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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!

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    $\begingroup$ 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 $\endgroup$ Jun 6 '20 at 12:20
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One from mathematical physics.

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 $\mathcal{X}$.

The procedure can be generalized by giving a planar bipartite graph and constructing with it the image of the moment map of a toric variety (see Quantum Calabi-Yau and Classical Crystals). In that context the enummeration of perfect matchings on the graph is the same problem as the ennumeration of plane partitions in the image of the moment map of the aforementioned toric variety.

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