  Motivated by Gr\"obner basis theory for finite point configurations, we
define and study the class of "standard complexes" associated to a matroid.
Standard complexes are certain subcomplexes of the independence complex that
are invariant under matroid duality. For the lexicographic term order, the
standard complexes satisfy a deletion-contraction-type recurrence. We
explicitly determine the lexicographic standard complexes for lattice path
matroids using classical bijective combinatorics.
