Chen B. Combinatorial Proofs Using Complex Weights

Harvey Mudd College, 2010. — 35 p.In 1961, Kasteleyn, Fisher, and Temperley gave a result for the number of possible tilings of a 2mx 2n checkerboard with dominoes. Their proof involves the evaluation of a complicated Pfaffian. In this thesis we investigate combinatorial strategies to evaluate the sum of evenly spaced binomial coefficients, and present steps towards a purely combinatorial proof of the 1961 result.Structure of This Document.
Sums of Binomial Coefficients.
Basics.
Weighted Enumeration.
Evenly Spaced Binomial Coefficients.
Shifted Coefficients.
Alternative Forms.
Checkerboards and Dominoes.
A Simplified Kasteleyn Proof.
The Combinatorial Idea.
m = 1.
m 1.FutureWork.