On a unimodality conjecture in matroid theory

A certain unimodal conjecture in matroid theory states the number of rank-r matroids on a set of size n is unimodal in r and attains its maximum at r=\lfloor n/2 \rfloor . We show that this conjecture holds up to r=3 by constructing a map from a class of rank-2 matroids into the class of loopless rank-3 matroids. Similar inequalities are proven for the number of non-isomorphic loopless matroids, loopless matroids and matroids.


Introduction
Let us first recall some elementary definitions (further details may be found in Oxley (1), Welsh (2) and the excellent appendix of White (3)).Let S n be a finite set of size n.A matroid M on the ground set S n is a collection of subsets I (M) of S n satisfying • if X ∈ I (M) and Y ⊆ X then Y ∈ I (M), • if X,Y ∈ I (M) with |X| = |Y | + 1, then there exists x ∈ X\Y such that Y ∪ {x} ∈ I (M).
Sets in I (M) are called independent sets.The rank of a set X ⊆ S n , denoted r(X), is the size of the largest independent set which it contains.The rank of the matroid r(M) := r(S n ).A set X is closed (or termed a flat) if r(X ∪ {x}) = r(X) + 1 for all x ∈ S n \X.We denote by F (M) the closed sets of M. The loops of M are the elements of the rank-0 flat.Also note that x ∈ S n is a loop if it is not contained in any of the independent sets of M.
A certain unimodal conjecture in matroid theory states that the sequence of the number of non-isomorphic rank-r matroids on S n , { f r (n) : 1 r n}, is unimodal in r and attains its maximum at r = ⌊n/2⌋ (see Oxley (1) or Welsh (4) p.300).It is easily seen that where p(n) is the number of integer partitions of n.The step between rank-2 and rank-3 is not as clear since the exact value of f 3 (n) remains unknown.We show, through construction of a map between a class of rank-2 matroids and loopless rank-3 matroids and known values of these numbers from the On-line Encyclopedia of Integer Sequences, that this unimodal conjecture holds for these rank-2 versus rank-3 matroids.Furthermore, we show the corresponding inequalities hold for the number of rank-2, 3 non-isomorphic loopless matroids, g 2 (n) g 3 (n), loopless matroids, c 2 (n) c 3 (n), and matroids, m 2 (n) < m 3 (n).
Let b i (n) be the number of partitions of the set S n .into i parts and b(n) be the n th Bell number.Let p i (n) the number of partitions of the integer n into i parts.The number of rank-2 matroids can be enumerated through considering the points and lines of the associated geometry.We have for proofs see Dukes (5)).The main results of this paper are given in Theorems 2.5, 2.6, 2.11 and 2.12.
2 Mapping rank-2 to rank-3 matroids Let M r (n) be the collection of rank-r matroids on S n .Let A r (n) be the collection of rank-r matroids on S n with at least one loop and B r (n) := M r (n)\A r (n).We define the map σ : It is easily checked that these collections of flats satisfy the axioms for a loopless rank-3 matroid.For M ∈ M r (n), let us write d(M) for the number of rank-1 flats of M (which we will refer to as the degree of M).Let us mention that for any loopless matroid M, the rank-1 flats of M partition the ground set.Similarly, for any matroid, the rank-1 flats partition the ground set less the set of loops.Also note that in the collection F 2 (M ′ ), there are precisely d(M) sets containing F 0 , 2 sets containing F i (for any 1 i d(M)) and one set containing F i ∪ F j (for all 0 i = j d(M)).
The following lemma shows that to each rank-2 matroid with at least one loop, there corresponds a rank-3 loopless matroid (although not necessarily unique).The following lemma classifies those matroids which map to a unique loopless matroid in B 3 (n) and those which do not.
1 , F (1) 1 , F (1) Proof: We show that the number of unique matroids in the image of A 2 (n) under σ is given by b(n + In the enumeration below, we divide the matroids to be counted in the image into two classes, those matroids M with d(M) = 2 and those with d(M) > 2. The former class projects different matroids to the same matroid in B 3 (n) and through the use of the previous lemma we take care of this over-counting, hence

✷
The corresponding inequality for the number of non-isomorphic loopless matroids is proved in Lemma 2.3.We do this in a similar manner as before, by showing that each rank-2 matroid (which is not a loopless matroid) of degree greater than 3 corresponds uniquely to a rank-3 loopless matroid.
Lemma 2.3 For all n 4, g 3 (n) Proof: We show the number of non-isomorphic matroids in the image of A 2 (n) under σ is given by Let us identify A ⋆ 2 (n) ⊆ A 2 (n) by placing an ordering on the elements of S n = {x 1 , . . .,

Let us now write
It is obvious that no two matroids in T (n) are isomorphic to one-another.Similarly with Ω k,l (n).We have simply reduced our class of matroids from A 2 (n) to A ⋆⋆ 2 (n) in the same manner as one moves from the set of partitions of a finite set of size n to the set of integer partitions of n.
The unions in the definition of A ⋆⋆ 2 (n) are strictly disjoint and no isomorphisms may occur between matroids in different classes or matroids in the same class.The same is true of the image of A ⋆⋆ 2 (n) under the map σ.We may directly enumerate the number of non-isomorphic matroids in B 3 (n) in the image of A ⋆⋆ 2 (n) under σ as The rightmost term is bounded below; for all n 2. As for p 3 (n), from Hall (6) [p.32], we have and so ✷

Matroids
The following lemma is needed in order to support the theorem which follows it.
and using Lemma 2.4, The problem has been reduced to showing 3 n−1 − (n + 3)2 n−2 + n 2 n−1 − 1 for all n 5, which is easily shown by induction.✷ We now show the number of rank-3 matroids dominates the number of rank-2 matroids by using two things: the first is the result proved previously, that the number of rank-3 loopless matroids is at least as large as the number of rank-2 loopless matroids; the second is the first few known values of the numbers c 2 (n) and c 3 (n).The latter knowledge makes the inequality strict.

Proof:
The number of rank-r matroids on S n is related to the number of loopless matroids on S n by In Theorem 2.5 we showed that c 3 (n) c 2 (n) for all n 5. Replacing r = 3 in the above expression and using the first few values of c 3 (n) (taken from row 3, table A058710, of Sloane ( 7)), A simple check shows that 830 n 6 + 75 n 5 − 3 n 4 − 3 n 3 − n 2 is greater than zero and increasing for all n 7. From Table 1 (see Appendix), the result is also seen to hold for n = 5, 6. Equality holds only for n = 5, for all other values of n the inequality is strict.✷

Non-isomorphic matroids
Proving the corresponding inequalities for the non-isomorphic numbers is more difficult.We first prove several lemmas related to the numbers p(n) which we will need in the proofs of the two remaining theorems. Lemma

Table 1 :
(7) number of partitions of the integer n + 1 whose first part contains the integer 1 is precisely p(n).The number beginning with i, for any 2 i n+1 2is at least 1 since we can have the partition n + 1 = i + (n + 1 − i).Also, the number n + 1 is a partition by itself, hence, Known values for the number of rank-2 and rank-3 matroids taken from Sloane(7).