Rattles with beads or necklace with beads?

Question

I came across this problem in a book called limits, sequences combinations great book for intro to combinatorics .

A rattle consists of a ring with $3$ white beads and $7$ red ones strung on it. Some rattles seemingly different can be made identical by arranging the rings and moving the beads in a suitable manner (rotation or flipping). Find the number of different rattles .

I of course thought polya enumeration on this one but was thinking it can be done case by case without being too messy . Can anyone help? Also this is essentially the same problem as a necklace with $n$ beads, $k$ colors is it not ?

Answer 1

A hand count is not hard. You have to find a way to organize it so you count each configuration only once. We get one configuration for each weak partition of $7$ into $3$ parts. We get $$7,0,0\\6,1,0\\5,2,0\\5,1,1\\4,3,0\\4,2,1\\3,3,1\\3,2,2$$ for eight possibilities. Clearly these are all distinct. You need to convince yourself that there are not two configurations for a partition, but rotation and flipping give six configurations, which matches the number of orders of a partition.

Answer 2

Let me just observe that with $(3,10) = 1$ and $(7,10) = 1$ Polya will be very simple in this case. The OEIS uses necklace for cyclic symmetry and bracelet for dihedral, so we have a bracelet here. The cycle index is from first principles

$$Z(D_{10}) = \frac{1}{20} \sum_{d|10} \varphi(d) a_d^{10/d} + \frac{1}{4} a_2^5 + \frac{1}{4} a_1^2 a_2^4.$$

With the two colors our answer is given by

$$[W^3 R^7] \frac{1}{20} \sum_{d|10} \varphi(d) (W^d+R^d)^{10/d} \\ + [W^3 R^7] \frac{1}{4} (W^2+R^2)^5 \\ + [W^3 R^7] \frac{1}{4} (W+R)^2 (W^2+R^2)^4.$$

From the first term only $d=1$ contributes, the second does not contribute at all and from the third the first term must contribute $WR.$ We get

$$[W^3 R^7] \frac{1}{20} (W+R)^{10} + [W^2 R^6] \frac{1}{2} (W^2+R^2)^4 \\ = [W^3 R^7] \frac{1}{20} (W+R)^{10} + [W^1 R^3] \frac{1}{2} (W+R)^4.$$

The desired count is thus

$$\frac{1}{20} {10\choose 3} + \frac{1}{2} {4\choose 1} = 8.$$