Internally Perfect Matroids

In 1977 Stanley proved that the $h$-vector of a matroid is an $\mathcal{O}$-sequence and conjectured that it is a pure $\mathcal{O}$-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.


Introduction
An ordered matroid is a matroid M = (E, B) together with a linear ordering of the ground set E. Given an ordered matroid M and a basis B ∈ B(M), an element e ∈ B is internally passive (with respect to B) if there is a basis B ∈ B such that B = B − {e} ∪ {e } where e < e in the linear ordering of the ground set. An element that is not internally passive is internally active.
The internal order of an ordered matroid is the poset P int (M) = (B ∪1, int ) on the set of bases of M together with an artifical maximal element1, where B int B if and only if every internally passive element of B is internally passive in B . In [10], Las Vergnas proves that the internal order of an ordered matroid is a graded lattice and that the height of a basis B in P int (M) is given by the number of internally passive elements of B. The unique minimal element of P int (M) is the lexicographically smallest basis of M and is denoted B 0 (M). (We use the modifier "internally" to stress that our definitions depend on the notion of internal activity as opposed to the dual notion of external activity. As we will have no cause to mention the external activity again in this paper, we typically drop the modifier when the additional stress is superfluous.) Though the definitions of perfect, abundant, and deficient bases as stated above are useful in computations, they each have a more intuitive characterization in terms of the join operator of P int (M); see Proposition 3.5. Other highlights of Section 3 include proofs of the existence of perfect bases in any ordered matroid and that all rank 2 matroids are perfect (see Proposition 3.4). These preliminary results give way in later sections to our two main structural results concerning perfect matroids.
The first central result details to what extent minors of a perfect matroid M are perfect when the ordering of the ground set of each minor is induced from that of M. Theorem 4.1. Let M = (E, B, φ) be an internally perfect ordered matroid with initial basis B 0 and let F 1 and F 2 be disjoint subsets of E such that any element of F 2 ∩ B 0 is a coloop. Then the minor M/F 1 \ F 2 is internally perfect with respect to the ordering of its ground set induced by the order of M.
The second of our main results pertains to a conjecture of R. Stanley concerning the structure of the h-vectors of matroids. Indeed, this conjecture provided the original motivation for our study of internally perfect matroids. As such we take a momentary diversion to sketch the background of the conjecture as well as the work done in recent years toward finding a proof.
The h-vector, h(M) = (h 0 , h 1 , . . . , h r ), of a rank r matroid M is defined to be the h-vector of the independence complex of M, that is, the h-vector of the simplicial complex on E consisting of all subsets I ⊆ B for some B ∈ B. Given such a matroid M, the entry h i can be computed in a number of ways. For example, it is the coefficient on x r−i in the evaluation of the Tutte polynomial, T M (x, y), of M at y = 1; see [2]. Equivalently, for any linear ordering of E, the entry h i is the number of bases of M with r − i internally active elements; see [1]. By a result of Las Vergnas, one can also obtain h i by counting the number of bases at height i in P int (M); see Theorem 2.2. Any of these results implies that h 0 = 1 and h i ≥ 0 for all i ∈ [r].
While a result of Kruskal [9] and Katona [7] gives an explicit description of the possible vectors that can occur as the h-vector of simplicial complexes, no such description is yet known when restricting to the matroid complex case. So when given a vector v ∈ N r+1 it is natural to ask for tests that would verify (or rebuff) v as the h-vector of a rank r matroid.
In [17], Stanley  ). An O-sequence is called pure if it is the O-sequence of a pure order ideal, that is, of an order ideal whose maximal elements all have the same coordinate sum. Stanley proved that the h-vector of any rank r matroid is the O-sequence of an order ideal in P and made the following conjecture: (Stanley 1977). The h-vector of a matroid is a pure O-sequence.
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 ).
