Question

A necklace is a circular arrangement of beads which can be rotated and flipped without being considered a different arrangement. For example, there is only one necklace strung together from three beads where all beads have different colors.

a) How many different necklaces with N beads can be made with N distinct colors?

b) How many different necklaces can you string together if you have to use 5 beads and up to 2 colors?

c) Now suppose we can only rotate necklaces, but not flip them, so rotations are considered the same arrangement but flips are considered different. Can you count the different necklaces with N beads that can be made from a supply of 2 types of beads, given that N is prime and the 2 types need not both be used?

Answer 1

a) Obscrve that if we consider the string of bcads as a straight string then tho number of diffcrent strings would be N!. But then as rotation is considered to produce the same arrangement, hence we are counting the same arrangement N times. Also, as flipping also docsn't change the arrangement the same ar rangement is counted twicc. Hence the number of different strings that can be created from N distinct colours and N distinct beads is 2N b) Let us compute every single possibility of the strings, we will use 1 and 2 to denote the colors: Case I (Using only one color): There are only 2 possibilitics in this casc Case III (Using only one beads of a color): In this case there can be only 1 1 or 1 2. We will calculate for 1 2. Then the only choice is 21111. Hence tho number of possibility is 2. Case III (Using only two beads of a color): In this casc There can only be 2 1's or 2 2's in the string. We will calculate for 2 bcads labelled 2. Now obscrve that only two possibilitics are there: 22111 and 21211 (bccause of tho flipping). Hence the number of possible choices is 4 Considering all the cases, the total number of possible necklaces (2 + 2 + 4) - c) Let us consider the rotation group of the necklace, say G. Then it is gener- ated by some element, say Q 1L. G-(a . Now for a 2 colouring of the necklace it has to be invariant under Qk y k E . Observe that order of Qk- as N is a prime. Hence, by Burnside's lemma, the number of indistinguishable k, N