Ferrers Rocher

Ferrers Rocher is an interactive web app where users can animate bijections of integer partitions with Ferrers diagrams. The web app is primarily aimed at applied mathematicians, especially professors and students in the combinatorics and probability fields. However, it is still engaging and fun for people who are not into math! 🙂

Background

This project started off as a 10-week senior math research project under the supervision of Prof. Stephen DeSalvo in the spring of 2014 at UCLA. The focus was to create an app which (1) generates random partitions of a given positive integer $n$ such that they have asymptotically $O(\sqrt{n} \log{n})$ parts with high probability, and (2) visualizes how certain bijections affect the overall limit shape of those partitions. The app was originally supposed to be written in C++ using Qt, but it ultimately became a Java applet instead due to time constraints.

Ferrers Rocher is intended to replace that applet, since Java applets were removed from Java SE 11 in September 2018.

Math Terms: What You Need to Know

• $\mathbb Z^+$ is the set of all positive integers.
• An (integer) partition of $n \in \mathbb Z^+$ is an expression of $n$ as a sequence of parts $\{\lambda_k \in \mathbb Z^+ : \sum\lambda_k = n\}$, which are conventionally in decreasing order. The notation $\lambda ⊢ n$ means that $\lambda$ is a partition of $n$. For example, $\lambda = (3, 2, 2, 2, 1) ⊢ 10$.
• A Ferrers diagram represents an integer partition $\lambda$ as patterns of dots, with the $k$-th row having the same number of dots as the $k$-th largest part in $\lambda$.
• A bijection is a function which is one-to-one and onto.
  • A function $f$ with domain $X$ is one-to-one if for all $a$ and $b$ in $X$, whenever $a \ne b$, then $f(a) \ne f(b)$.
  • A function $f$ with domain $X$ and range $Y$ is onto if for all $y \in Y$, there is at least one $x \in X$ such that $f(x) = y$.

Building and Running Locally

git clone https://github.com/kristorres/ferrers-rocher
cd ferrers-rocher
pnpm install
pnpm build/dev

Research Papers

• The Nature of Partition Bijections II: Asymptotic Stability
• Probabilistic Divide-and-Conquer: A New Exact Simulation Method, with Integer Partitions as an Example
• Improvements to Exact Boltzmann Sampling Using Probabilistic Divide-and-Conquer and the Recursive Method