Exponential Generating Function

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MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 29, Monday, November 10

Exponential Generating Function An exponential generating function g(x) for ar, [the number of arrangements of n objects] is a function with the power series expansion: g(x) = a0 + a1x + a2 x2/2! + ... + arxr/r! + ...

Example 1 Find the exponential generating function for ar, the number of r arrangements without repetitions of n objects. Answer: g(x) = (1 + x)n = 1 + P(n,1)x/1! + ... + P(n,r)xr/r! + ... + P(n,n)xn/n!.

Example 2 Find the exponential generating function for ar, the number of different arrangements of r objects chosen from four different types of objects with each type of objects appearing at least two and no more than five times. Answer: (x2/2! + x3/3! + x4/4! + x5/5!)4.

Example 3 Find the exponential generating function for the number of ways to place r distinct people into three different rooms with at least one person in each room. (Repeat with an even number of people in each room: Answer: (x + x2/2! + x3/3! + ...)3 = (ex – 1)3. (1 + x2/2! + x4/4! + ...)3 = [(ex + e -x)/2]3

Example 4 Find the number of different r arrangements of objects chosen from unlimited supplies of n types of objects. Answer: enx. ar = nr.

Example 5 Find the number of ways to place 25 people into three rooms with at least one person in each room. Answer: g(x) = (ex – 1)3 = e3x – 3e2x + 3ex – 1. a25 = 325 – 3 £ 225 + 3.

Example 6 Find the number of r-digit quaternary sequences (digits 0,1,2,3) with an even number of 0s and an odd number of 1s. Answer: g(x) = (1/2)(ex + e-x)(1/2)(ex – e-x)exex = (1/4)(e2x – e-2x)e2x. ar = 4r-1.