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, be the entire set of all permutations of size n. (a) Find the cycle index Ps, (21, 22, 23, 24). (b) Explain why Ps, (k,,..., k) is the number of sequences #{(a,..., ak): a; 20, a + +ak = n} (c) Find a formula for Ps, (k..... k).