In each of the other papers referenced above, general properties of either matroids or order ideals are studied and then used to prove Stanley's Conjecture for a particular class of matroids. The second of our main results is of this type. It states that, up to a relabeling of the nodes, the internal order of an internally perfect matroid is a pure order ideal.
Theorem 5.1. The internal order of an internally perfect matroid is isomorphic to a pure order ideal.
In particular this result implies that if M = (E, B) is a matroid such that there exists a linear ordering of the E making M an internally perfect matroid, then M satisfies Stanley's Conjecture.
This article is organized as follows. In Section 2 we fix notation and give the necessary background on matroids and the internal order. In Section 3 we define and prove preliminary properties of perfect, abundant, and deficient bases of an order matroid. We turn to the proof of Theorem 4.1 concerning the minors of perfect matroids in Section 4. We also provide an example showing that the theorem is the best possible result when the linear order on the ground set of the minor is induced from that of the original matroid. Moreover, we conjecture that one can always find a reordering of the ground set of a minor of perfect matroid such that the minor is perfect with respect to the new order. In Section 5 we prove Theorem 5.1 by explicitly giving the poset isomorphism between the internal order of a perfect matroid and a pure order ideal. Finally, in Section 6 we construct a variety of examples of internally perfect matroids including an infinite family of cographic matroids as well as an example of an internally perfect matroid that is not contained in any of the classes for which Stanley's Conjecture is known to hold.

Preliminaries: Matroids and the Internal Order
First we fix some notation. For a positive integer n we write [n] := {1, 2, . . . , n}. We use standard basic set theory notation with the following exception: for a singleton {f } we typically suppress the set braces and simply write f .
We now review some basic matroid terminology following [15]. Then we recall the facts we need concerning activities and the internal order essentially following [10]. We assume a basic familiarity with posets and simplicial complexes at the level of Chapter 3 of [18] and Chapter 1 of [13], respectively.
be a matroid and let F ⊆ E. Then F is an independent set of M if it is a subset of some basis. The set of all independent sets of M is denoted I. A subset of E that is not independent of M is a dependent set and a dependent set that is minimal with respect to inclusion is a circuit. The set of all circuits of M will be denoted C = C(M). A loop of M is a circuit consisting of one element. We write L(M) for the set of all loops. If two elements e, f ∈ E form a two-element circuit then they are said to be parallel . A maximal collection of elements of E containing no loops such that the elements are pairwise parallel in M is called a parallel class of M.
The rank of a subset S ⊆ E, denoted rank M (S), is the cardinality of any maximal independent set of M contained in S. It is easy to see from the definition that every basis of M has the same rank r, called the rank of M and written rank(M). When the matroid under consideration is clear from the context, we typically drop it from the notation.
The dual matroid, M * = (E, B * ), of M is the matroid whose bases are the complements of bases in M. The bases of the dual matroid are called cobases. More generally, we prepend the prefix "co-" to any object associated to a matroid to indicate that we are discussing the corresponding dual object. In particular, the cocircuits of M are the circuits of M * and will be denoted C * . For example, a coloop of M is a loop in the dual matroid M * . Equivalently, a coloop is an element of the ground set that is in every basis of M.
Let M = (E, B) be a matroid, B be a basis of M, and suppose e ∈ E is not an element of B. Then there is a unique circuit, C(B; e), of M contained in the set B ∪ e called the fundamental circuit of B with respect to e. Similarly, for an element f ∈ B the fundamental cocircuit of B with respect to f is the unique cocircuit C * (B; f ) of M contained in the set E \ B ∪ f . It is a basic fact that for b ∈ B and b / ∈ B the following are equivalent: The set B above is said to be obtained from B by pivoting .
Let M be a rank r matroid. As the set I of independent sets of M is nonempty and closed under taking subsets, it forms a simplicial complex ∆(M) on E, called the matroid (or independence) complex of M. The dimension of the matroid complex ∆(M) is the rank of M. The f -vector and h-vector of a matroid M are the f -and h-vector of its matroid complex, respectively.
