MATH 310 FALL 2003 Combinatorial Problem Solving Lecture
MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 29, Monday, November 10
Exponential Generating Function n n 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) = a 0 + a 1 x + a 2 x 2/2! +. . . + arxr/r! +. . .
Example 1 Find the exponential generating function for ar, the number of r arrangements without repetitions of n objects. n Answer: g(x) = (1 + x)n = 1 + P(n, 1)x/1! +. . . + P(n, r)xr/r! +. . . + P(n, n)xn/n!. 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. n Answer: (x 2/2! + x 3/3! + x 4/4! + x 5/5!)4. n
Example 3 n n 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 + x 2/2! + x 3/3! +. . . )3 = (ex – 1)3. (1 + x 2/2! + x 4/4! +. . . )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. n Answer: enx. ar = nr. n
Example 5 Find the number of ways to place 25 people into three rooms with at least one person in each room. n Answer: g(x) = (ex – 1)3 = e 3 x – 3 e 2 x + 3 ex – 1. n a 25 = 325 – 3 £ 225 + 3. n
Example 6 Find the number of r-digit quaternary sequences (digits 0, 1, 2, 3) with an even number of 0 s and an odd number of 1 s. n Answer: n n g(x) = (1/2)(ex + e-x)(1/2)(ex – e-x)exex = (1/4)(e 2 x – e-2 x)e 2 x. ar = 4 r-1.
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