If you want to revisit this material with a fresh copy of the handouts, you can print everything from a single file.

The final and correct table of contents is also available.

Welcome. We start with definitions and combinatorial interpretations. Discuss the techniques of constructive combinatorial proof. And start applying them right away to binomial identities.

Our focus will be on Fibonacci identities, which we will tackle using tilings of 1xn boards with squares and dominoes. Once we get a good handle we can begin to generalize. Probabilistic tilings will allow us a new angle to prove Binet's formula.

Weighting our square-domino tilings will allow us to general the identities and arguments to other sequences satisfying the Fibonacci recurrence  (the so-called Gibonacci numbers) as well as general linear recurrences of order 2. One Gibonacci identity can be used to create some mathemagic!

Finite continued fractions can be used to approximate irrational numbers. We begin the day with an introduction to contiued fractions followed by a combinatorial interpretation using weighted square and domino tilings (where the weights on the squares depend on the position of the square).  It brings a concrete slant to irrational numbers.

We will focus on identities where the terms alternate between positive and negative signs using the Method of D.I.E. (description, involution, exception). By the end, I hope that you agree with the statement, ``matching is easier than counting.''  A final visit to Binet's formula will involve finite colored, weighted tilings and show the true power of this technique.

After a brief introduction to determinants, we will see that the big formula (a.k.a. the determinantal expansion over all permutations) is really just an alternating sum that can be handled with the Method of D.I.E.  Our last hurrah will be a combinatorial proof of the determinant for Vandermonde's matrix. References for the material presented this week is also included in this material.