Necklaces, self-reciprocal polynomials, and \(q\)-cycles
type: Item
instance of: scholarly article
title: Necklaces, self-reciprocal polynomials, and \(q\)-cycles
zbMATH Open document ID: 1414.11041
DOI: 10.1155/2014/593749
author: Umarin Pintoptang
author: Vichian Laohakosol
author: Suton Tadee
published in: International Journal of Combinatorics
publication date: 2014-11-17T00:00:00Z
review text: Summary: Let \(n\ge 2\) be a positive integer and \(q\) a prime power. Consider necklaces consisting of \(n\) beads, each of which has one of the given \(q\) colors. A primitive \(C_n\)-orbit is an equivalence class of \(n\) necklaces closed under rotation. A \(C_n\)-orbit is self-complementary when it is closed under an assigned color matching. In the work of \textit{R. L. Miller} [Discrete Math. 22, 25--33 (1978; Zbl 0395.05004)], it is shown that there is a 1-1 correspondence between the set of primitive, self-complementary \(C_n\)-orbits and that of self-reciprocal irreducible monic (srim) polynomials of degree \(n\). Let \(N\) be a positive integer relatively prime to \(q\). A \(q\)-cycle \(\bmod N\) is a finite sequence of nonnegative integers closed under multiplication by \(q\). In the work of \textit{Z.-X. Wan} [Lectures on finite fields and Galois rings. River Edge, NJ: World Scientific (2003; Zbl 1028.11072)], it is shown that \(q\)-cycles \(\bmod N\) are closely related to monic irreducible divisors of \(x^N-1\in \mathbb F_q[x]\). Here, we show that: (1) \(q\)-cycles can be used to obtain information about srim polynomials; (2) there are correspondences among certain \(q\)-cycles and \(C_n\)-orbits; (3) there are alternative proofs of Miller's results in the work of Miller (1978, loc. cit.) based on the use of \(q\)-cycles.
Mathematics Subject Classification ID: 11C08
Mathematics Subject Classification ID: 05C31
zbMATH DE Number: 6370094
zbMATH Keywords: correspondence
zbMATH Keywords: set of primitive self-complementary orbits
zbMATH Keywords: set of self-reciprocal irreducible monic polynomials of degree n
zbMATH Keywords: q-cycles
Wikidata QID: Q59043555
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2014/593749
OpenAlex ID: W2067590569
cites work: Self-reciprocal irreducible polynomials over finite fields
cites work: Q3509721
cites work: Q4435447
cites work: Necklaces, symmetries and self-reciprocal polynomials
description: scientific article; zbMATH DE number 6370094