Concrete mathematics : a foundation for computer science

316 GENERATING FUNCTIONS
determine all the necessary coefficients:
n 0 5 10 15 20 25 30 35 40 45 50
P, 1 1 1 1 1 1 1 1 1 1 1
NTI 12345 6 7 8 9 10 11
D, 12 4 6 9 1216 25 36
Qn 1 13 49
G 1 50
The final value in the table gives us our answer, COO: There are exactly 50 ways
to leave a 50-cent tip. (Not counting the
How about a closed form for C,? Multiplying the equations together
gives us the compact expression
Option ofchar@ng
1 1 1 1 1
c = ----~~
1 --z 1 --5 1 -zz~o 1 -z25 1 -z50 1 (7.11)
but it’s not obvious how to get from here to the coefficient of zn. Fortunately
there is a way; we’ll return to this problem later in the chapter.
More elegant formulas arise if we consider the problem of giving change
when we live in a land that mints coins of every positive integer denomination
(0, 0, 0, . . . ) instead of just the five we allowed before. The corresponding
generating function is an infinite product of fractions,
1
(1 -z)(l -22)(1 -23)..1'
and the coefficient of 2” when these factors are fully multiplied out is called
p(n), the number of partitions of n. A partition of n is a representation of n
as a sum of positive integers, disregarding order. For example, there are seven
different partitions of 5, namely
5=4+1=3+2=3+11-1=2+2+1=2+1+1+1=1+1+1+1+1;
hence p(5) = 7. (Also p(2) =: 2, p(3) = 3, p(4) = 5, and p(6) = 11; it begins
to look as if p(n) is always a prime number. But p( 7) = 15, spoiling the
pattern.) There is no closed form for p(n), but the theory of partitions is a
fascinating branch of mathematics in which many remarkable discoveries have
been made. For example, Ramanujan proved that p(5n + 4) E 0 (mod 5),
p(7n + 5) s 0 (mod 7), and p(1 In + 6) E 0 (mod 1 l), by making ingenious
transformations of generating functions (see Andrews [ll, Chapter lo]).
the tip to a credit
card.)