For the n-dimensional Torus, the k-th homology group (with integer coefficients) is isomorphic to the direct sum of $n \choose k$ copies of $\mathbb{Z}$. Poincaré duality thus gives us a somewhat indirect way to show that ${n \choose k} = {n \choose n-k}$.

I am looking for results (or a starting point in the literature) where less obvious combinatorial results are derived by Poincaré duality.

I think that this is roughly related to the following questions:

Combinatorial results without known combinatorial proofs

From what I have learned from the comments, the Dehn–Sommerville equations would be an example of a combinatoric result based on Poincaré duality.

symmetryof a sequence to be provable most easily by a topological duality theorem, rather than by a direct combinatorial argument. Even the Dehn--Sommerville equations have a relatively straightforward combinatorial proof. $\endgroup$ – Timothy Chow Jun 18 '15 at 16:13