The following question solves the existence of such a partition: Partitioning $\{1,\cdots,k\}$ into $p$ subsets with equal sums. However, I'm trying to count how many such 2-partitions exist. Specifically,
Let $X = \{1, \cdots, 2n\}$. How many unique $S \subset X$ has cardinality $n$ and $\sum S = 2n^2+n$?
I'm trying to count the number of exact sequences on finite dimensional vector spaces $\mathbb R^1, \cdots, \mathbb R^{2n}$.