Internally Perfect Matroids

Abstract

In 1977 Stanley proved that the h-vector of a matroid is an Osequence and conjectured that it is a pure O-sequence. In the subsequent years the validity of this conjecture has been shown for a variety of classes of matroids, though the general case is still open. In this paper we use Las Vergnas' internal order to introduce a new class of matroids which we call internally perfect. We prove that these matroids satisfy Stanley's Conjecture and compare them to other classes of matroids for which the conjecture is known to hold. We also prove that, up to a certain restriction on deletions, every minor of an internally perfect ordered matroid is internally perfect.

Dual matroid polytopes and internal activity of independence complexes

“…Inspired by Stanley's h-vector conjecture, we focus on a special kind of broken line shellings. These have the potential to extend Dall's work on internally perfect matroids [11] and prove the main Conjecture 3.10 of [20], perhaps in some special cases. Our goal is to make sure that the poset is ranked, that the rank of an element of the poset corresponds to the size of the restriction set, and to hope for some additional structure.…”

“…In [11], Dall proves that for a large class of ordered matroids pM, ăq, which he calls internally perfect, the restriction set poset Int ă pM q is the divisibility poset of a multicomplex, thereby showing Stanley's pure O-sequence conjecture for this class of matroids.…”

“…However, varying the order of the ground set produces significant changes in the above-mentioned structures and changing shelling orders yields even more, as we explore in this article. Examples of these differences are elucidated by Dall [11], who defines internally perfect matroids (a class of ordered matroids).…”

“…Recall that a multi-complex is a family of monomials closed under divisibility. It has received significant attention and has numerous connections to combinatorial commutative algebra, chip-firing games, matching theory, and more [22,24,16,28,11,20,14]. In Section 7, we show how pinned broken line shellings witness an h-vector decomposition studied in [20].…”

Triconed Graphs, weighted forests, and h-vectors of matroid complexes

“…The above conjecture has been open for over four decades, and there are some specific classes of matroids for which the above conjecture has been established to be true. In particular, these include cographic matroids by Merino in [11], lattice-path matroids by Schweig in [14], cotransversal matroids by Oh in [13], paving matroids by Merino, Noble, Ramirez-Ibanez, and Villarroel-Flores [12], internally perfect matroids by Dall in [4], rank 3 matroids by Há, Stokes, and Zanello in [6], rank 3 and corank 2 matroids by DeLoera, Kemper, and Klee in [5], rank 4 matroids by Klee and Samper in [7], and rank d matroids with h d ≤ 5 by Constantinescu, Kahle, and Varbaro in [2].…”