I understand that the Bell number $B_n$ is the number of partitions of a set of size $n$. Despite my incredible ineptitude at combinatorics, I also understand most of how the binomial coefficient works (formulas, basic properties, a bound or two). However, I'm looking for something that describes the number of ways you can cut a set $S$ of size $n$ into, say, 2 disjoint partitions, or 3 partitions, or $k<n$ partitions. Is there terminology to describe this, a nice expression, or nice upper bounds?
I tried looking for bounds on the sum of binomial coefficients from $2$ to $\lfloor n/2 \rfloor$ (since every combination is a disjoint subset), which led me to this paper, but those bounds are exponential in $n$ (would be nice to find polynomial), and I think I'm doing more work than I need to (but I have bad intuition for combinatorics, so I'm not sure).
Can I just replace 3 with 2 and finagle $k$ in this answer?
Suppose you want to find the number of ways to partition a set of n elements into k subsets or less, all of these non-empty and without the order of the partitions mattering.
Then this is the same as $\sum_{j=1}^k S(n,j)$ where $S(n,j)$ denotes the number of way so partition a set of n elements into j subsets without the partitions being "labelled".
the numbers $S(n,m)$ are called the Stirling numbers of the second kind.