Stirling numbers of the second kind

The Stirling numbers of the second kind, denoted as \( S(n, k) \), are a set of combinatorial numbers that count the ways to partition a set of \( n \) objects into \( k \) non-empty subsets. In other words, \( S(n, k) \) gives the number of different ways to group \( n \) distinct items into \( k \) groups, where groups can have different sizes but cannot be empty.