1. List all the inequivalent ways to color the vertices of the graph from the
last problem set with 2 colors. How many are there with 3 colors?
2. Let $G = S_n$ be the entire set of all permutations of size $n$.
(a) Find the cycle index $P_{S_4}(z_1, z_2, z_3, z_4)$.
(b) Explain why $P_{S_n}(k, \dots, k)$ is the number of sequences
$\#\{(a_1, \dots, a_k) : a_i \ge 0, a_1 + \dots + a_k = n\}$
(c) Find a formula for $P_{S_n}(k, \dots, k)$.
