Combinatorics Through Guided Discovery, 2004a

Appendix C
Exponential Generating
Functions
C.1 Indicator Functions
When we introduced the idea of a generating function, we said that the formal sum
n∑
a i x i
i=0
may be thought of as a convenient way to keep track of the sequence a i . We then did
quite a few examples that showed how combinatorial properties of arrangements
counted by the coefficients in a generating function could be mirrored by algebraic
properties of the generating functions themselves. The monomials x i are called
indicator polynomials. (They indicate the position of the coefficient a i .) One
example of a generating function is given by
∞∑ ( ) n
(1 + x) n = x i .
i
Thus we say that (1 + x) n is the generating function for the binomial coefficients
( n ).
i
The notation tells us that we are assuming that only i varies in the sum on the
right, but that the equation holds for each fixed integer n. This is implicit when
we say that (1 + x) n is the generating function for ( n ),
i
because we haven’t written
i anywhere in (1 + x) n , so it is free to vary.
Another example of a generating function is given by
∞∑
x n = s(n, i)x i .
i=0
Thus we say that x n is the generating function for the Stirling numbers of the
first kind, s(n, i). There is a similar equation for Stirling numbers of the second
kind, namely
∞∑
x n = S(n, i)x i .
i=0
i=0
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