Solving Functional Equations Dear to W.T. Tutte using the Naive (yet fullly rigorous!) Guess And Check Method
arxiv(2024)
摘要
In his seminal paper “A census of planar triangulations", published in 1962,
the iconic graph theorist (and code-breaker), W.T. Tutte, spent a few pages to
prove that a certain bi-variate generating function that enumerates
triangulations, satisfies a certain functional equation. He then used his
genius to actually solve it, giving closed-form solutions to the enumerating
sequences. While the first part, of deriving the functional equation, still
needs human ingenuity, the second part, of solving it, can nowadays be fully
automated. Our Maple program, accompanying this paper, Tutte.txt, can not only
solve Tutte's original equation in a few seconds, it can also solve many, far
more complicated ones, way beyond the scope of even such a giant as W.T. Tutte.
We use our favorite method of “guess and check" and show how it can always be
made fully rigorous (if desired).
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