In 1977 Stanley conjectured that the h-vector of any matroid is a pure O-sequence, that is, is the Hilbert function of an Artinian monomial algebra whose socle is concentrated in a single degree. Though this conjecture has motivated a great deal of research in combinatorial commutative algebra in the mean time, the general case is still wide open.
In the article Internally Perfect Matroids (recently published here) we study a new class of matroids having highly structured internal orders. Indeed, we show that these are precisely the matroids whose internal order is isomorphic to the divisor poset on the standard monomials of an Artinian ideal over a polynomial ring, or equivalently, whose internal orders are certificates for the validity of Stanley’s Conjecture. We show that internally perfect matroids are not contained in any of the classes of matroids for which Stanley’s Conjecture was previously known to hold and prove, up to mild condition, that they form a minor closed class. The many examples in the paper can be verified with the MatroidActivities package and the computations in Macaulay2 that went in to working out the example in the last section of the paper are also worked out here.
Sweeping out the Pantheon 