Stanley’s Matroid Conjecture and a Recursive
Construction of Order Ideals
Jorge Alberto Olarte
Contents
1 Introduction 3
1.1 Notation . . . 4
2 Simplicial Complexes 5 2.1 Basic Concepts . . . 5
2.2 Stanley-Reisner Ideals and Hilbert Series . . . 5
2.3 Shellings . . . 8
3 Matroids 11 3.1 Basic Concepts . . . 11
3.2 Matroid Examples . . . 16
3.3 Duals, Minors and Tutte Polynomial . . . 20
3.4 Ordered Matroids and Shellings . . . 25
4 Order Ideals and Stanley’s Conjecture 31 5 Cographic Matroids 35 6 Generalized Construction 44 6.1 The Construction . . . 44
Acknowledgements
I would like to thank Tristram Bogart for all of his guidance and useful suggestions. I would also like to thank Federico Ardila for showing me the problem and motivating me to start working on it. I also want to thank Jos´e Alejandro Samper for many useful discussions and for giving me early access to his work. I finally want to thank my parents and my sisters for their everyday support.
Chapter 1
Introduction
The class of simplicial complexes is studied by various branches of math-ematics such as algebraic topology, combinatorics and commutative alge-bra. From triangulation of topological space as a tool to study of homotopy groups using simplicial homology to the study of monomial ideals using Stanley-Reisner rings, simplicial complexes appear as a fundamental tool in mathematics. The h-vector of a simplicial complex is an invariant that encodes much of the important information of these objects. It is strongly related to homology and the Hilbert functions.
Matroids form a particular family of simplicial complexes, very rich in its combinatorial structure. They appear naturally in many branches of mathe-matics, such as graph theory, matching theory, linear algebra, commutative algebra, model theory, geometry, etc. Theh-vector of matroids seem to be particularly interesting and in the 70’s Richard Stanley gave a conjecture on how to characterize these sequences [20]. While the problem remains open in general, many partial results have been developed in recent years [6, 7, 8, 10, 11, 12, 15, 16, 18, 21].
The goal of this thesis, is to present a construction that shows a new proof that the conjecture holds for the class of cographic matroids (which was already known to satisfy the conjecture) and to show a possible way to extend this construction to any matroid in general.
In chapter 2 we review basic concepts of simplicial complexes and their connection to commutative algebra. We define and motivate theh-vector of simplicial complexes.
We review basic results on matroid theory in chapter 3 and we study fur-ther the structure ofh-vectors for this partcular class of simplicial complexes.
In chapter 4 we study order ideals and O-sequences. We state the con-jecture of Stanley and we review previous work on it.
We present a new proof of the conjecture for cographic matroids in chap-ter 5 and we study this construction.
Finally, in chapter 6 we show how to extend this technique and study a possible way to prove the conjecture for any matroid in general. We also show how the construction applies to corank 2, uniform matroids and paving matroids.
1.1
Notation
We will use the following notation
• Instead of writingA\Bfor the set of all elements inAand not inB, we useA−B to avoid confusion with the restrcition operator in matroid theory denoted by\.
• Ifn∈N, [n] :={1,2, . . . n}.
• Ifn∈N, then-analog ofx is [x]n:={1 +x+x2+· · ·+xn−1}
• IfA andB are sets of monmoials,AB={mn | m∈A n∈B}.
• For any setA,P(A) is the set of all subset ofA.
• Ifx= (x1. . . xr) and a= (a1. . . ar),xa =xa11. . . xarr.
• If G is a graph then E(G) is the set of edges of E and V(G) the set of vertices. IfA ⊆ V(G), G[A] is the graph induced by A, that is, a graph with vertex set A and as edges every edge of E(G) that goes between any two elements ofA.
Chapter 2
Simplicial Complexes
2.1
Basic Concepts
Definition 2.1. A simplicial complex consists of a set ∆ of subsets of a finite set V such that if A ⊆ B ∈ ∆ then A ∈ ∆. The elements in V are calledvertices and it is assumed that{v} ∈∆ for eachv∈V. The elements of ∆ are called faces and faces which are maximal by inclusion are called facets. For F ∈ ∆ the dimension of F is |F| −1. The dimension of ∆ is the maximal dimension of its faces. If all the facets of ∆ are of the same dimension we say that ∆ ispure.
