Gabriel Lamé's Counting of Triangulations

Abstract

This curricular module outlines Gabriel Lamé's solution to the problem stated in the title of his 1838 publication “Given a convex polygon, in how many ways can one partition it into triangles by means of diagonals?" written in response to a challenge posed by Joseph Liouville.  Liouville published several solutions to this problem including Lamé’s highly original and clever reasoning that involved an averaging argument over certain symmetries of a polygon.  The solution yielded what are known today as the Catalan numbers, after Eugène Catalan, whose solution to the problem, also published by Liouville in the same volume, was difficult to follow. The Catalan numbers appear today for other uses in mathematics and computer science and enumerate the number of rooted, binary trees.  The project is designed for a course in combinatorics, advanced undergraduate discrete mathematics, or algorithm design.