1. THE BALLOT PROBLEM OF MANY CANDIDATES

5. Objective To find the formula and proof of Ballot problem for many candidates.

6. In the case of two candidates (The Original Ballot problem)

7. Suppose is the ballots of the 1 st candidate. is the ballots of the 2 nd candidate, when . Define “1” as the ballot given to 1 st candidate. “ -1” as the ballot given to 2 nd candidate. In the case of two Candidates

8. The number of ways to count the ballots for required condition. The number of permutation of the sequence: such that the partial sum is always positive. The number of ways to walk on the lattice plane starting at (0,0) and finish at (a,b), and not allow pass through the line y=x except (0,0) = =

9. In the case of two Candidates

10. Reflection Principle “ Reflection principle” is use to count the number of path, one can show that the number of illegal ways which begin at (1,0) is equal to the number of ways begin at (0,1). It implies that, if we denote as the number of ways as required:

11. Reflection Principle Figure 1 Figure 2

12. In the case of three Candidates

13. In the case of three Candidates

14. In the case of three Candidates Figure 3 Figure 4

15. How to count ?

16. How to count ?

18. Example Counting Front View F(1) F(2) F(3) F(4) F(5)

19. Example Counting Side View S(1) S(2)

20. Example Counting Matching 2 2 1 1 1 S(2) 1 1 1 1 1 S(1) F(5) F(4) F(3) F(2) F(1) F(i) * S(j) F(1) F(2) F(3) F(4) F(5) S(1) S(2) 5 7 12 Total

21. Dynamic Programming

22. Counting (5,4) with D.P. y x x+y 1 1 1 1 1 1 1 4 3 2 1 0 0 9 5 2 0 0 0 14 5 0 0 0 0 14 0 0 0 0 0

23. Formula for three candidates

24. Definition is the number of ways to count the ballot that correspond to the required condition Lemma 1.1 Lemma 1.2 Formula for three candidates

25. Formula for three candidates Conjecture

26. Let Consider Use strong induction; given is the “base” therefore Proof Hence, the base case is true. Formula for three candidates

28. Formula for three candidates

29. By strong induction, we get that, Formula for three candidates

30. Formula for n candidates

31. is the number of ways to count the ballots of the n candidates such that, while the ballots are being counted, the winner will always get more ballots than the loser. Definition Lemma 3 Formula for n candidates

32. (base) n=2 n=k Mathematical Induction Sub - Strong Mathematical Induction (base) n=k-1 Proof

33. Formula for n candidates

34. Formula for n candidates

35. Formula for n candidates

36. By mathematical induction, Formula for n candidates

37. 1. Think about the condition “never less than” instead of “always more than.” Development 2. What if we suppose that the ballots of the 3 rd candidate don’t relate to anyone else?

39. Application In Cryptography Define the plaintext (code) used to send the data Increases the security of the system.

40. Reference Chen Chuan-Chong and Koh Khee-Meng, Principles and Techniques in Combinatorics , World Scientific, 3rd ed., 1999. Marc Renault, Four Proofs of the Ballot Theorem, U.S.A., 2007. Michael L. GARGANO, Lorraine L. LURIE Louis V. QUINTAS, and Eric M. WAHL, The Ballot Problem, U.S.A.,2005. Miklos Bona, Unimodality, Introduction to Enumerative Combinatorics, McGrawHill, 2007. Sriram V. Pemmaraju, Steven S. Skienay, A System for Exploring Combinatorics and Graph Theory in Mathematica, U.S.A., 2004.