Some narcissistic power-sequence Z(n+1) terraces with n an odd prime power

Anderson, I. and Preece, D.A. (2010) Some narcissistic power-sequence Z(n+1) terraces with n an odd prime power. Ars Combinatoria, 97A, pp. 33-57.

Abstract

A terrace for Z(m), is an arrangement (a(1), a(2), ... , a(m)) of the m elements of Z(m) such that the sets of differences a(i+1) - a(i) and a(i) - a(i+1) (i = 1, 2, ..., m - 1) between them contain each element of Z(m) \ {0} exactly twice. For m odd, many procedures are available for constructing power-sequence terraces for Z(m); each such terrace may be partitioned into segments one of which contains merely the zero element of Z(m) whereas each other segment is either (a) a sequence of successive powers of a non-zero element of Z(m) or (b) such a sequence multiplied throughout by a constant. For n an odd prime power satisfying n equivalent to 1 or 3 (mod 8), this idea has previously been extended by using power-sequences in Z(n) to produce some Z(m) terraces (a(1), a(2), ... , a(m)) where m = n + 1 = 2 mu, with a(i+1) - a(i) = -(a(i+1+mu) - a(i+mu)) for all i is an element of [1, mu-1]. Each of these "da capo directed terraces" consists of a sequence of segments, one containing just the element 0 and another just containing the element n, the remaining segments each being of type (a) or (b) above with each of its distinct entries x from Z(n) \ {0} evaluated so that 1 <= x <= n - 1. Now, for many odd prime powers n satisfying n equivalent to 1 (mod 4), we similarly produce narcissistic terraces for Z(n+1); these have a(i+1) - a(i) = a(m-i+1) - a(m-i) for all i is an element of [1, mu-1].

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Anderson, Dr Ian
• Authors: Anderson, I., and Preece, D.A.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Ars Combinatoria
• ISSN: 0381-7032