Lexicographic shellability, matroids, and pure order ideals

Abstract

In 1977 Stanley conjectured that the h-vector of a matroid independence complex is a pure O-sequence. In this paper we use lexicographic shellability for matroids to motivate a new approach to proving Stanley's conjecture. This suggests that a pure O-sequence can be constructed from combinatorial data arising from the shelling. We then prove that our conjecture holds for matroids of rank at most four, settling the rank four case of Stanley's conjecture. In general, we prove that if our conjecture holds for all rank d matroids on at most 2d elements, then it holds for all matroids of rank d.

Algebraic and Geometric Methods in Enumerative Combinatorics

“…This conjecture has been proved, using rather different methods, for several families: duals of graphic matroids, [Mer01], lattice path matroids [Sch10] cotransversal matroids [Oh13], paving matroids [MNRIVF12], and matroids up to rank 4 or corank 2 [DLKK12,KS14]. The general case remains open.…”

Internally Perfect Matroids

“…We write IP for the set of all internally perfect matroids and F i for the family of all matroids satisfying property (i) above. The interesting perfect matroid in Example 6 shows that IP F i for all i ∈ [8]. We will now show that none of the opposite inclusions hold by providing a matroid in each family F i that is not internally perfect for any linear order of its ground set.…”

“…Moreover, it is not hard to show by induction on n and |D| that M has no spanning circuit and that the last entry of the h-vector of M is at least 6. Thus M does not satisfy either (7) or (8).…”

“…Following a 23-year period during which no partial results were published, the last fifteen years have seen a flurry of research concerning Stanley's Conjecture [3,4,5,8,11,12,14,16]. These results are essentially of two types.…”

“…In each of [5,11,12,14,16] a certain class of matroids is considered and Stanley's Conjecture is shown to hold by exploiting properties of the class. In particular, Stanley's conjecture is known to hold for cographic matroids (Merino [11]), paving matroids (De Loera et al [5]) and matroids with rank no more than four (Klee-Samper [8]).…”