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The Polyhedral Index Partition (PIP) and the Discovery of Pascal's Dimensions: Enabling Computational Retrieval and Reversibility in High-Index Partition Arrays

Listen to the paper: here


Author: Andrew Lehti

Disciplines: Cognitive Psychology, Linguistics, Mathematics, and their Histories

DOI: 10.6084/m9.figshare.27642783

Abstract

The Polyhedral Index Partition (PIP) framework introduces an innovative approach to calculating integer partitions by leveraging the mathematical structures of Pascal's Triangle, specifically through the concepts of Pascal's Dimensions and Pascal's Laterals. This repository contains the Python implementation of PIP, which offers significant performance improvements over traditional partition enumeration methods.


Getting Started

Prerequisites

  • Python 3.8 or later
  • mpmath library for arbitrary-precision arithmetic

Install mpmath using:

pip install mpmath

Cloning the Repository

Clone this repository to your local machine:

git clone https://github.com/andylehti/Polyhedral-Index-Partition.git

Usage

Run the Python script to compute partitions and their indices:

Example: Partition Calculation

n = 55234587678685685685663467263573
p = 7
a = partition(n, p)
print("Partition:", a)

Example: Reverse Mapping

r = getInverse(*a)
print("Index:", r)

For more examples, see the notebook.


References

  1. Pascal, B. (1654). Traité du triangle arithmétique.
  2. Stanley, R. P. (1999). Enumerative Combinatorics.
  3. Sloane, N. J. A. (2003). The On-Line Encyclopedia of Integer Sequences.

Polyhedral Index Partition © Andrew Lehti, 2024

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