9.1 Introductory Examples

2. 9.1 Introductory Examples Ex. 9.1 Find the number of integer solutions to

3. 9.1 Introductory Examples Ex. 9.3 How many integer solutions are there for the equation

4. 9.2 Definitions and Examples: Calculational Techniques

5. 9.2 Definitions and Examples: Calculational Techniques

6. 9.2 Definitions and Examples: Calculational Techniques Ex. 9.5 (continued)

7. 9.2 Definitions and Examples: Calculational Techniques

8. 9.2 Definitions and Examples: Calculational Techniques Extension of binomial coefficient

9. 9.2 Definitions and Examples: Calculational Techniques

10. 9.2 Definitions and Examples: Calculational Techniques Ex. 9.11 In how many ways can we select, with repetitions allowed, r objects from n distinct objects?

11. 9.2 Definitions and Examples: Calculational Techniques

12. 9.2 Definitions and Examples: Calculational Techniques

13. 9.2 Definitions and Examples: Calculational Techniques Ex. 9.15 Use generating functions to determine how many four- element subsets of S={1,2,3,...,15} contain no consecutive integers.

14. 9.2 Definitions and Examples: Calculational Techniques

15. 9.3 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=1+1+1+1 p(5)=7: 5=4+1=3+2=3+1+1=2+2+1=2+1+1+1 =1+1+1+1+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.

16. 9.3 Partitions of Integers

17. 9.3 Partitions of Integers Ex. 9.18 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.

18. 9.3 Partitions of Integers Ex. 9.19 Find the generating function for pd(n), the number of partitions of a positive integer n into distinct summands.

19. 9.3 Partitions of Integers Ex. 9.21 Partition into odd summands but each such odd summands must occur an odd number of times-or not at all. 14=4+3+3+2+1+1 =6+4+3+1 Ferrer's graph The number of partitions of an integer n into m summands is equal to the number of partitions where m is the largest summands.

20. 9.4 The Exponential Generating Function ordinary generating functions: selections (order is irrelevant)

21. 9.4 The Exponential Generating Function Ex. 9.23 In how many ways can four of the letters in ENGINE be arranged?

22. 9.4 The Exponential Generating Function Ex. 9.25 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.

23. 9.4 The Exponential Generating Function Ex. 9.25 (continued) (a) How many of these signals use an even number of blue flags and odd number of black flags?

24. 9.4 The Exponential Generating Function Ex. 9.25 (continued) (b) How many of these signals have at least three white flags or no white flags at all?

25. Ex. 9.26 Assign 11 new employees to 4 subdivisions. Each subdivision will get at least one new employees.. the number of onto functions

26. 9.5 The Summation Operator

27. 9.5 The Summation Operator

28. Summaries (m objects, n containers) Objects Containers Some Number Are Are Containers of Distinct Distinct May Be Empty Distributions Yes Yes Yes nm 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

29. Exercise: P390: 6 P399: 18,20 P403: 9,10 P408: 6