UMA FÓRMULA EXPLÍCITA PARA OS NÚMEROS DE STIRLING DO PRIMEIRO TIPO ATRAVÉS DE PARTIÇÕES INTEIRAS

Abstract

The aim of this article is to present an explicit formula for the Stirling numbers of the first kind and prove it through counting arguments. This is only possible because these numbers have a strong combinatorial appeal, since we can define them as the number of ways to distribute n people around k identical circular tables, without leaving any empty tables. In order to establish the proof for the main theorem, we will explore some identities involving the Stirling numbers of the first kind with the binomial coefficient, as well as introduce the concept of partitioning positive integers and use it as the main tool for combinatorial arguments in the proof of the main result. Additionally, we prove new identities and others found in the literature through this theorem.