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

TL;DR

This paper discusses the remarkable theorem that counts directed animals with a compact source using the number 3^n, and presents an even more astonishing proof along with a comprehensive Maple package for implementation.

Contribution

It introduces a new proof of the 3^n enumeration theorem for directed animals and provides a Maple package for further exploration.

Findings

The number of directed animals with n+1 points equals 3^n.

The proof by Bétrema and Penaud is notably elegant and innovative.

A Maple package is provided for computational verification and exploration.

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)