Definition 2.2. The f-vector of a simplicial complex ∆, isf(∆) = (f0, f1 . . . fd) wherefi is the number of i−1 dimensional faces of ∆.
Example 1. LetV ={a, b, c, d, e}and let ∆ be the set of all the subsets of
{a, b},{a, c, d} and {b, d, e}. ∆ is not pure as {a, b} is a facet of dimension 1 while{a, c, d}and{b, d, e}are facets of dimension 2. Simplicial complexes are often interpreted geometrically. That is, each simplicial complex rep-resents a topological space. To do that, each vertex is placed in Rd where
d is the dimension of ∆ (each vertex is placed in different points), and if
A∈∆ then the convex hull ofA is drawn. Figure 2.1 shows the geometric representation of ∆. We have thatf(∆) = (1,5,7,2).
2.2
Stanley-Reisner Ideals and Hilbert Series
In this section we review the connection of simplicial complexes with com-mutative algebra. We use [14] as reference. Let ∆ be a simplicial complex
a b
c
d e
Figure 2.1: The geometric represantation of ∆
with vertex set [n], letkbe any field of characteristic 0,x= (x1, . . . xn) and
S=k[x]. S is a vector space over k, and we have
S = M
a∈Nn Sa
Where Sa is the vector subspace generated by xa. As SaSb = Sa+b,
we say that S is a Nn graded k-algebra. It can also be considered as an N
gradedk-algebra as
S=M
a∈N
Sa
where Sa is the subspace generated by all monomials of degree a. Again,
SaSb =Sa+b.
Definition 2.3. A monomialxa ∈S is square-free if all the entries ofaare 0 or 1.
Now for everyσ⊆[n] we can associateσ with its characteristic function and hence to the square-free monomialxσ :=Q
i∈σxi.
Definition 2.4. The Stanley-Reisner ideal of a simplicial complex ∆ is the idealI∆=< xσ | σ /∈∆>. The Stanley-Reisner ring isk[∆] =S/I∆.
Via this correspondence, it is easy to show that square-free monomial ideals, that is, ideals generated by square-free monomials, are in bijection with simplicial complexes. Thus the study of squafree ideals can be re-duced to the study of simplicial complexes.
Definition 2.5. LetM be anS-module. We say that M isNn graded if
M = M
a∈Nn
as a vector space overk and SaMb ⊆Ma+b. For any Nn graded S module
we define theNn-graded Hilbert series as
H(M;x) = X
a∈Nn
dimk(Ma)xa
If we replacexi =tfor alli∈[n] we get theN-graded Hilbert series,H(M;t).
Hilbert series are one of the most important notions in algebraic geom-etry and combinatorial commutative algebra. Hilbert series live in the ring of formal power seriesk[[x]] and it can be expressed as a rational function. For example, using the fact that 11−t = 1 +t+t2. . ., the Hilbert series for
S itself is
H(S;x) =
n
Y
i=1
1 1−xi
Which equals the sum over all monomials of S. Note that k[∆] is an S -module. But moreover, it is alsoNn graded as partitioningS by any
mono-mial ideal, an ideal generated by monomono-mials, clearly yields an Nn-graded
module. So we focus the rest of the section on calculating the Hilbert series for Stanley-Reisner rings of simplicial complexes.
Proposition 2.6. TheNn-graded Hilbert series of k[∆]is
H(k[∆],x) =X
σ∈∆
Y
i∈σ
xi
1−xi and the N-graded Hilbert series is
H(k[∆], λ) = 1 (1−λ)d
d
X
i=0
fiλi(1−λ)d−i
where d=dim(∆) + 1.
