Connecting the Stirling numbers and $k$-bonacci sums

TL;DR

This paper explores the connection between Stirling numbers and $k$-bonacci sums, establishing a formal link through algebraic proof and bijection, enhancing understanding of their combinatorial relationships.

Contribution

It introduces a bijection linking a known summation formula for $k$-bonaccis with an experimentally derived formula, providing a new algebraic proof of their relationship.

Findings

Stirling numbers appear in $k$-bonacci sum formulas

A bijection between two $k$-bonacci formulas is established

An algebraic proof confirms the connection

Abstract

This paper proves why the Stirling numbers show up in a experimentally determined formula for the $k$-bonaccis. We develop a bijection between a previously determined summation formula for $k$-bonaccis and an experimentally determined formula, proven algebraically.