
{
0
} of size at least
2
k
−
8
k in all but at most a bounded number of cases.
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.
In 1980, about 20 years after sequenceable groups were introduced by Gordon to construct row-complete latin squares, Keedwell published a survey of all the available results concerning sequencings. This was updated (jointly with Dénes) in 1991 and a short overview, including results about complete mappings and R-sequencings, was given in the CRC Handbook of Combinatorial Designs in 1995. In Sections 1 and 2 we give a survey of the current situation concerning sequencings, including details of the most important constructions. In Section 3 we consider some concepts closely related to sequenceable groups: R-sequencings, harmonious groups, supersequenceable groups (also known as super P-groups), terraces and the Gordon game. We also look at constructions for row-complete latin squares that do not use sequencings.
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