Proof. I∆ contains all of monomials whose support is not in ∆. In other
words, ifa= (a1. . . an) we definesupp(a) ={i∈[n] | ai 6= 0}, and we have
thatxa ∈I
∆ if and only ifsupp(a)∈/ ∆. Then k[∆] is generated by all the
monomials whose support is in ∆. So
H(k[∆],x) = X{xa | supp(a)∈∆}
= X
σ∈∆
X
{xa | supp(a) =σ}
= X
σ∈∆
Y
i∈σ
xi
Now if we letxi =λfor all i∈[n], we get for every faceσ such that|σ|=i
a term of the form λi
(1−λ)i. Having (1−λ)d as common denominator gives
the second equation.
Consider the numerator of the polynomial of theN-graded Hilbert series,
as polynomialh∆(λ), that is
d
X
i=0
fiλi(1−λ)d−i=h0+h1λ1+· · ·+hdλd (2.1)
We call h∆(λ) the h-polynomial and the vector h(∆) = (h0, . . . hd) the h
-vector of ∆. From the above equation we get that the h-vector and the
f-vector uniquely determines each other via the following relations
hk= k
X
i=1
(−1)k−i
d−i k−i
fi
and
fk= k
X
i=1
d−i k−i
hi
Theh-vector will be the main object of study throughout this thesis.
2.3
Shellings
In this section we show how theh-vector may be computed for a special case of simplicial complexes. We use [1] as reference.
Definition 2.7. Let ∆ be a pure simplicial complex. Ashelling of ∆ is an ordering of the facetsF1, F2, . . . , Ft such that, for each 1≤i < j ≤tthere
are 1≤k < j and x ∈Fj such thatFi∩Fj ⊂Fk∩Fj =Fj− {x}. A pure
complex is said to beshellable, if such shelling exists.
In other words, a shelling is an ordering of the facets such that each facet
Fi meets the complex generated by its predecessors, ∆i−1 := S{G | G ⊆ Fj j < i}, in a non-void union of dim(∆)−1-dimensional proper faces. Let R(Fi) ={x∈Fi | Fi− {x} ∈∆i−1}, this is the unique minimal face ofFi
which lies on ∆i−∆i−1. If A1 ⊆ A2 ∈ ∆, we define the Boolean interval
[A1, A2] :={G∈∆ | A1 ⊆G⊆A2}.
Proposition 2.8. The intervals [R(Fi), Fi] = ∆i−∆i−1. In particular they
Proof. First letG∈∆i−∆i−1. Its is clear thatG⊆Fi. Now suppose there
is a vertex x ∈ R(Fi) such that x /∈ G. Then G ⊆ Fi − {x} ∈ ∆i−1, so G ∈ ∆i−1 which is a contradiction. Then G ∈ [R(Fi), Fi]. We have that
∆i−∆i−1 ⊆[R(Fi), Fi]
Now suppose G ∈ [R(Fi), Fi]. As G ⊆ Fi, we have G ∈ ∆i. Now if
G∈∆i−1, we have G ⊆Fi− {x} ∈∆i−1 for some x. But thenx ∈R(Fi)
and so R(Fi)6⊆ G, which again is a contradiction. So G ∈∆i−∆i−1 and
we have ∆i−∆i−1 = [R(Fi), Fi]
Now for a shellable complex we define the shelling polynomial:
h0∆(λ) =
t
X
i=1
λ|Fi−R(Fi)|
And we let the face enumerator polynomial
f∆(λ) =
d
X
i=0 fd−iλi
Proposition 2.9. Let∆be a shellable complex with shelling polynomialh0
∆.
Then
h0∆(λ+ 1) =f∆(λ)
Hence h0∆ does not depend on the shelling. Proof. By proposition 2.8, we have that
fd−i= t
X
j=1
|Fj−R(Fj)|
i
as any face belongs to an interval [R(Fj), Fj] and there are
|Fj−R(Fj)|
i
ways to chooseivertices out of Fj. Hence
h0∆(λ+ 1) =
t
X
i=1
(λ+ 1)|Fi−R(Fi)|
= t X i=1 d X j=0
|Fi−R(Fi)|
j λj = d X j=0 t X i=1
|Fi−R(Fi)|
j ! λj = d X j=0 fd−jλj
= f∆(λ)
Now using equation 2.1 we get
λdh0
∆(
1
λ) = λ
dh0
∆(1 +
1−λ λ
= λdf∆0 (1−λ
λ )
=
d
X
i=0
fiλi(1−λ)d−i
= h∆(λ)
soh∆(λ)0 =Pdi=0hd−iλi. In other words
h∆(λ) =
t
X
i=1
Chapter 3
Matroids
Throughout this chapter we provide basic knowledge of matroid theory. For sections 3.1, 3.2 and 3.3 we refer the reader to [17] or [22]. For section 3.4 we use [1] as reference.
