https://github.com/kristorres/ferrers-rocher

🤓 A web app that animates bijections of integer partitions.
https://github.com/kristorres/ferrers-rocher

javascript mathematics nodejs rollup svelte web-app

Last synced: about 1 month ago
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🤓 A web app that animates bijections of integer partitions.

• Host: GitHub
• URL: https://github.com/kristorres/ferrers-rocher
• Owner: kristorres
• Created: 2022-08-18T23:15:13.000Z (over 2 years ago)
• Default Branch: main
• Last Pushed: 2023-11-03T03:18:34.000Z (about 1 year ago)
• Last Synced: 2023-11-03T04:24:11.275Z (about 1 year ago)
• Topics: javascript, mathematics, nodejs, rollup, svelte, web-app
• Language: JavaScript
• Homepage: https://kristorres.github.io/ferrers-rocher/
• Size: 9.42 MB
• Stars: 1
• Watchers: 2
• Forks: 0
• Open Issues: 0
• Metadata Files:
  • Readme: README.md

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README

        

    

    

    

Ferrers Rocher

==============



    



![The Ferrers Rocher web app animating the Sylvester/Glaisher bijection.](/images/sylvester-glaisher-bijection-example.gif)

**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](http://kristorres.weebly.com/partitions.html) 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

----------------------------

```sh

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](https://www.math.ucla.edu/~pak/papers/stab5.pdf)

  * [Probabilistic Divide-and-Conquer: A New Exact Simulation Method, with Integer Partitions as an Example](https://arxiv.org/pdf/1110.3856.pdf)

  * [Improvements to Exact Boltzmann Sampling Using Probabilistic Divide-and-Conquer and the Recursive Method](https://arxiv.org/pdf/1608.07922.pdf)