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Standard complexes of matroids and lattice paths. (English) Zbl 1493.05300

A matroid is a combinatorial structure that abstracts and generalizes the notion of dependence in graphs and vector spaces. Matroids come with a rich enumerative theory and have found applications in geometry, topology, combinatorial optimization, network theory, and coding theory. In this article, the authors define and study a new class of simpicial complexes, called standard compelxes, associated to a matroid. The motivation behind the study of these complexes comes from the Groöbner basis theory for finite point configurations. The standard compelxes are certain subcomplexes of the independence complexes that are invariant under matroid duality. The authors prove that, for lexicographic term order, the standard compelxes satisfy a deletion-contraction-type recurrence. They also explicitly determine the lexicographic standard complexes for lattice path matroids.

MSC:

05E45 Combinatorial aspects of simplicial complexes
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13P25 Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.)
05B35 Combinatorial aspects of matroids and geometric lattices
52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
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