Combinatorial Proofs Using Complex Weights
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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.Contents. Introduction. 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. Conclusion. FutureWork. Bibliography.
