International Journal of Combinatorics

Volume 2014, Article ID 593749, 4 pages

http://dx.doi.org/10.1155/2014/593749

## Necklaces, Self-Reciprocal Polynomials, and -Cycles

^{1}Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand^{2}Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand^{3}Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand^{4}Department of Mathematics and Statistics, Faculty of Science and Technology, Thepsatri Rajabhat University, Lopburi 15000, Thailand

Received 16 June 2014; Revised 16 October 2014; Accepted 16 October 2014; Published 9 November 2014

Academic Editor: Toufik Mansour

Copyright © 2014 Umarin Pintoptang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Let be a positive integer and a prime power. Consider necklaces consisting of beads, each of which has one of the given colors. A primitive -orbit is an equivalence class of necklaces closed under rotation. A -orbit is self-complementary when it is closed under an assigned color matching. In the work of Miller (1978), it is shown that there is a 1-1 correspondence between the set of primitive, self-complementary -orbits and that of self-reciprocal irreducible monic (srim) polynomials of degree . Let be a positive integer relatively prime to . A -cycle mod is a finite sequence of nonnegative integers closed under multiplication by . In the work of Wan (2003), it is shown that -cycles mod are closely related to monic irreducible divisors of . Here, we show that: (1) -cycles can be used to obtain information about srim polynomials; (2) there are correspondences among certain -cycles and -orbits; (3) there are alternative proofs of Miller's results in the work of Miller (1978) based on the use of -cycles.