International Journal of Combinatorics
Volume 2014 (2014), Article ID 593749, 4 pages
http://dx.doi.org/10.1155/2014/593749

Research Article

Necklaces, Self-Reciprocal Polynomials, and -Cycles

Umarin Pintoptang,1,2 Vichian Laohakosol,³ and Suton Tadee⁴

¹Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
²Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand
³Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand
⁴Department 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.

1. Introduction

1.1. Necklaces

Let , , and let be a prime power. Consider the set

of all necklaces (or seating arrangements) consisting of beads, each of which is colored with one of colors denoted by . Clearly, . Partition the colors into pairs with one extra color if is odd. Two colors in the same pair are called complementary, and if is odd the extra color is called self-complementary. The complement of color is so arranged as the color . Let

be the cyclic group of order ; the group acts on via rotating each bead -times. Let be the group which acts on via , the identity, preserving each color and replacing each color by its complement. Each equivalence class of elements in under the action of is called a -orbit. For a -orbit , the element is an equivalence class derived from through the action . A -orbit is called a self-complementary if it is invariant under ; that is, . The remaining -orbits fall into pairs called complementary pairs. A necklace in is called primitive if its -orbit has cardinality .

1.2. Self-Reciprocal, Irreducible, Monic Polynomials

Let denote the finite field of elements. The reciprocal polynomial of

is defined by

A polynomial is called self-reciprocal if .

The following characterization is well known.

Proposition 1 (see [1, page 275]). Let be irreducible and monic of degree . Then is self-reciprocal if and only if its set of roots (each of which is evidently nonzero) is closed under inversion (and so its degree must be even).

We are interested here in self-reciprocal, irreducible, monic (srim) polynomials in . Since there is only one first degree srim-polynomial, namely, , from now on we treat only srim-polynomials of even degree .

Let be a generator (primitive element) of the group and let be an irreducible monic polynomial of degree . It is well known [2, Theorem 2.14] that all distinct roots of are of the form

for some integer . Let

be its base expansion. For an irreducible, monic of degree , we associate a -orbit containing the necklace .

1.3. -Cycles

Let be a fixed positive integer relatively prime to . Let be distinct numbers chosen from . If

then we say that forms a -cycle with leading element , and call the length of this -cycle. The notion of -cycles was introduced by Wan in his book [3, page 203]. Since , where is the order of in (the multiplicative group of nonzero integers modulo ), it clearly follows that each -cycle always has a unique length which is the least positive integer for which . The concept of -cycles is important because of the following connections with irreducible polynomials in [3, Theorem 9.11].

Let be a primitive th root of unity (if the order of in is , then there exists a primitive th root of unity in ). If is a -cycle, then

is a monic irreducible factor of in .

Conversely, if is a monic irreducible factor of in , then all the roots of are powers of whose exponents form a -cycle.

We henceforth refer to these two facts as the cycle-polynomial correspondence.

2. Connection between -Cycles and Irreducible Polynomials

We start with a basic result.

Lemma 2. Let be all the (positive integer) divisors of .

(i)If is such that for some , then each -cycle with leading element has length .(ii)The polynomial constructed via (8) from a -cycle with leading element , , is a monic irreducible polynomial with , and .(iii)Each irreducible polynomial with , and is an irreducible factor of arising, through (8), from a -cycle with leading element , .

Proof. (i) Note from the definition of -cycle that a -cycle , with leading element and , has length if and only if divides but does not divide (if , only the first divisibility needs to be considered). This defining condition of is indeed the meaning of .
(ii)-(iii) From part (i), each -cycle with leading element , , has length . Through the cycle-polynomial correspondence, such a -cycle gives rise to a monic irreducible , , of degree and conversely.

By Lemma 2(i), a -cycle with leading element has length . Since , we have . From the observation that , we immediately obtain , and hence, a -cycle with leading element has length .

The case is of particular interest for it shows that -cycles are closely related to srim-polynomials.

