A problem on partial sums in abelian groups

Costa, Simone; Morini, Fiorenza; Pasotti, Anita; Pellegrini, Marco Antonio

doi:10.1016/j.disc.2017.11.013

Cited by 36 publications

(72 citation statements)

References 16 publications

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“…Similarly, the constructive methods of Section give a partial result for composite

n

. The construction method in the proof of Theorem (in conjunction with the computational results for small groups of ) is sufficient to give an ordering for

S = Z_{n} ⧹ {0, x, y}

for arbitrary

n

provided that at least one of

x

and

y

is coprime to

n

.…”

Section: Discussionmentioning

confidence: 99%

“…In , a greedy algorithm approach to Problem is used. The preceding discussion allows us to slightly improve this result in the case where

n

is prime.…”

Section: Sequences Taken From Given Subsetsmentioning

confidence: 99%

“…Costa et al proved Conjecture when

∣ A ∣ \leq 9

and verified by computer that it holds for abelian groups of order at most 27. They stated that the conjecture arose during the study of Heffter systems (see , Section 2).…”

Section: Introductionmentioning

confidence: 99%

“…In proving Conjecture for just the case

∣ A ∣ = 6

, Archdeacon et al break the proof into

(∣ A ∣ / 2) + 1

cases depending upon the number of pairs

{x, - x}

contained in

A

and each of these cases breaks into between three and nine subcases. There is a similarly large amount of casework done in for Conjecture for

∣ A ∣ = 9

. In each paper, the proofs are constructive.…”

Section: Introductionmentioning

confidence: 99%

“…If Conjecture 1.2 is not true, then we might consider the following question, which was posed in [8]. [12]). For any abelian group G ( , +) and any k-subset…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Distinct partial sums in cyclic groups: polynomial method and constructive approaches

Hicks¹,

Ollis

Schmitt

2019

J of Combinatorial Designs

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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.

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“…Similarly, the constructive methods of Section give a partial result for composite

n

. The construction method in the proof of Theorem (in conjunction with the computational results for small groups of ) is sufficient to give an ordering for

S = Z_{n} ⧹ {0, x, y}

for arbitrary

n

provided that at least one of

x

and

y

is coprime to

n

.…”

Section: Discussionmentioning

confidence: 99%

“…In , a greedy algorithm approach to Problem is used. The preceding discussion allows us to slightly improve this result in the case where

n

is prime.…”

Section: Sequences Taken From Given Subsetsmentioning

confidence: 99%

“…Costa et al proved Conjecture when

∣ A ∣ \leq 9

and verified by computer that it holds for abelian groups of order at most 27. They stated that the conjecture arose during the study of Heffter systems (see , Section 2).…”

Section: Introductionmentioning

confidence: 99%

“…In proving Conjecture for just the case

∣ A ∣ = 6

, Archdeacon et al break the proof into

(∣ A ∣ / 2) + 1

cases depending upon the number of pairs

{x, - x}

contained in

A

and each of these cases breaks into between three and nine subcases. There is a similarly large amount of casework done in for Conjecture for

∣ A ∣ = 9

. In each paper, the proofs are constructive.…”

Section: Introductionmentioning

confidence: 99%

“…If Conjecture 1.2 is not true, then we might consider the following question, which was posed in [8]. [12]). For any abelian group G ( , +) and any k-subset…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Distinct partial sums in cyclic groups: polynomial method and constructive approaches

Hicks¹,

Ollis

Schmitt

2019

J of Combinatorial Designs

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show abstract

A generalization of Heffter arrays

Costa

Morini

Pasotti

et al. 2019

J of Combinatorial Designs

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In this paper we define a new class of partially filled arrays, called relative Heffter arrays, that are a generalization of the Heffter arrays introduced by Archdeacon in 2015. Let v = 2nk + t be a positive integer, where t divides 2nk, and let J be the subgroup of Zv of order t. A Ht(m, n; s, k) Heffter array over Zv relative to J is an m × n partially filled array with elements in Zv such that: (i) each row contains s filled cells and each column contains k filled cells; (ii) for every x ∈ Z 2nk+t \ J, either x or −x appears in the array; (iii) the elements in every row and column sum to 0. In particular, here we study the existence for t = k of integer (i.e. the entries are chosen in ± 1, . . . , 2nk+t 2 and the sums are zero in Z) square relative Heffter arrays.

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