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Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.
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Geometric motivation for the Stanley-Reisner correspondence
The Stanley-Reisner ring of an abstract simplicial complex $\Delta$ on the vertex set $\{1,...,n\}$ is the $k$-algebra
$$
k[X_1,...,X_n]/I_\Delta
$$
where $I_\Delta$ is the ideal generated by the $X_{ …