Combinatorics Through Guided Discovery, 2004a

4.4. Supplementary Problems 91
Write down a recurrence for the total length a n of all the branches of the tree at
the end of growing season n. Find the general solution to your recurrence relation.
Assume that we begin with a one meter cutting of new wood (from the previous
year) which branches out and grows a total of two meters of new wood in the first
year. What will the total length of all the branches of the tree be at the end of n
years?
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8. (Relevant to Appendix C) We have some chairs which we are going to paint with
red, white, blue, green, yellow and purple paint. Suppose that we may paint any
number of chairs red or white, that we may paint at most one chair blue, at most
three chairs green, only an even number of chairs yellow, and only a multiple of
four chairs purple. In how many ways may we paint n chairs?
9. What is the generating function for the number of partitions of an integer in
which each part is used at most m times? Why is this also the generating function
for partitions in which consecutive parts (in a decreasing list representation) differ
by at most m and the smallest part is also at most m?