Discrete Math by R.S. Chang, Dept. CSIE, NDHU1 Chapter 9

Discrete Math by R.S. Chang, Dept. CSIE, NDHU2 Chapter Introductory Examples Ex. 9.1 Find the number of integer solutions to

Discrete Math by R.S. Chang, Dept. CSIE, NDHU3 Chapter Introductory Examples Ex. 9.3 How many integer solutions are there for the equation

Discrete Math by R.S. Chang, Dept. CSIE, NDHU4 Chapter Definitions and Examples: Calculational Techniques

Discrete Math by R.S. Chang, Dept. CSIE, NDHU5 Chapter Definitions and Examples: Calculational Techniques

Discrete Math by R.S. Chang, Dept. CSIE, NDHU6 Chapter Definitions and Examples: Calculational Techniques Ex. 9.5 (continued)

Discrete Math by R.S. Chang, Dept. CSIE, NDHU7 Chapter Definitions and Examples: Calculational Techniques

Discrete Math by R.S. Chang, Dept. CSIE, NDHU8 Chapter Definitions and Examples: Calculational Techniques Extension of binomial coefficient

Discrete Math by R.S. Chang, Dept. CSIE, NDHU9 Chapter Definitions and Examples: Calculational Techniques

Discrete Math by R.S. Chang, Dept. CSIE, NDHU10 Chapter Definitions and Examples: Calculational Techniques Ex In how many ways can we select, with repetitions allowed, r objects from n distinct objects?

Discrete Math by R.S. Chang, Dept. CSIE, NDHU11 Chapter Definitions and Examples: Calculational Techniques

Discrete Math by R.S. Chang, Dept. CSIE, NDHU12 Chapter Definitions and Examples: Calculational Techniques

Discrete Math by R.S. Chang, Dept. CSIE, NDHU13 Chapter Definitions and Examples: Calculational Techniques Ex Use generating functions to determine how many four- element subsets of S={1,2,3,...,15} contain no consecutive integers.

Discrete Math by R.S. Chang, Dept. CSIE, NDHU14 Chapter Definitions and Examples: Calculational Techniques

Discrete Math by R.S. Chang, Dept. CSIE, NDHU15 Chapter Partitions of Integers Partition a positive integer n into positive summands and seeking the number of such partitions, without regard to order. For example, p(1)=1: 1 p(2)=2: 2=1+1 p(3)=3: 3=2+1=1+1+1 p(4)=5: 4=3+1=2+2=2+1+1= p(5)=7: 5=4+1=3+2=3+1+1=2+2+1= = We should like to obtain p(n) for a given n without having to list all the partitions. We need a tool to keep track of the numbers of 1's, 2's,..., n's that are used as summands for n.

Discrete Math by R.S. Chang, Dept. CSIE, NDHU16 Chapter Partitions of Integers

Discrete Math by R.S. Chang, Dept. CSIE, NDHU17 Chapter Partitions of Integers Ex Find the generating function for the number of ways an advertising agent can purchase n minutes of air time if time slots for commercials come in blocks of 30, 60, or 120 seconds.

Discrete Math by R.S. Chang, Dept. CSIE, NDHU18 Chapter Partitions of Integers Ex Find the generating function for p d (n), the number of partitions of a positive integer n into distinct summands.

Discrete Math by R.S. Chang, Dept. CSIE, NDHU19 Chapter Partitions of Integers Ex Partition into odd summands but each such odd summands must occur an odd number of times-or not at all. Ferrer's graph 14= = The number of partitions of an integer n into m summands is equal to the number of partitions where m is the largest summands.

Discrete Math by R.S. Chang, Dept. CSIE, NDHU20 Chapter The Exponential Generating Function ordinary generating functions: selections (order is irrelevant)

Discrete Math by R.S. Chang, Dept. CSIE, NDHU21 Chapter The Exponential Generating Function Ex In how many ways can four of the letters in ENGINE be arranged?

Discrete Math by R.S. Chang, Dept. CSIE, NDHU22 Chapter The Exponential Generating Function Ex A ship carries 48 flags, 12 each of the colors red, white, blue, and black. Twelve of these flags are placed on a vertical pole in order to communicate a signal to other ships.

Discrete Math by R.S. Chang, Dept. CSIE, NDHU23 Chapter The Exponential Generating Function Ex (continued) (a) How many of these signals use an even number of blue flags and odd number of black flags?

Discrete Math by R.S. Chang, Dept. CSIE, NDHU24 Chapter The Exponential Generating Function Ex (continued) (b) How many of these signals have at least three white flags or no white flags at all?

Discrete Math by R.S. Chang, Dept. CSIE, NDHU25 Chapter 9 Ex Assign 11 new employees to 4 subdivisions. Each subdivision will get at least one new employees.. the number of onto functions

Discrete Math by R.S. Chang, Dept. CSIE, NDHU26 Chapter The Summation Operator

Discrete Math by R.S. Chang, Dept. CSIE, NDHU27 Chapter The Summation Operator

Discrete Math by R.S. Chang, Dept. CSIE, NDHU28 Chapter 9 Summaries (m objects, n containers) Objects Containers Some Number Are Are Containers of Distinct Distinct May Be Empty Distributions Yes Yes Yes n m Yes Yes No n!S(m,n) Yes No Yes S(m,1)+S(m,2)+...+S(m,n) Yes No No S(m,n) No Yes Yes No Yes No No No Yes (1) p(m), for n=m No No No (2) p(m,1)+p(m,2)+...+p(m,n), n<m p(m,n) p(m.n):number of partitions of m into exactly n summands

Discrete Math by R.S. Chang, Dept. CSIE, NDHU29 Chapter 9 Exercise: P390: 6 P399: 18,20 P403: 9,10 P408: 6