Constructions for Terraces and R-Sequencings, Including a Proof That Bailey's Conjecture Holds for Abelian Groups

Ollis, M. A.; Willmott, Devin

doi:10.1002/jcd.21396

Cited by 5 publications

(6 citation statements)

References 14 publications

(39 reference statements)

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“…This construction covers ∈ d r ( , ) { (3,9), (4,12), (5,15), (6,18), (7,21)}. Examples for the remaining 14 cases are given in Table 2, constructed along similar lines to the general construction of the previous paragraph.…”

Section: − 1… (3 + 1) 2 )mentioning

confidence: 99%

See 1 more Smart Citation

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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Section: − 1… (3 + 1) 2 )mentioning

confidence: 99%

“…Clearly, the directed rotational terrace determines the rotational sequencing; the reverse is also true. (Note that there are several different but equivalent definitions in the literature and the less descriptive terms “ R ‐sequencing” and “directed R ‐terraces” are often used instead; see . )…”

Section: The Constructive Approachmentioning

confidence: 99%

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

Hicks¹,

Ollis

Schmitt

2019

J of Combinatorial Designs

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“…(Note: there are several different but equivalent definitions in the literature; see, for example, [2,13,20,26]. We avoid the more common, but less descriptive, names "Rsequencing" and "directed R-terrace" for these concepts to bypass confusion with our S-sequencings and directed S-terraces.)…”

Section: Large Kmentioning

confidence: 99%

Sequences in Dihedral Groups with Distinct Partial Products

Ollis

2019

Preprint

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Given a subset S of the non-identity elements of the dihedral group of order 2m, is it possible to order the elements of S so that the partial products are distinct? This is equivalent to the sequenceability of the group when |S| = 2m − 1 and so it is known that the answer is yes in this case if and only if m > 4. We show that the answer is yes when |S| ≤ 9 and m is an odd prime other than 3, when |S| = 2m − 2 and m is even or prime, and when |S| = 2m − 2 for many instances of the problem when m is odd and composite. We also consider the problem in the more general setting of arbitrary non-abelian groups and discuss connections between this work and the concept of strong sequenceability.

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“…Clearly, the directed rotational terrace determines the rotational sequencing; the reverse is also true. (Note: there are several different but equivalent definitions in the literature and the less descriptive terms "R-sequencing" and "directed R-terraces" are often used instead; see, for example, [2,14,16,18]. )…”

Section: The Constructive Approachmentioning

confidence: 99%

Distinct Partial Sums in Cyclic Groups: Polynomial Method and Constructive Approaches

Hicks¹,

Ollis²,

Schmitt³

2018

Preprint

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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 = j i=1 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 non-constructive 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 2k − √ 8k in all but at most a bounded number of cases.

show abstract

Constructions for Terraces and R-Sequencings, Including a Proof That Bailey's Conjecture Holds for Abelian Groups

Cited by 5 publications

References 14 publications

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

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

Sequences in Dihedral Groups with Distinct Partial Products

Distinct Partial Sums in Cyclic Groups: Polynomial Method and Constructive Approaches

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