# Bounds on these numbers

Let $$[n]$$ be the set of natural numbers $$1,2,3 \cdots n$$ and $$k$$ be a natural number. Define $$S(n,k) = \# \{ A \subset [n] \mid \displaystyle\sum_{i \in A} i =k \}$$. My question is; Are there any known good bounds (both upper and lower) on these numbers? Instead of taking the subsets in a special set such as $$[n]$$, if we take from a given subset $$A$$ of the set of integers, then it is the famous subset sum problem to decide whether $$S(A,k) = 0$$ or not.

• The first place to check would be the references of OEIS A053632. Nov 10, 2022 at 13:54
• Thanks for the pointer! Nov 10, 2022 at 15:21
• It is a number of strict partitions of $k$ with parts not exceeding $n$, that is, the coefficient of $x^k$ in the polynomial $(1+x)(1+x^2)\cdots(1+x^n)$. A lot is known about asymptotics in various regimes. Nov 10, 2022 at 16:44
• Thanks, I looked at a few articles by S. Finch (following the OEIS pointer of @PeterTaylor) on related objects but haven't found any good asymptotic bounds. These numbers seem to be related to various problems in Physics and Biology as well. Nov 11, 2022 at 7:47