3.1
Basic Concepts
Definition 3.1. A matroid M is a pair (E,I), whereE is a finite set and
I is a family of subsets of E such that (I1) ∅ ∈ I.
(I2) IfA⊂B and B∈ I, thenA∈ I.
(I3) IfA, B∈ I and|A|>|B|, there existse∈A−Bsuch thatB∪{e} ∈ I.
E is called the ground set of the matroid. If A ∈ I we say that A is independent. Note that in particular,I is a simplicial complex with vertex set E. So matroids are a particular class of simplicial complexes. For any simplicial complex ∆ with vertex set V if (V,∆) is a matroid, we call ∆ a matroid complex. A set that is not independent will be naturally called dependent. This idea of independent set comes from linear algebra and the concept of linear independence. More explicitly:
Proposition 3.2. Let V be a vector space and let E ⊆ V be a finite set of vectors. Let I be the set of linearly independent subsets of E. Then
Proof. It is clear thatI satisfies (I1) and (I2). Now letA an B inI such that |A| < |B|. Consider U to be the span of A∪B. As A is indepen-dent dim(U) ≥ |A|. Now suppose that for every v ∈ A−B, B ∪ {v} is linearly dependent. Then v ∈ span(B). Then A ⊆ span(B) which means
U ⊆ span(B) but |B| = dim(span(B)) ≥ dim(U) ≥ |A| > |B which is a contradiction. Then there is a v ∈ A−B such that B∪ {v} is linearly independent. ThenI satisfies (I3) and M is a matroid.
Definition 3.3. Two matroids M = (E,I) and M0 = (E0,I0) are
isomor-phic if there is a bijectionf fromE toE0 such thatf(A)∈ I0 if and only if
A∈ I.
Definition 3.4. A matroidM islinear over a fieldkif there exists a vector space V on k and E0 ⊆V such that the matroid generated by the linearly
independent sets ofE0 is isomorphic toM.
Linearity of a matroid depends on the field k, or more explicitly, on the characteristic of k. A matroid may be linear over a field of certain characteristic but not on a field of different characteristic (see example 3). In linear algebra we also have the notions of bases, dimension and span. These concepts can also be generalized for any matroid in the following way.
Definition 3.5. An independent set B which is maximal in I under con-tainment is called a basis
For linear matroids, bases correspond precisely to the bases ofspan(E).
Example 2. Consider the vectors given by the columns of the following matrix
A=
a b c d e f
1 0 0 0 0 0 0 1 0 1 2 0 0 0 1 1 2 0
We have that E = {a, b, c, d, e, f} and the bases are {a, b, c}, {a, b, d},
{a, b, e}, {a, c, d} and {a, c, e}. The matroid associated to the matrix A is denoted as M[A].
Proposition 3.6. All bases of a given matroid have the same cardinality. Proof. Suppose that we haveB1 and B2 bases ofM and |B1|<|B2|. Then
by (I3) we have that there exists e∈B2−B1 such that B1∪ {e} ∈ I. But
thenB1 is not maximal in I, which is a contradiction to the fact thatB1 is
Note that the bases are the facets ofIwhen seen as a simplicial complex. This means that in particular all matroids are pure simplicial complexes. The set of bases will be denoted as B. Note that as ∅ ∈ I, B 6= ∅. An interesting fact of matroids is that they can be defined criptomorphically in many ways. We already saw one definition, by setting axioms on the set of independent sets. We can also define matroids by its set of bases.