Theorem 3. Let . Then each -cycle of length gives rise through the cycle-polynomial correspondence to a srim-polynomial in of degree and conversely.

Proof. Let be a -cycle of length . By Lemma 2, this -cycle gives rise, through (8), to a monic irreducible polynomial in of degree and conversely. There remains only to check that such polynomial is self-reciprocal, that is, to check that, for being a primitive th root of unity, the set is closed under inversion. Let . Putting , where , is such that . Since is also a primitive th root of unity, it suffices to treat only the case . The assertion that the set is closed under inversion is immediate from and the observation that for .

3. Connection between Necklaces and -Cycles

We start with a characterization of primitive necklaces. In the proof of our next theorem, we make use of a 1-1 correspondence between a necklace and a base representation of the form .

Theorem 4. Let be a primitive necklace in a -orbit. Then this -orbit is self-complementary if and only if there exists such that

Proof. The -orbit containing a primitive necklace is of the form

where all the necklaces in are distinct. From the 1-1 correspondence mentioned above, the base representations of the elements in are

From the definition, is self-complementary if and only if

where

If is self-complementary, then there exists such that ; that is,

yielding

which is (9). On the other hand, if (9) holds, reversing the above steps and appealing to the action of , we see that is self-complementary.

By the proof of Lemma 2(i), a -cycle with leading element has length . The case shows that -cycles are related to -orbits containing a primitive necklace.

Let be a -cycle of length . Then are distinct numbers in satisfying

Writing with respect to base representation, we have

and associate with the necklace . Working , we get

showing that the necklace associated with is . Proceeding in the same manner, we see that the necklace associated with is

Since are distinct, the necklaces associated with each are distinct, and so the necklace is primitive. The above steps can evidently be reversed and we have thus proved.

Theorem 5. For , there is a 1-1 correspondence between the set of -cycles of length and the set of -orbits containing a primitive necklace.

We proceed next to consider self-complemantary orbits and -cycles. Let be a -cycle of length so that

By Theorem 3, the polynomial

is a srim-polynomial of degree over , where is a primitive th root of unity. Let be a primitive th root of unity. Then we may take

Writing with respect to base representation, we get

Now working , we have either

provided (if , the above congruences can be adjusted accordingly), or

Letting

be the right-hand expression in (24) or (25), we see that

that is, the relation (9) holds with , . Proceeding in the same manner, we see that the digits in also satisfy the relation (9). Theorem 4 shows then that represents a primitive necklace in a self-complementary -orbit.

Conversely, given a self-complementary -orbit containing a primitive necklace whose first element is of the form (26), being self-complementary, we may assume without loss of generality that the digits in the first element are so arranged that the relations in (27) hold. Reversing the arguments, we get a -cycle of length of the form , where . We have thus proved.

Theorem 6. For , there is a 1-1 correspondence between the set of -cycles of length and the set of self-complementary -orbits containing a primitive necklace.

An immediate consequence of Theorem 6 is the following result of Miller [4] mentioned in the introduction.

Corollary 7. For , there is a 1-1 correspondence between the set of self-complementary -orbits containing a primitive necklace and the set of srim polynomials of degree .

Proof. From the cycle-polynomial correspondence, each -cycle of length gives rise, through (21), to a srim-polynomial of degree and conversely. The corollary follows at once from Theorem 6.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand, and by Kasetsart University and Faculty of Science through Research Cluster Fund (KU SciRCF, Cluster 4, no. 1).

References

J. L. Yucas and G. L. Mullen, “Self-reciprocal irreducible polynomials over finite fields,” Designs, Codes and Cryptography, vol. 33, no. 3, pp. 275–281, 2004.

R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, Cambridge, UK, 1997.

Z.-X. Wan, Lectures on Finite Fields and Galois Rings, World Scientific, Singapore, 2003.

R. L. Miller, “Necklaces, symmetries and self-reciprocal polynomials,” Discrete Mathematics, vol. 22, no. 1, pp. 25–33, 1978.