Two useful constructions for creating new matroids are deletion and contraction defined, respectively, as follows. Let T ⊆ E. The deletion of M at T , written as M \ T , is the matroid whose independent sets are I \ T for each I ∈ I, while the contraction of M at T is the matroid defined by M/T := (M * \ T ) * . A minor of M is any matroid that can be obtained from M by a sequence of deletions and contractions.
Let M = (E, B) be a matroid with |E| = n. If φ : E → [n] is a bijection, then the usual ordering on [n] induces a linear order on E where e ≺ φ f if φ(e) < φ(f ). The matroid M together with such a bijection φ is called an ordered matroid . When the map φ is clear from the context, we write e < f in place of e ≺ φ f . In other words, e is internally active in M with respect to F if e ∈ F and there is a cocircuit C * of M contained in E − F ∪ e such that e = min C * . When the underlying ordered matroid is clear from the context we will simply say that e is internally active in F . Any element of F that is not internally active with respect to F is internally passive. We write IP M (F ) for the set of all internally passive elements of F .
In this paper we typically focus on the case when F = B is a basis of M. Notice that in this case an element f ∈ B is internally active in B if and only if f is the minimum element of the fundamental cocircuit C * (B; f ). As M is an ordered matroid, there is a lexicographically-smallest basis which we denote by B 0 and call the initial basis of M. It is trivial to check that IA(B 0 ) = B 0 and IP(B 0 ) = ∅. Now consider an arbitrary basis B ∈ B. By Corollary 2.3 below, any internally active element of B is also an element of the initial basis B 0 , though the converse need not hold. On the other hand, if e ∈ B − B 0 , then e is internally passive in B. This motivates us to partition B into three sets defined as follows: We call the elements of S = S(B) perpetually passive (with respect to M and B), while the elements of T are called provisionally passive. The reason for this choice of nomenclature will become obvious in the next paragraph. We will often write S T A for B when we want to emphasize this partition of the elements of B.   The h-vector of M is (1, 3, 5, 5). The natural ordering on the ground set yields an ordered matroid whose internal order is shown on the left in Figure 1. In this internal order we have highlighted the three principal chains in P int : the 5principal chain has length 2, while the 6- The internal order of this ordered matroid is shown on the right in Figure 1, where we have highlighted the clean bases in red. The two posets in Figure 1 are not isomorphic as there are two height-3 bases in the first poset that cover exactly one element while there is only one height-3 basis in the second poset that covers one element. Nonetheless, both internal orders have a number of features in common. For example, both are graded lattices with h i (M) bases at height i.
Next we give a structural result due to Las Vergnas for the internal order of an ordered matroid. For this we need the following definition: Given an independent set I of a rank r ordered matroid M, the minimum basis containing I, written MinBas(I), is defined to be the lexicographically-least basis in B containing I.
The following corollaries will be useful later and are direct consequences of Theorems 2.1 and 2.2. The first verifies our previous claim that internally active elements of a basis B are always elements of the initial basis B 0 , while the second computes the height of the internal order in terms of a matroid's rank and its number of coloops. Proof. By Theorem 2.2 the internal order of a matroid is a graded lattice and B 0 is the unique minimal element, so for any basis B in the matroid there is a saturated chain saturated chain B 0 int B 1 int · · · int B i int B from B 0 to B in the internal order. But then we can apply part (iii) of Theorem 2.1 which assures us that for any basis B and any such saturated chain in the internal order, we have  We now prove two results concerning principal bases. For the first, recall that a (finite) saturated chain in a poset P is a subposet whose ground set consists of elements p 1 , p 2 , . . . , p k such that p i+1 covers p i for all i ∈ [k − 1].
, the set of f -principal bases forms a saturated chain.
where e i < e i+1 with respect to the order on the ground set of M. Then every f -principal basis of M is of the Hence the set of f -principal bases forms a saturated chain.

