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Power-sequence terraces for Zn where n is an odd prime power
2003
Discrete Mathematics
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Many elegant families of such Zn terraces are constructed for values of n that are odd prime powers. ...
A power-sequence terrace for Zn is deÿned to be a terrace which can be partitioned into segments one of which contains merely the zero element of Zn, whilst each other segment is either (a) a sequence ...
Ollis (Queen Mary, University of London) for discussions that helped to stimulate the work reported in this paper. ...
doi:10.1016/s0012-365x(02)00459-4
fatcat:pvhksbkqfbdz3mrisjq2wxqvle
SOME $\mathbb{Z}_{n-1}$ TERRACES FROM $\mathbb{Z}_{n}$ POWER-SEQUENCES, $n$ BEING AN ODD PRIME POWER
2007
Proceedings of the Edinburgh Mathematical Society
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We now refine this idea to show that, for m = n − 1, where n is an odd prime power, there are many ways in which power-sequences in Zn can be used to arrange the elements of Zn \ {0} in a sequence of distinct ...
Our constructions provide terraces for Z n−1 for all prime powers n satisfying 0 < n < 300 except for n = 125, 127 and 257. ...
Ollis (Marlboro College, Marlboro, VT) for providing the stimulus for the work reported in this paper. ...
doi:10.1017/s0013091504000045
fatcat:3uq7imkcfffhtpsrbsn2dt4lpa
Some da capo directed power-sequence Zn+1 terraces with n an odd prime power
2008
Discrete Mathematics
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We now extend this idea by using power-sequences in Z n to produce some terraces for Z n+1 where n is an odd prime power satisfying n ≡ 1 or 3 (mod 8). ...
A terrace for Z n is a particular type of sequence formed from the n elements of Z n . ...
Z n+1 terraces with n = p 2 where p is an odd prime The following three theorems provide constructions for Z n+1 terraces with n = p 2 for certain odd primes. ...
doi:10.1016/j.disc.2006.11.033
fatcat:4luerrwwsvce5nocf77qdf76a4
A general approach to constructing power-sequence terraces for Zn
2008
Discrete Mathematics
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It also yields terraces for some groups Z n with n = p 2 where p is an odd prime, and for some Z n with n = pq where p and q are distinct primes greater than 3. ...
We now present a new general power-sequence approach that yields Z n terraces for all odd primes n less than 1000 except for n = 601. ...
Acknowledgement We are grateful to Dr Wilson Stothers (University of Glasgow) for informing us of Theorem 1.2. ...
doi:10.1016/j.disc.2007.07.051
fatcat:an3hs5q4m5btrgij6usf7v7wvi
Some Zn+2 terraces from Zn power-sequences, n being an odd prime
2008
Discrete Mathematics
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We now extend this idea by using power-sequences in Z n , where n is an odd prime, to obtain terraces for Z m where m = n + 2. ...
We provide Z n+2 terraces for all odd primes n satisfying 0 < n < 1000 except for n = 127, 601, 683. ...
Acknowledgement The authors warmly thank a referee who, having read the paper with very great care, made various helpful suggestions for correction and enhancement. ...
doi:10.1016/j.disc.2007.07.110
fatcat:elsrpuxiwvaalgts56hfiu7iyq
Narcissistic half-and-half power-sequence terraces for Zn with n=pqt
2004
Discrete Mathematics
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Constructions are provided for narcissistic half-and-half power-sequence terraces for Zn with n = pq t where p and q are distinct odd primes and t is a positive integer. ...
If n is odd, a Zn terrace (a1; a2; : : : ; an) is a narcissistic half-and-half terrace if ai − ai−1 = an+2−i − an+1−i for i = 2; 3; : : : ; (n + 1)=2. ...
Bailey (Queen Mary, University of London) for discussion seminal to the derivation of results in this paper, and to P.J. Cameron (Queen Mary, University of London) for insights into primitive -roots. ...
doi:10.1016/s0012-365x(03)00261-9
fatcat:t4vraf2vtbctlj7lkv35kihnaq
SOME n−2 TERRACES FROM n POWER-SEQUENCES, n BEING AN ODD PRIME POWER
2009
Glasgow Mathematical Journal
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We now adapt this idea by using power-sequences in ޚ n , where n is an odd prime power, to obtain terraces for ޚ m , where m = n − 2. ...
We also provide ޚ n−2 terraces for n = 3 r (r > 1) and for some values n = p 2 , where p is prime. 2000 Mathematics Subject Classification. ...
