Applications of homological algebra in combinatorics I have started learning homological algebra recently. It looks like the most abstract subject I've seen so far. The most concrete one is without doubts is combinatorics. So I have very specific reference request -
Can you provide a reference to a book/research paper/whatever that illustrates how abstract homological tools allow to compute something concrete/finite/nontrivial or at least prove it exists.
Thanks a lot for your time!
 A: This isn't exactly an application of bare homological algebra, but it does involve cohomology: the hard Lefschetz theorem in algebraic geometry implies that the sequences of even and odd Betti numbers of a smooth projective variety over $\mathbb{C}$ are both unimodal, meaning that they first increase and then decrease. A simple example is a product of $n$ copies of the complex projective line $\mathbb{CP}^1$; here the even Betti numbers are binomial coefficients ${n \choose k}$. 
A more interesting example is the Grassmannian $\text{Gr}_d(\mathbb{C}^n)$, whose even Betti numbers count the number of partitions fitting into a $d \times (n-d)$ box. No purely combinatorial proof that this sequence is unimodal is known (edit: it seems my information is out of date! See this survey by Zeilberger of the result, which is due to O'Hara). See this survey by Stanley for more. 
A: In this paper Power series representing Posets the problem of counting in how many ways can you label a Poset (while preserving the order) is solved for a family of Posets called Wixárika Posets (they look like Wixárika collars). The main results are proven using homological algebra ideas. As a consequence several combinatorial identities were discovered (we checked on books of identities and we coulnd't find them).