For a basis B = S T
A and an f ∈ S we define the f -part of T to be the set T (B; f ) consisting of provisionally passive elements of B that are also provisionally passive in MinBas(f ∪ T ), that is, T (B; f ) := T (MinBas(f ∪ T )). For a given basis B, one can read off the f -part of T easily from the internal order using the following proposition.
A be a basis of an ordered matroid and let f ∈ S. Then MinBas(f ∪ T ) is the maximal f -principal basis that is less than (or equal to) B in the internal order. We write D, A, and P for the set of deficient, abundant, and perfect bases of M, respectively. Clearly, these sets partition the bases of M, that is, B = D A P. Moreover, the set of perfect bases P is never empty, as the next proposition shows.
Now suppose M is a rank-r ordered matroid with c coloops such that r − c = 2. Then, by Corollary 2.4, the height of P int (M) is two. Thus, if B = S T A is a basis of M, then |S| ∈ {0, 1, 2}. In any case, the previous paragraph implies that B is perfect since either |S| = 1, or |S| ∈ {0, 2} in which case T = ∅.
In particular, the previous proposition tells us that if M is an unordered matroid of rank 2, then M is internally perfect for any linear ordering of its ground set. This is far from the case for matroids of higher rank. In Example 2 we see that the matroid M of Example 1 is not perfect with respect to the natural ordering of its ground set but is perfect with respect to the ordering (2, 3, 1, 4, 5, 6). Furthermore, in Example 3 we supply a matroid that is not perfect with respect to any linear ordering of its ground set. Thus B is deficient with respect to the natural order on [6]. On the other hand, one can verify that every basis of the ordered matroid whose internal order is given by the poset on the right is perfect.
Example 3. In this example we provide a matroid that is not internally perfect for any linear ordering of the ground set. Let M be the vector matroid of the columns of the matrix M (over Q) given by Then M is a rank-5 matroid with 42 bases and h-vector (1,3,6,9,12,11). The internal order of M(M ) (with respect to the natural order on the ground set) is given in Figure 2 where the f -principal chains are colored green and the perfect (respectively abundant, deficient) bases are black (respectively blue, red). Using, for example, the Macaulay2 package Posets (see [6]), one can perform a brute force computation to show that none of the 8! linear orders on the ground set of M yields a perfect ordered matroid.
We now use this example to make some observations in order to motivate upcoming results. First notice that if B = S T A is a perfect basis of M, then B covers exactly |S| bases in P int and that B can be expressed as the join of principal bases in a unique way. For example, the perfect basis 567 13 covers the bases 56 13 4 , 567 We now proceed to prove that the properties verified in the previous example hold in general. We begin by characterizing the three types of bases in terms of the interplay between principal bases and the join operator in the internal order.   (4) is the unique way to write B as the join of f -principal bases for f ∈ S, and multi-decomposable otherwise. Since every basis of M is of exactly one of these three types, we obtain a partition of B(M) which we call the decomposability partition. With this terminology in hand we can now state our characterizations of perfect, abundant, and deficient bases. If the basis B is deficient, then IP(B ) = S ∪ T IP(B) which proves that B is strictly smaller than B in the internal order. Now suppose there is a collection of principal bases whose join is B. Then this collection must be of the form is the maximal f -principal smaller than B in the internal order. Hence B cannot be written as the join of principal bases. Now fix a basis B that is not deficient. Then IP(B ) = IP(B) and so the bases B and B coincide. Note that in this case we can write B = f ∈S P f for any collection of principal bases {P f | S(P f ) = f } such that T (P f ) = T . If B is a perfect basis, we must have P f = MinBas(f ∪ T ) and so (4) is a unique expression for B as the join of principal bases. On the other hand, if B is abundant then we can write B = P f where S(P f ) = f and As the bases P f are all principal and the sets T (P f ) partition T it follows that P f is an expression of B as the join of principal bases and that this expression is different from (4), completing the proof.  This union is disjoint, so it follows that B is perfect in this case.
