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### Power-sequence terraces for Zn where n is an odd prime power

Ian Anderson, D.A. Preece
2003 Discrete Mathematics
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.  ...

### SOME $\mathbb{Z}_{n-1}$ TERRACES FROM $\mathbb{Z}_{n}$ POWER-SEQUENCES, $n$ BEING AN ODD PRIME POWER

Ian Anderson, D. A. Preece
2007 Proceedings of the Edinburgh Mathematical Society
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.  ...

### Some da capo directed power-sequence Zn+1 terraces with n an odd prime power

Ian Anderson, D.A. Preece
2008 Discrete Mathematics
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.  ...

### A general approach to constructing power-sequence terraces for Zn

Ian Anderson, D.A. Preece
2008 Discrete Mathematics
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.  ...

### Some Zn+2 terraces from Zn power-sequences, n being an odd prime

Ian Anderson, D.A. Preece
2008 Discrete Mathematics
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.  ...

### Narcissistic half-and-half power-sequence terraces for Zn with n=pqt

Ian Anderson, D.A. Preece
2004 Discrete Mathematics
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.  ...

### SOME n−2 TERRACES FROM n POWER-SEQUENCES, n BEING AN ODD PRIME POWER

IAN ANDERSON, D. A. PREECE
2009 Glasgow Mathematical Journal
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.  ...

### Terraces for Small Groups [article]

M. A. Ollis
2016 arXiv   pre-print
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.  ...

### The spectrum of group-based Latin squares [article]

M. A. Ollis, Christopher R. Tripp
2018 arXiv   pre-print
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.  ...

### Some power-sequence terraces for Zpq with as few segments as possible

Ian Anderson, D.A. Preece
2005 Discrete Mathematics
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 .  ...

### The Spectrum of Group-Based Complete Latin Squares

M. A. Ollis, Christopher R. Tripp
2019 Electronic Journal of Combinatorics
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.  ...

### Sequenceable Groups and Related Topics

M. A. Ollis
2013 Electronic Journal of Combinatorics
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.  ...

### Combinatorially fruitful properties of 3⋅2−1 and 3⋅2−2 modulo p

Ian Anderson, D.A. Preece
2010 Discrete Mathematics
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.  ...

### On terraces for abelian groups

M.A. Ollis
2005 Discrete Mathematics
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.  ...

### Sectionable terraces and the (generalised) Oberwolfach problem

M.A. Ollis, D.A. Preece
2003 Discrete Mathematics
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.  ...
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