In the technical Combinatorial sense, an -ary necklace of length is a string of characters, each of possible types. Rotation is ignored, in the sense that is equivalent to for any , 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 of beads, each of 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 and label them , , ...,
where is the number of Divisors of . Then

where is the Totient Function. For and an Odd Prime, this simplifies to

A table of the first few numbers of necklaces for and follows. Note that is larger than for . For , the necklace 110100 is inequivalent to its Mirror Image 0110100, accounting for the difference of 1 between and . Similarly, the two necklaces 0010110 and 0101110 are inequivalent to their reversals, accounting for the difference of 2 between and .

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 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.

**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.

© 1996-9

1999-05-25