A combinatorial proof of Sun's "curious" identity

TL;DR

This paper presents a simple combinatorial proof of Sun's binomial coefficient identity using involutions on configurations with dominos and colorings, and extends the method to a generalization.

Contribution

It introduces a novel combinatorial proof technique for Sun's identity and its generalization, simplifying previous analytic approaches.

Findings

The proof employs weight-reversing involutions on combinatorial configurations.

The method extends to a broader generalization of the original identity.

The approach simplifies understanding of Sun's binomial coefficient identity.

Abstract

A binomial coefficient identity due to Zhi-Wei Sun is the subject of half a dozen recent papers that prove it by various analytic techniques and establish a generalization. Here we give a simple proof that uses weight-reversing involutions on suitable configurations involving dominos and colorings. With somewhat more work, the method extends to the generalization also.