Sheaves in combinatorics and discrete geometry

I am looking for examples for the application of sheaves, sheaf-like constructions or the (co)homology of sheaves to problems in combinatorics and discrete geometry.

For example given a poset $$(P,\leq)$$ one can look at the topology given by declaring that the open sets are order filters $$U \subseteq P$$, i.e. if $$x \in U$$ and $$x \leq y$$ then $$y \in U$$. Now, any functor $$\mathcal{F}$$ from $$P$$ to some category, e.g. $$\mathcal{F}:P \to \mathbf{Ab}$$ to the category of abelian groups, gives a sheaf (e.g. of abelian groups) on the topological space described before. This is called a sheaf on $$P$$.

I am aware of the following applications of sheaves on posets:

I suppose there should be plenty more examples and I am looking forward to your answers. Thanks!

• Maybe this is more combinatorial commutative algebra than pure combinatorics per se, but in this article sheaves on posets are used to extend the idea of monomial ideals/squarefree monomial ideals to a more general context than polynomial rings: sciencedirect.com/science/article/pii/S0022404900000955 Jun 6 '20 at 12:20

The enummeration of plane partitions in the image of the moment map of a toric variety compute the Donaldson-Thomas theory of the toric variety $$\mathcal{X}$$ by identifying plane partitions with monomial ideals of the structural sheaf of $$\mathcal{X}$$, namely ideal sheaves of $$\mathcal{X}$$.