Partition Approximation via the Cube Root of Binomial-Partition Ratios: A First-Principles Derivation with Stirling-Based Correction

This work presents a novel approximation method for the integer partition function based on a cube root transformation of the ratio between central binomial coefficients and partition values. This paper provides a full theoretical foundation for this relationship, which the author first identified through empirical analysis in a previous technical note [here]. The method is derived rigorously from first principles using Stirling’s formula with Euler-Maclaurin corrections and the saddle point method applied to partition generating functions. A theoretically motivated, empirically optimized Stirling-based correction factor is introduced, yielding significant improvements in approximation accuracy across a wide range of input sizes. Extensive computational experiments demonstrate error reductions by up to 4.5× compared to classical Hardy-Ramanujan asymptotics, with stable and efficient performance validated up to n= 80,000.

Key Contributions

• First-principles derivation of a cube root ratio-based approximation combining binomial coefficient and partition function asymptotics.

• Introduction of an adaptive correction factor α(n) = 3 + 1/(120n) motivated by Stirling’s approximation error terms.

• Empirical optimization of the correction coefficient that substantially improves accuracy across all tested ranges.

• Rigorous computational validation using Euler’s recurrence relation for exact partitions up to n=80,000.

• Demonstration of superior accuracy and numerical stability compared to classical Hardy-Ramanujan formulas.

• Insights suggesting deeper mathematical structure linking the adaptive correction to classical Stirling series that warrant further theoretical investigation.

Included Files

• Venkat2025_PartitionApproximation.pdf:  The full academic paper detailing the first-principles derivation, methodology, and performance analysis of the Stirling-corrected partition approximation formula.
• partition_forms_analyzer.py : Core Python script implementing the Stirling-corrected partition function approximation.
• partition_error_summary_adaptive.csv : Error summary table result output of the python script.
• fig_partition_error_comparison.png : Plot comparing the relative approximation errors (%) of the Stirling-corrected cube root partition function approximation against the fixed-factor theoretical formula and the Hardy-Ramanujan asymptotic.
• requirements.txt: A text file listing the Python dependencies required to run the code.
  
  

Licensing

This project uses a dual-license model:

• Source Code (.py file): MIT License
• All Other Files (Paper, documentation, results): Creative Commons Attribution 4.0 International (CC BY 4.0)