In the subsequent sections we will be interested in perfect ordered matroids, that is, ordered matroids whose bases are all perfect. Note that Proposition 3.6 supplies us with a useful computational tool to verify that a given ordered matroid is perfect insofar as it implies that to check that every basis of an ordered matroid is perfect, it is enough to check that all coatoms of the internal order (i.e., all bases covered by the artificial top element1) are perfect. Also note that implicit in the proof of Proposition 3.6 is the fact that a perfect basis B = S T A covers at most |S| many bases in the internal order. In [10], Las Vergnas shows that the number of bases covered by an arbitrary basis B in the internal order is no greater than the number of internally passive elements of B. In the next proposition we compute the exact number of bases covered by a perfect basis B in the internal order. As Equation (5)  Given distinct f, g ∈ S, the bases A f and A g are incomparable in the internal order of M and their join is B. If we write B g for the basis in [A g , B] that is covered by B, then it follows that B f = B g for distinct f, g ∈ S. This implies that B covers at least |S| bases in P int . Since the proof of the previous proposition implies that B covers at most |S| bases, the result follows.

Minors of Perfect Matroids
In the previous section we studied local properties of internally perfect bases of an ordered matroid. In this and subsequent sections, we take a global perspective and investigate internally perfect ordered matroids, that is, ordered matroids whose bases are all internally perfect. For brevity's sake we typically call such an ordered matroid a perfect matroid. Moreover, we call an unordered matroid (internally ) perfect if there is some linear ordering of its ground set so that the resulting ordered matroid is internally perfect. Our goal in this section is to study the minors of perfect matroids in order to prove the following theorem. For the remainder of this section let us fix an ordered internally perfect matroid M = (E, B, φ) and an element e ∈ E. In the trivial case where M has exactly one basis, Proposition 3.4 implies that M is internally perfect. Moreover, the same proposition implies that every minor of M is internally perfect. So without loss of generality we assume hereafter that |B(M)| > 1.
We prove Theorem 4.1 in three main steps in order of increasing complexity. First we show that deleting or contracting a loop or coloop of M preserves perfection. Then we prove that M − e is perfect whenever e / ∈ B 0 (M); see Section 4.1. Finally, in Section 4.2 we demonstrate that M/e is perfect whenever M is.
We begin with the case when e ∈ E is either a loop or a coloop. It is a straightforward exercise to prove that in this case the posets P int  Our goal is to prove that N is perfect whenever e is not in the initial basis of M. We will need the following well-known (and easy to prove) facts characterizing the bases, circuits and cocircuits of N in terms of those of M. As any set B ⊆ E − e that is a basis of N is also a basis of M, it is helpful to introduce notation to clarify the matroid of which we are considering B a basis.   Taken together, Corollary 4.4 and the remarks at the end of the introduction to this section show that N = M − e is perfect whenever e is a (co)loop or is not an element of the initial basis of M. This result is the best possible in the sense that there are internally perfect ordered matroids for which deleting an element of their initial bases yields an ordered matroid that is not perfect with respect to the order on the original matroid, as the next example shows.

, 6, 7) is internally perfect
It is natural to ask whether the phenomenon witnessed in the previous example is borne out in the general case. More precisely, is there always a reordering of the ground set of the ordered matroid N = M − e (where e ∈ B 0 (M)) such that N is internally perfect? Computational evidence verifies that this is true for matroids with small rank (r(M) ≤ 4). Establishing this in the general case would prove the following conjecture.

4.2.
Contractions of Perfect Matroids. We have seen that internal perfection is preserved when deleting elements from a perfect matroid as long as the elements deleted do not belong to the initial basis. Our present goal is to show that internal perfection and matroid contraction are more compatible in the sense that any contraction of an internally perfect is internally perfect. As before, we fix an ordered internally perfect matroid M = (E, B, φ) and an element e ∈ E. The proof presented here is essentially a detailed analysis of STA-decompositions of the minimal bases MinBas N (I) and MinBas M (I), where N = M/e and I is an independent set of N .