Theorem 3.7. Let E be a set and B a subset of P(E). Then B is the set of bases of a matroid if and only if the following conditions are satisfied:
(B1) B 6=∅
(B2) If A, B ∈ B and a ∈ A−B, then there exists b ∈ B −A such that (A− {a})∪ {b} ∈ B
Proof. First let M = (E,I) a matroid and B its set of bases. As ∅ ∈ I we have that B 6= ∅ so B satisfies (B1). Now consider A, B ∈ B and let
a∈A−B. ThenA− {a}is an independent set and by proposition 3.6 we have that|A|=|B|so|A− {a}|<|B|. Then by (I3) we have that there is an elementb∈B−(A− {a}) =B−Asuch that (A− {a})∪ {b} ∈ I. But as|(A− {a})∪ {b}|=|A|we have that (A− {a})∪ {b} ∈ B, so we have that
Bsatisfies (B2).
Now supposeBsatisfies (B1) and (B2). First we prove that all elements ofBare equicardinal. SupposeB1, B2∈ Bare such that|B2−B1|is minimal while |B1| < |B2|. Let x ∈ B2 −B1. Then there is y ∈ B1 −B2 such
that (B2 − {x})∪ {y} ∈ B. But |(B2 − {x})∪ {y}| = |B2| > |B1| but
|(B2−{x})∪{y}−B1|=|B2−{x}−B1|<|B2−B1|which is a contradiction to the minimality of|B2−B1|. Hence all bases have the same size.
Now letI be the set of subsets of element ofB. AsB 6=∅,∅ ∈ I and we have (I1). IfI1 ∈ I and I2 ⊆I1, then there exists B∈ B such thatI1⊆B, but thenI2 ⊆B and we have thatI2 ∈ I and (I2) holds. Now to proof (I3),
suppose we have I1 and I2 such that |I1| < |I2| but for every x ∈ I2−I1
we have I1∪ {x}∈ I/ . LetB1 and B2 elements ofB such that I1 ⊆B1 and
I2 ⊆B2. Moreover, chooseB2such that|B2−(B1∪I2)|is minimal. If there is
an elemente∈B1∩(I2−I1) then{e}∪I1∈ I which is a contradiction. Then I2−I1 =I2∩B1. IfB2−(B1∪I2) is not empty, choosex∈B2−(B1∪I2). By (B2) we have that there is an elementy ∈B1−B2such that (B2−{x})∪{y}
is a basis. Then |(B2 − {x})∪ {y} −(B1 ∪I2)| < |B2 −(B1∪I2)| which
contradicts the minimality of|B2−(B1∪I2)|. ThenB2−(B1∪I2) =∅and
we have B2−B1 =I2−B1 =I2−I1.
Suppose B1 −(I1∪B2) 6= ∅. Let x ∈ B1−(I1 ∪B2), then there is an
soI1∪{y} ⊆(B1−{x})∪{y}soI1∪{y} ∈ I which is a contradiction. Then B1−(I1∪B2) =∅. Then B1−B2=I1−B2 ⊆I1−I2. But as |B2|=|B1|
we have that |I2−I1|= |B2 −B1| = |B1 −B2| ≤ |I1−I2| which implies
|I1| ≥ |I2|which is again a contradiction. Then (I3) holds.
Definition 3.8. A circuit C of a matroid M is a minimal dependent set.
Proposition 3.9. Let C1 andC2 be distinct circuits ofM and e∈C1∩C2.
Then there is a circuit C3⊆C1∪C2− {e}.
Proof. Suppose that there is no such circuit, then C1 ∪C2− {e} ∈ I. As C1 andC2 are minimal dependent sets, they are incomparable. Thus, there
is an element a∈ C2−C1. By minimality of C2 we have that C2− {a} is
independent. Let I be a maximal independent set that contains C2 − {a}
but is a subset ofC1∪C2. Clearly a /∈I asC2 6⊆I. Also,C1 6⊆I so there
is an element b∈C1 such that b /∈I. As a∈ C2−C1 we have thata6=b.
Then|I| ≤ |(C1∪C2)− {a, b}|<|(C1∪C2)− {e}|then by (I3) we have that
there is an element of (C1∪C2)− {e} that can be added to I and still be independent, which is a contradiction to the maximality ofI.
