# Explanation and exact formula of Zipfs law evaluated from rank-share combinatorics

## TL;DR

This paper derives an exact formula for Zipfs law using combinatorics, demonstrating the statistical dependence of ranks and shares, and validating the model with real-world data and simulations.

## Contribution

It provides a new combinatorial formula for rank-share distribution and explains the origin of Zipfs law from expected rank values.

## Key findings

- High correlation (0.99899) with Bureau of Labor Statistics data.
- The formula accurately models Zipfs law across different datasets.
- Monte-Carlo simulations support the theoretical results.

## Abstract

This work proves that ranks and shares are statistically dependent on one another, based on simple combinatorics. It presents a formula for rank-share distribution and illustrates that Zipfs law, is descended from expected values of various ranks in the new distribution. All conclusions, formulas and charts presented here were tested against publicly available statistical data in different areas. The correlation coefficient between the calculated values and statistical numbers provided by Bureau of Labor Statistics was 0.99899. Monte-Carlo simulations were performed as additional evidence.

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Source: https://tomesphere.com/paper/1705.07890