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Abstract:
The goal of this investigation is to provide combinatorial insight for
some known determinantal identities involving the Fibonacci numbers.
The method, due to Lindström-Gessel-Viennot, is to associate a
determinantal value with the number of nonintersecting n-routes in a
carefully chosen digraph. The advantage of this method is the ability
to gain better insight into the nature of the identity, thereby allowing
room to generalize the identity or even discover new related identities.
In this presentation, I intend to (i) provide the necessary background
about basic graph theory and the Fibonacci numbers, (ii) discuss the
theorem of Lindström-Gessel-Viennot which explains a relationship
between non-intersecting path systems and determinants, (iii) present
some known Fibonacci identities, including Cassini's identity, and (iv)
show how one can understand and generalize these identities with
creative applications of the Lindström-Gessel-Viennot theorem.
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