Necklace

In the technical Combinatorial sense, an $a$-ary necklace $N(n,a)$ of length $n$ is a string of $n$ characters, each of $a$ possible types. Rotation is ignored, in the sense that $b_1 b_2 \ldots b_n$ is equivalent to $b_k b_{k+1} \cdots b_n b_1 b_2 \cdots b_{k-1}$ for any $k$, but reversal of strings is respected. Necklaces therefore correspond to circular collections of beads in which the Fixed necklace may not be picked up out of the Plane (so that opposite orientations are not considered equivalent).

The number of distinct Free necklaces $N'(n,a)$ of $n$ beads, each of $a$ possible colors, in which opposite orientations (Mirror Images) are regarded as equivalent (so the necklace can be picked up out of the Plane and flipped over) can be found as follows. Find the Divisors of $n$ and label them $d_1\equiv 1$, $d_2$, ..., $d_\nu(n)\equiv n$ where $\nu(n)$ is the number of Divisors of $n$. Then

$$N'(n,a)={1\over 2n}\cases{ \sum_{i=1}^{\nu(n)} \phi(d_i)a^{... ...\over 2}}n(1+a)a^{n/2}\cr \quad{\rm for\ } n {\rm\ even},\cr}$$

where $\phi(x)$ is the Totient Function. For $a=2$ and $n=p$ an Odd Prime, this simplifies to

$$N'(p,2)={2^{p-1}-1\over p}+2^{(p-1)/2}+1.$$

A table of the first few numbers of necklaces for $a=2$ and $a=3$ follows. Note that $N(n,2)$ is larger than $N'(n,2)$ for $n\geq 6$. For $n=6$, the necklace 110100 is inequivalent to its Mirror Image 0110100, accounting for the difference of 1 between $N(6,2)$ and $N'(6,2)$. Similarly, the two necklaces 0010110 and 0101110 are inequivalent to their reversals, accounting for the difference of 2 between $N(7,2)$ and $N'(7,2)$.

$n$	$N(n,2)$	$N'(n,2)$	$N'(n,3)$
Sloane	Sloane's A000031	Sloane's A000029	Sloane's A027671
1	2	2	3
2	3	3	6
3	4	4	10
4	6	6	21
5	8	8	39
6	14	13	92
7	20	18	198
8	36	30	498
9	60	46	1219
10	108	78	3210
11	188	126	8418
12	352	224	22913
13	632	380	62415
14	1182	687	173088
15	2192	1224	481598

Ball and Coxeter (1987) consider the problem of finding the number of distinct arrangements of $n$ people in a ring such that no person has the same two neighbors two or more times. For 8 people, there are 21 such arrangements.

See also Antoine's Necklace, de Bruijn Sequence, Fixed, Free, Irreducible Polynomial, Josephus Problem, Lyndon Word

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 49-50, 1987.

Dudeney, H. E. Problem 275 in 536 Puzzles & Curious Problems. New York: Scribner, 1967.

Gardner, M. Martin Gardner's New Mathematical Diversions from Scientific American. New York: Simon and Schuster, pp. 240-246, 1966.

Gilbert, E. N. and Riordan, J. ``Symmetry Types of Periodic Sequences.'' Illinois J. Math. 5, 657-665, 1961.

Riordan, J. ``The Combinatorial Significance of a Theorem of Pólya.'' J. SIAM 4, 232-234, 1957.

Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, p. 162, 1980.

Ruskey, F. ``Information on Necklaces, Lyndon Words, de Bruijn Sequences.'' http://sue.csc.uvic.ca/~cos/inf/neck/NecklaceInfo.html.

Sloane, N. J. A. Sequences A000029/M0563, A000031/M0564, A001869/M3860, and A027671 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.