Proposition 3.10. Let I ∈ I ande∈E−I be such thatI∪ {e}∈ I/ . Then there is a unique circuit contained inI∪ {e}.
Proof. Clearly, there is at least one circuit C ⊆ I∪ {e}. Suppose there is a different circuit C0 ⊆ I ∪ {e}. We have that e ∈ C ∩C0. But then by
proposition 3.9 we have that there is a circuit C00 ⊆ (C∪C0)− {e} ⊆ I, which is a contradiction toI ∈ I.
Definition 3.11. LetM = (E,I) a matroid. Therank function of M is a functionr:P(E)→Nsuch that forA⊆E,r(A) :=max({|I| | I ∈ I I ⊆
A}). In particular the rank ofM isr(E), that is, the cardinality of the bases ofM and is denoted as simply r.
The rank function for linear matroids correspond to the dimension of the span. That is, ifA is a set of vectors,r(A) =dim(span(A)). Note that the rank is a non decreasing function, that is, if A ⊆B then r(A) ≤r(B). In general, 0 ≤ r(A) ≤ |A|. Now if v ∈ V, r(A∪ {v}) = r(A) if and only if
v∈span(A). That observation motivates the following definition:
Definition 3.12. The closure function of a matroidM = (E,I) is a func-tioncl:P(E)→ P(E) such thatcl(A) :={e∈E | r(A∪ {e}) =r(A)}
From this definition it is clear that A⊆ cl(A). As the rank function is non decreasing, we also have that if A ⊆B then cl(A) ⊆cl(B). Now sup-pose that r(cl(A)) > r(A), this would imply the existence of B ⊆ cl(A) independent such that |B| > |A| but then there is a b ∈ B − A such that A∪ {b} is independent but b ∈ cl(A) which is a contradiction. So
r(cl(A)) = r(A). Additionally, if A ⊆ B and B 6⊆ cl(A), r(B) > r(A). So
cl(A) = max({B | A ⊆ B r(B) = r(A)}). From this observation we get thatcl(cl(A)) =cl(A). This gives rise to the following definition:
Definition 3.13. A set F is called a flat ifcl(F) =F. If a flat F is such thatr(F) =r−1, then F is called ahyperplane.
For linear matroids flats correspond to subspaces (or more precisely, the intersection of a subspace with E). Matroids can be criptomorphically de-fine by all of this concepts. We already reviewed the axioms for independent set and bases, but there are also sets of axioms that define a matroid by imposing conditions on the rank function, circuits, closure function, flats, hyperplanes and many more. There are even different set of axioms for the same concept that define the same structure, see corollary 3.24.
Note that for example 2, {f} is itself dependent. Dependent singletons like this receive a special name:
Definition 3.14. An elementesuch that{e} is dependent is called aloop Note that an element is a loop if and only if it belongs tocl(∅). Another important situation is when D and E are parallel as vectors. This means that they belong to the same 1-dimensional vector subspace. This notion can be easily generalized in the following way for matroids in general.
Definition 3.15. Two different elementseande0 such that neither of them are loops but{e, e0} is dependent are calledparallel. The setcl(e)−cl(∅) =
cl(e0)−cl(∅) is called the parallelism class of e(and of e0).
It is easy to see that parallelism classes partition the set of non-loop ele-ments. It is also easy to check that ifeand e0 are parallel ande∈I whereI
is independent, then (I− {e})∪ {e0} is also independent. Thus, parallel ele-ments really behave identically. For this reason, many choose to work with matroids without loops or parallel elements, and are called combinatorial geometries. One can always get a combinatorial geometry from any matroid by removing all of the loops and contracting under the parallelism equiva-lence relation, a process called simplification. However, parallel classes play
an important role for us, so we are not going to do this.
3.2
Matroid Examples
We start by looking at perhaps the most simple class of matroids, the uni-form matroids.
Definition 3.16. Let n and r integers such that 0≤r ≤n. Theuniform matroid Ur,n is a the matroid with ground set E such that |E|= n and a
subset ofE is independent if and only if it has at mostr elements.
