Stirling Numbers are a set of mathematical numbers that count the ways to partition a set of objects. Specifically, there are two types: the first kind, denoted as S(n, k), counts the ways to arrange n objects into k cycles, while the second kind, denoted as S(n, k), counts the ways to partition n objects into k non-empty subsets. These numbers have applications in combinatorics, algebra, and even computer science. They help in understanding permutations and combinations, and they appear in various mathematical problems, including those related to Bell Numbers and Polynomial Expansions.