Proof of bijection for combinatorial number system

TL;DR

This paper proves that every non-negative natural number can be uniquely represented as a sum of binomial coefficients, establishing a fundamental property of the combinatorial number system.

Contribution

It provides an induction proof confirming the existence and uniqueness of the combinatorial number system representation for all non-negative integers.

Findings

Unique representation of natural numbers as binomial sums

Induction proof validating the combinatorial number system

Foundational result for combinatorial number system theory

Abstract

Combinatorial number system represents a non-negative natural numbers as sum of binomial coefficients. This paper presents an induction proof that there exists unique representation of every non-negative natural number $m$ as sum of $r$ binomial coefficients.