First we collect some well-known results concerning matroid contractions (of unordered matroids).
Since any basis is the disjoint union of its sets of internally active and internally passive elements, we have the shown the following proposition.    To prove the first case of the proposition, let e ∈ S . Then S = S − {e} and the perfection of M implies that T = T − T e and A = A ∪ T e . We now apply one of the two last cases of (6) to obtain S = S − e = S , proving the lemma in the case when e ∈ S . Now suppose e ∈ T . The prefection of M implies that there is a unique f ∈ S such that e ∈ T f . It follows that in this case S = S , T = T − {t ∈ T f | t ≤ e}, and A = A ∪ {t ∈ T f | t < e}. Using these facts and the second case of (6), we obtain S = S = S , as desired.
We now have all of the ingredients necessary to show that B is perfect when e ∈ S : Combining Corollary 4.4 and Theorem 4.11 proves Theorem 4.1. We saw in Example 4 that if we wish to preserve the ordering of the ground set of a perfect matroid M when passing to a minor N , then in general we may only delete or contract elements of B 0 that are coloops. If, on the other hand, we allow a reordering of the ground set of N , then we have found no example of a minor of a perfect matroid that is not perfect. Thus we close by offering the following conjecture that would follow directly from our results here and Conjecture 4.5.
Conjecture 4.12. The family of internally perfect unordered matroids is minorclosed.

Internally Perfect Matroids and Stanley's Conjecture
As mentioned in the introduction, our study of internally perfect matroids is motivated by the desire to prove Stanley's Conjecture using the internal order of a matroid as follows: Let M = (E, B, φ) be an ordered matroid and let P be the internal order of M less the artificial top element1. Write F = E−(B 0 (M)∪L(M) and let S be the monoid over N generated by F , that is, Call a map µ : P → (S, dom ) valid if µ is a height-preserving map whose image is an order ideal of (S, dom ). Then since P is pure and the number of bases of M at height i in P equals h i (M), the existence of a valid map µ implies that M satisfies Stanley's Conjecture.
The main result of this article is that internally perfect matroids satisfy Stanley's Conjecture: In order to prove that such a matroid satisfies Stanley's Conjecture we need to produce a pure order ideal whose O-sequence equals the h-vector of the matroid. We will prove the theorem via a sequence of lemmas that actually imply a stronger result: the internal order of an internally perfect matroid is isomorphic to a pure order ideal.
For the remainder of this section we fix a rank-r internally perfect matroid M on the ground set E. If e ∈ E is a loop in M, then e does not occur in any basis. This in turn implies that the h-vector of M − {e} equals the h-vector of M. Therefore, we assume without loss of generality that M contains no loops. The first step is to define the monoid in which we will construct the appropriate order ideal. Let It is evident that for any basis B the coordinate sum of µ(B) is equal to the number of internally passive elements of B, and hence µ is height-preserving.
The first lemma we will prove is that µ injective. for all f ∈ F . Since M is internally perfect, so are both B and B . So applying Proposition 3.5 for perfect bases, we see that which shows µ is injective.
We will now prove that if M is an internally perfect matroid, then M satisfies Proof. To see that the image of µ is an order ideal we need to check that for any basis B = S T A of M and any e f in the support of µ(B), the vector µ(B) − e f is in the image of µ. Note that for any element f ∈ F the generator e f is in the support of µ(B) if and only if the element f ∈ S. Since B is a perfect basis it covers |S| bases in the internal order by Proposition 3.7. By Lemma 5.2, the map µ is injective, so the |S| bases covered by B in P int get mapped to |S| distinct vectors in S with support contained in S whose coordinate sum is one less than that of µ(B). As µ(B) has support |S|, it follows that for every f ∈ S there is a unique basis B ∈ B such that µ(B ) = µ(B) − e f . Now we check that the image of µ is a pure order ideal. By Theorem 2.2, the internal order P int is a graded lattice. It follows that P int (the internal order with the top element removed) is a pure graded poset, and so the image of µ is a pure order ideal.
