1. For example,pi2 is (a b c d)

2. pi2 and pi3

3. 1/4[2**64 + 2*2**16 + 2**32]

4. n even: 1/2(3*2^{n-1}); n odd 1/2(3*2^{n-1} + 3*2^{(n-1)/2)})

5. 1/9[3**N + 5*3 + 2*3**3]

6. In the style of burnside’s lemma pick a transformation and look for subsets that are invariant under the transformation. Look at the transformation cycle structure. For a subset to be invariant, every cycle must be uniform with respect to subset: all elements in the subset, or all not in the subset. If there are -say - k cycles in all in the transformation, then there are 2**k ways of building a subset (and its complement) to yield an invariant.

7. a): 1/6[(b+w)^{6} +(b^{6} + w^{6})+2(b^{3} + w^{3})**2 + (b^{2} + w^{2})**3] ... b) 1/9[(b+w)^{9} +6(b^{9} + w^{9})+2(b^{3} + w^{3})**3]

8. for no two colors adjacent, you must have colors in the pattern 2-2-1. 3-1-1, for example, will put two colors next to each other. So generate the pat inventory and select the 2-2-1 patterns, and distinguish between the consecutive ones and the non- consecutive ones. for each pattern there's only one, so: b^{2}w^{2}r + b^{2}wr^{2} + bw^{2}r^{2 }

-- RobbieMoll - 2012-05-07

Topic revision: r3 - 2014-12-04 - RobbieMoll

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