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

MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 29, Monday, November 10. Exponential Generating Function. An exponential generating function g(x) for a r , [the number of arrangements of n objects] is a function with the power series expansion:

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

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

  2. 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! + ...

  3. 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!.

  4. 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.

  5. 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

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

  7. 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.

  8. 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.

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