Finally, since µ is an injective map sending the h i (M) bases in P int at height i to vectors with coordinate sum i in S, it follows that the h-vector of M is a pure O-sequence completing the proof.
The preceding proof shows that the internal order of a perfect matroid M is isomorphic to the corresponding valid order ideal given by the map µ, and hence that M satisfies Stanley's Conjecture. Stanley's Conjecture is known to hold for a number of families of matroids, and in the next section we compare these families to the family of internally perfect matroids. In particular, in Example 5 of the next section we provide a perfect matroid that is not in any of the families for which Stanley's Conjecture is known to hold.

Perfect Matroids: Constructions, Examples and Counterexamples
We now turn to the construction of internally perfect matroids. We have already seen in Proposition 3.4 that every rank-2 matroid is internally perfect. Other matroids that are trivially internally perfect include the graphic matroid M(C n ) where C n is the the cycle on n vertices, as well as the 0-and 1-sums of perfect matroids.
The smallest graphic matroid that is not internally perfect for any ordering of its ground set is M(K 4 ). As K 4 is planar and self-dual, we see that the cographic matroids are not contained in the set of internally perfect matroids. On the other hand we now construct an infinite family of nontrivial perfect cographic matroids as follows. Let where PE and PL denote the principal extension and principal lift, respectively. It is easy to see that, for all positive integers r, the matroid M r is a graphic. Indeed, for r > 1 if we let G r = (V r , E r ) be defined by V r−1 ∪ {r + 1} and E r = E r−1 ∪ {e, f } where e = (r, r + 1) and f = (1, r + 1), then we have M(G r ) = M r . For r = 1, 2, 3, the graphs corresponding to the M r are given in Figure 3. The graph corresponding to M r is planar and self-dual, so M r is also cographic.  Figure 3. graphs giving rise to M r for r = 1, 2, 3 Proposition 6.1. For any positive integer r, the matroid M r is internally perfect with respect to the natural ordering on its ground set.
Proof. We proceed by induction on r and note that the base case is trivial. For r greater than 1, each basis of M r takes exactly one of the following three forms: M is a truncation [3], or (8) h s ≤ 5 [3].
This list represents the state of the art concerning Stanley's Conjecture at the writing of this article. For the sake of brevity let us call a matroid M interesting if it satisfies none of the above properties. A priori it is not clear that there are any interesting internally perfect matroids. Thus, to show that Theorem 5.1 is of theoretical interest, we must exhibit such a matroid. In the next example we describe an interesting, perfect, rank-7 matroid on 10 elements. Then we generalize this matroid to obtain an infinite family of interesting matroids and we conjecture that every such matroid is perfect. We first show that M is interesting and then prove that M is internally perfect. The restriction of M * = N to {e 1 , e 2 , e 3 , e 5 } is isomorphic to the uniform matroid U 2,4 which shows that M * is not graphic. So M does not satisfy Property (1) above. To see that M doesn't satisfy (2) above note that the restriction of M * to the set {e 5 , e 6 , e 7 , e 8 , e 9 , e 10 } is the matroid obtained from the cycle C 3 by adding a parallel element to each edge. This restriction is not transversal by Theorem 14.3.1 in [19]. It is easy to see that transversal matroids are closed under taking restrictions, and it follows that M * is not transversal. Also, M * is a rankthree matroid on ten elements with six parallel classes, and so satisfies neither (3) nor (4)  where, for example, 78910 refers to the set {e 7 , e 8 , e 9 , e 10 }. Since 1234 is a circuit with fewer than rank(M) elements, M is not paving. Moreover, one can show that for any circuit C of M the cardinality of the closure of C is at most 8. So M contains no spanning circuits. By Remark 1.12 in [3], this shows that M is not a