Abstract
In this methodological article on experimental-yet-rigorous enumerative combinatorics, we use two instructive case studies, to show that often, just like Alexander the Great before us, the simple, “cheating” solution to a hard problem is the best. So before you spend days (and possibly years) trying to answer a mathematical question by analyzing and trying to ‘understand’ its structure, let your computer generate enough data, and then let it guess the answer. Often its guess can be proved by a quick ‘hand-waving’ (yet fully rigorous) ‘meta-argument’. Since our purpose is to illustrate a methodology, we include many details, as well as Maple source-code.
This article is dedicated to Peter Paule, one of the great pioneers of experimental mathematics and symbolic computation. In particular, it is greatly inspired by his masterpiece, co-authored with Manuel Kauers, ‘The Concrete Tetrahedron’ [ 3 ], where a whole chapter is dedicated to our favorite ansatz, the C-finite ansatz
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Acknowledgements
Many thanks are due to a very careful referee that pointed out many minor, but annoying errors, that we hope corrected. Also thanks to Peter Doyle for permission to include his elegant electric argument.
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Yao, Y., Zeilberger, D. (2020). Untying the Gordian Knot via Experimental Mathematics. In: Pillwein, V., Schneider, C. (eds) Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-44559-1_18
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