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

• Published in 2012
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)

### BibTeX entry

@article{TheAmazing3nTheoremanditsevenmoreAmazingProofDiscoveredbyXavierGViennotandhiscoleBordelaisegang,
title = {The Amazing {\$}3^n{\$} Theorem and its even more Amazing Proof [Discovered by  Xavier G. Viennot and his {\'{E}}cole Bordelaise gang]},
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)},
url = {http://arxiv.org/abs/1208.2258v1 http://arxiv.org/pdf/1208.2258v1},
year = 2012,
author = {Doron Zeilberger},
comment = {},
urldate = {2021-03-15},
archivePrefix = {arXiv},
eprint = {1208.2258},
primaryClass = {math.CO},
collections = {attention-grabbing-titles,combinatorics,fun-maths-facts,things-to-make-and-do}
}