Anderson and Preece [2] [3] [4] [5] gave general constructions for 'power-sequence' terraces for ޚ m , where m is odd. ...
doi:10.1017/s0017089509990164
fatcat:4z5fnvqkljh2npxkeciedjxqzm
Terraces for Small Groups
[article]
2016
arXiv
pre-print
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We show that there is a sequenceable group for each odd order up to 555 at which there is a non-abelian group. ...
q_2 are prime powers with 3 ≤ q_1 ≤ 11 and 3 ≤ q_2 ≤ 8. ...
to one of A 7 , PSL(2, q) or PGL(2, q) for odd prime powers q > 3. ...
arXiv:1603.01496v1
fatcat:mxahhhtmc5erxm46zgmtu2xeie
The spectrum of group-based Latin squares
[article]
2018
arXiv
pre-print
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We construct sequencings for many groups that are a semi-direct product of an odd-order abelian group and a cyclic group of odd prime order. ...
It follows from these constructions that there is a group-based complete Latin square of order n if and only if n ∈{ 1,2,4} or there is a non-abelian group of order n. ...
We construct sequencings for some semi-direct products Z q ⋉ A where A is an abelian group of odd order and q is an odd prime, including all possible such groups when A is cyclic. ...
arXiv:1812.05526v1
fatcat:d5amr4aahng35pwpmmh4wgti64
Some power-sequence terraces for Zpq with as few segments as possible
2005
Discrete Mathematics
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If n = pq, where p and q are distinct odd primes, the minimum number of segments for such a terrace is 3+ (n), where (n) is the ratio (n)/ (n) of the number of units in Z n to the maximum order of a unit ...
A power-sequence terrace for Z n is a Z n terrace that can be partitioned into segments one of which contains merely the zero element of Z n whilst each other segment is either (a) a sequence of successive ...
London) for very helpful discussions about primitive -roots and the properties of the units of Z n . ...
doi:10.1016/j.disc.2004.08.020
fatcat:7jcqgwk7urdh5favqwkwkvlrda
The Spectrum of Group-Based Complete Latin Squares
2019
Electronic Journal of Combinatorics
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We construct sequencings for many groups that are a semi-direct product of an odd-order abelian group and a cyclic group of odd prime order. ...
It follows from these constructions that there is a group-based complete Latin square of order $n$ if and only if $n \in \{ 1,2,4\}$ or there is a non-abelian group of order $n$. ...
We construct sequencings for some semi-direct products Z q A where A is an abelian group of odd order and q is an odd prime, including all possible such groups when A is cyclic. ...
doi:10.37236/8542
fatcat:s7bcqahoxvfzxdsmantbetti3i
Sequenceable Groups and Related Topics
2013
Electronic Journal of Combinatorics
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We also look at constructions for row-complete latin squares that do not use sequencings. ...
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. ...
(ii) Dihedral groups D 2n where n is twice an odd prime. ...
doi:10.37236/30
fatcat:kbspizkvlrfytperbtf3niarmu
Combinatorially fruitful properties of 3⋅2−1 and 3⋅2−2 modulo p
2010
Discrete Mathematics
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Searches for such constructions have revealed that, if we wish to produce terraces for Z p−1 and Z p+1 where p is an odd prime, then use can be made of sequences of powers, modulo p, of a and b where a ...
Write a ≡ 3 · 2 −1 and b ≡ 3 · 2 −2 (mod p) where p is an odd prime. Let c be a value that is congruent (mod p) to either a or b. ...
Let p be an odd prime; let a = (p + 3)/2 and q = (p − 1)/2. ...
doi:10.1016/j.disc.2008.09.046
fatcat:wuteglrp4zh4zmj2vghufk2trm
On terraces for abelian groups
2005
Discrete Mathematics
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We adapt a construction for R-sequencings to give new terraces for abelian groups. This enables the construction of terraces for all groups of the form Z k 2 ×Z t where k ≥ 4 and t > 5 is odd. ...
We also present an extendable terrace for Z 3 2 × Z 5 . ...
Theorem 3 [3, 4] Let G be a group with a normal subgroup N. If N has odd order and G/N is terraced then G is terraced. Alternatively, if N has odd index and N is terraced then G is terraced. ...
doi:10.1016/j.disc.2005.07.007
fatcat:yzetgxyvpvepxjljofcre37afi
Sectionable terraces and the (generalised) Oberwolfach problem
2003
Discrete Mathematics
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It was known in 1892, though di erent terminology was then used, that a directed terrace with a symmetric sequencing for the cyclic group of order 2n can be used to solve OP(2n + 1). ...
We show how terraces with special properties can be used to solve OP(2; l1; l2) and OP(l1; l1; l2) for a wide selection of values of l1, l2 and v. ...
The other essential ingredient of that proof is the following: Example 12. If n is odd then the LWW terrace for Z n is an st-terrace. Theorem 11. ...
doi:10.1016/s0012-365x(02)00822-1
fatcat:rbtdw22bavb2ri2zao4cruwoaq
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