What is the Ballot problem? <ul><li>Suppose A and B are candidates and there are a+b voters, a voting for A and b for B. In how many ways can the ballots be counted so that A is always ahead of B ? </li></ul>The formula comes up to be . a voters for A b voters for B
Why is it interesting? <ul><li>The Ballot problem is one of very well-known problems. This problem had been proved for two candidates. </li></ul><ul><li>What about any n, which is more than 2, candidates? </li></ul>
Objective To find the formula and proof of Ballot problem for many candidates.
In the case of two candidates (The Original Ballot problem)
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
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) = =
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:
<ul><li>1. Consider the path seen at the front view. Find all possible ways to “walk” within the allowed plane. </li></ul><ul><li>2. Consider the path seen at the side view. Find all possible ways to “walk” within the allowed plane. </li></ul><ul><li>3. Match each way of 1. to each of 2. (Sometimes it yields more than one complete path.) </li></ul>How to count ?
Example Counting Front View F(1) F(2) F(3) F(4) F(5)
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
By mathematical induction, Formula for n candidates
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?
Application In Biology <ul><li>Lead to finding the number of random graph models for angiogenesis in the renal glomerulus. </li></ul><ul><li>Steps to form a vascular network </li></ul><ul><li>1. Budding </li></ul><ul><li>2. Spliting </li></ul><ul><li>3. Connecting </li></ul>
Application In Cryptography Define the plaintext (code) used to send the data Increases the security of the system.
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.