For W9
(7.48) each of Parts (a) and (g), do only Steps (i) and (ii) below: Find
the generating function H(x) for the sequence determined by the recurrence and
the initial conditions, as on p. 235. (We will not discuss the partial fractions
technique, which could then be used to find an explicit solution to the
recurrence.) Here is an added part to this exercise:
(g) hn = 7hn1 10hn2 for n 2 with h0 = 4 and h1 = 7.
(i) First find a rational function expression for H(x) whose numerator is
expressed in terms of h0, h1, , hk1.
(ii) Then use the specified initial conditions to rewrite the numerator as a
specific polynomial in x. (Consider photocopying your work for use with Step
(iii) of this exercise.)
(8.81) Suppose we have two identical 3, three identical 5, and two identical
7 stamps available. Use a generating function to find all of the postage amounts
which we can form using some of these stamps, and the number of different
ways each possible postage amount can be formed.
(8.82) Suppose we want to form a basket of fruit subject to the following
requirements: The number of Apples must be even, there between are 3 and
17 Bananas inclusive, and the number of Cantaloupes is a multiple of 5 and
nonzero. Further suppose that Apples cost 4, Bananas cost 5, and
Cantaloupes cost 9. Find an identity for the generating function that describes
how many baskets can be formed that have a total cost of n cents for n 0.
(8.83) Let k 1. Find an identity for the generating function for the counts
pk(n), n 0, of partitions of n into exactly k parts. (Hint: The diagrams
for the partitions counted by pk(n) have at least one column of length k. So
one approach would be to indicate how lecture's proof of the identity for the
generating function for pk(n) can be slightly modified to produce a slightly
different identity.)
(8.25) Remark: My proof of the identity presented in lecture for the
generating function for pk(n) tracked the numbers of columns of various
lengths in the Ferrers diagrams of the partitions. Hint: Imitate that proof, but
now track the numbers of rows of various lengths in the Ferrers diagrams of the
partitions as I did when proving the generating function identity for partitions
into odd parts.