S(n, k) represents the Stirling number of the second kind, which counts the ways to partition a set of n elements into k non-empty subsets. For example, if you have a set of 3 elements and want to divide it into 2 groups, S(3, 2) would give you the number of different ways to do that. These numbers are useful in combinatorics and have applications in various fields, including mathematics, computer science, and probability theory. They can also be calculated using a recursive formula or through specific combinatorial identities, making them a fundamental concept in understanding partitions.