# The Amazing $3^n$ Theorem and its even more Amazing Proof [Discovered by Xavier G. Viennot and his École Bordelaise gang]

- Published in 2012
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The most amazing (at least to me) result in Enumerative Combinatorics is Dominique Gouyou-Beauchamps and Xavier Viennot's theorem that states that the number of so-called directed animals with compact source (that are equivalent, via Viennot's beautiful concept of heaps, to towers of dominoes, that I take the liberty of renaming xaviers) with n+1 points equals 3^n. This amazing result received an even more amazing proof by Jean B\'etrema and Jean-Guy Penaud. Both theorem and proof deserve to be better known! Hence this article, that is also accompanied by a comprehensive Maple package http://www.math.rutgers.edu/~zeilberg/tokhniot/BORDELAISE that implements everything (and much more)

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## Other information

- key
- TheAmazing3nTheoremanditsevenmoreAmazingProofDiscoveredbyXavierGViennotandhiscoleBordelaisegang
- type
- article
- date_added
- 2021-03-15
- date_published
- 2012-07-11

### BibTeX entry

@article{TheAmazing3nTheoremanditsevenmoreAmazingProofDiscoveredbyXavierGViennotandhiscoleBordelaisegang, key = {TheAmazing3nTheoremanditsevenmoreAmazingProofDiscoveredbyXavierGViennotandhiscoleBordelaisegang}, type = {article}, title = {The Amazing {\$}3^n{\$} Theorem and its even more Amazing Proof [Discovered by Xavier G. Viennot and his {\'{E}}cole Bordelaise gang]}, author = {Doron Zeilberger}, abstract = {The most amazing (at least to me) result in Enumerative Combinatorics is Dominique Gouyou-Beauchamps and Xavier Viennot's theorem that states that the number of so-called directed animals with compact source (that are equivalent, via Viennot's beautiful concept of heaps, to towers of dominoes, that I take the liberty of renaming xaviers) with n+1 points equals 3^n. This amazing result received an even more amazing proof by Jean B\'etrema and Jean-Guy Penaud. Both theorem and proof deserve to be better known! Hence this article, that is also accompanied by a comprehensive Maple package http://www.math.rutgers.edu/{\~{}}zeilberg/tokhniot/BORDELAISE that implements everything (and much more)}, comment = {}, date_added = {2021-03-15}, date_published = {2012-07-11}, urls = {http://arxiv.org/abs/1208.2258v1,http://arxiv.org/pdf/1208.2258v1}, collections = {attention-grabbing-titles,combinatorics,fun-maths-facts,things-to-make-and-do}, url = {http://arxiv.org/abs/1208.2258v1 http://arxiv.org/pdf/1208.2258v1}, year = 2012, urldate = {2021-03-15}, archivePrefix = {arXiv}, eprint = {1208.2258}, primaryClass = {math.CO} }