truncation. Finally, the h-vector of M is (1, 3, 6, 10, 13, 15, 14, 6) which shows that M does not satisfy (8) above. So M is interesting. Now we show that M is internally perfect with respect to the linear order induced by the order of the columns of the matrix N . By Proposition 3.6, it is enough to check that the maximal bases of M (with respect to the internal order) are perfect. Each of these bases is of the form S T A where S = {3, 7, 9} and A = ∅. We give these bases below together with the partition 3 T (B i ;3) 7 T (B i ;7) 9 T (B i ;9) : B 1 = 379 1256 = 3 12  Building on the previous example, we now define an infinite family of interesting matroids. For n ≥ 3, let G = G n,D be a graph consisting of the double cycle C 2 n (the n-cycle with two copies of each edge) together with some subset D of the edges {(1, i) | i ∈ [n] \ 1}. Orient each edge {i, j} so that i → j if i < j, and order the set of all edges lexicographically. Write N = N (n, D) for the vertex-edge incidence matrix of G and let N be the (n × 4)-matrix with first two rows given by 2 1 3 3 1 1 1 1 and all other rows having all entries equal to zero. Let N = N n,D be the ordered vector matroid given by the matrix [N |N ] and let M = N * . Note that N is a rank-n matroid on 2n+|D|+4 elements with n+|D|+3 parallel classes. Moreover, the matroids U 2,4 and C 2 n can be obtained as restrictions of N . This shows that the matroid M = N * does not satisfy any of the properties (1)-(4) above. The matroid M has rank n + |D| + 4 ≥ 6 and the set {e 1 , e 2 , e 3 } is always a circuit of M, and so M does not satisfy (5) or (6). 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).
Computer experiments have shown that for n ≤ 7 and any collection of diagonals D the matroid M = (N n,D ) * is internally perfect. We conjecture that this is always the case. Conjecture 6.2. For n ≥ 3, let M be the dual of the ordered matroid N n,D defined above. Then M is internally perfect.
We now compare the family of internally perfect matroids to each of the families of matroids for which Stanley's Conjecture is known to hold. 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.
Example 7. As noted at the beginning of this section the matroid M = M(K 4 ) is a cographic matroid that is not internally perfect for any linear ordering of the ground set. One easily verifies that M is a rank-3, self-dual, paving matroid on six elements. Moreover, it satisfies the first condition of property (4). It follows that F i IP for i ∈ {1, 4, 5, 6}.
Example 8. Next we consider the rank-r uniform matroid on n elements, M := U r,n . We will show that if r > 2 and n ≥ r + 1, that M is not internally perfect with respect to the natural ordering of the ground set, from which it will follow that M is not perfect for any such ordering. Note that, since M * is also uniform, if  In particular, the rank-3 uniform matroid U 3,5 is not perfect. It is well-known (and easy to prove) that (U 3,5 ) * = U 2,5 is a rank-2 transversal matroid. Moreover, U 3,5 is obviously a truncation of U 4,5 . Thus U 3,5 ∈ F i for i ∈ {2, 4, 5, 6, 7} and we have shown that F i IP for any such i. Example 9. Let N be the rank-2 linear matroid on eight elements defined by the matrix 1 1 0 0 1 1 1 1 0 0 1 1 −1 −1 1 1 and let M = N * . Then M is a rank-6 matroid on eight elements whose dual has four parallel classes. Moreover, the h-vector of M is {1, 2, 3, 4, 5, 6, 3}. It follows that M ∈ F i for i ∈ {3, 4, 8}c . Checking all 8! linear orders of the ground set, one can show that M is not internally perfect for any such ordering. This implies that IP F i for i ∈ {3, 4, 8}.
The previous examples show that none of the families of matroids for which Stanley's Conjecture is known to hold consist entirely of perfect matroids. Moreover, note that the duals of the matroids U 3,5 and M from Example 8 and 9, respectively, are rank-2 matroids. This implies that (U 3,5 ) * and M * are internally perfect for any ordering of their respective ground sets, and hence that the set of internally perfect matroids is not closed under matroid duality.