For n, m ∈ Z+, let f(n, m) count the number of partitions of n where the summands form a non increasing sequence of positive integers and no summand exceeds m. With n = 4 and m = 2, for example, we find that f(4, 2) = 3 because here we are concerned with the three partitions
4 = 2 + 2, 4 = 2+1 +1, 4 = l + l + l + l.
(a) Verify that for all n, m ∈ Z+,
f(n, m) = f(n - m, m) + f(n, m - 1).
(b) Write a computer program (or develop an algorithm) to compute f(n, m) for n, m ∈ Z+.
(c) Write a computer program (or develop an algorithm) to compute p(n), the number of partitions of a given positive integer n.