We have that its independent set are I = {I ⊆E | |I| ≤ r}, its bases are B = {B ⊆ E | |B| = r}, its rank is r(A) = min(r,|A|) and its flats are E and every subset of size at most r−1. Uniform matroids may be fairly simple, however, they provide useful examples. The matroidU2,4, for
example, is not a graphic matroid.
Definition 3.17. LetM a matroid of rankr. If every circuit ofM has size at leastr, thenM is called paving matroid.
Note that for a uniform matroid, all circuits are of size r+ 1, so they are in particular paving. Now we see more ways to represent matroids. For small rank matroids, we often use the affine representation. In this geo-metric representation, elements of M are placed in the Rr−1 just like the
geometric representation of simplicial complexes. However, we do not draw the convex hull of the independent sets, but of the flats of the matroid.
Example 3. Figure 3.1 shows the affine representation of two matroids on [7]. M1 called the non-Fano matroid, whose circuits are {{1,2,7},{1,3,5},
{1,4,6},{2,4,5},{3,4,6},{5,6,7}} and every set of 4 elements that do not contain these. M2, called the Fano matroid, whose circuits of 3 elements are
the same ofM1plus{2,3,4}and every set of 4 elements that do not contain
these. Note that a matroid is uniquely determined by their circuits. It is easy to check that both M1 and M2 are matroids. The non-Fano matroid
is linear over every field except in characteristic 2. Interestingly, the Fano matroid can only be represented in a field of characteristic 2.
1
2 3
4
5 6 7
M1
1
2 3
4
5 6 7
M2
Figure 3.1: M1 the non-Fano matroid andM2 the Fano matroid
In this representation, the elements are assumed to be independent them-selves. If one chooses to include loops, they should be drawn separated and marked as loops. Figure 3.2 shows the affine representation ofM[A] where
A is the matrix of example 2. Note that e and d are parallel, so they are represented by the same point.
Loops
a
b c d
e
f
M[A]
Figure 3.2: Affine representation of M[A]
Now we turn our attention to a special class of matroids that arises from graphs. For now, we assume graphs can have repeated edges and loops. Recall that ifG= (V, E) and A⊆E thenG[A] is the subgraph induced by
A.
Proposition 3.18. Let G= (V, E)be a graph with vertices V and edges E. Let I be the set of all forestsI of E, that is, all I such that I has no cycles. ThenM(G) = (E,I) is a matroid.
Proof. Clearly ∅ has no cycles so ∅ ∈ I. Suppose I1 ∈ I and I2 ⊆I1, then
asI1 does not have cycles I2 can not have cycles either, soI2∈ I. Now let I1, I2 ∈ I and suppose that|I2|<|I1|. Then there is a connected component
of G[A] with A⊆V where I1 has more edges than I2. Then I1 is incident
to more vertices of A than I2. Suppose v ∈ A is a vertex such that I1 is
incident to it but I2 has not. Let e∈ I1 be an edge that is incident to v.
Then clearly I2 ∪ {e} has no cycles, as I2 did not have cycles and e has
incident on a vertex that I2 did not have incidence. Then I2∪ {e} ∈ I and
we have thatI satisfies (I1)-(I3).
Let us review what the previously defined concepts of matroid theory represent in graphs. By definition, the independent sets are forests. Then the bases are spanning forests. The circuits would be cycles (hence their name). The rank ofA⊆E would be the number of verticesAhas incidence on minus the number of connected componentsAhas edges on. The closure ofAwould beE(G[U]), the edges ofG[U] whereU is the set of vertices that are incident toA. The flats would be any set of the formE(G[U]) for some
U ⊆ V. An edge is a loop in the matroid if and only if it is a loop in the graph (hence their name) and parallel edges are edges that are incident the same two vertices.
Example 4. Consider again the matrix A from example 2, and the graph
Gfrom figure 3.3. It is easy to check thatM[A] =M(G). MatroidsM such that there is a graph G such that M = M(G) are called graphic. Hence
M[A] is a graphic matroid.
a
b c
d
e
f
Different graphs may produce the same matroid. Hence, if we want to study a graphic matroidM, we can assume the graph that represents M is nice in the following sense.
Proposition 3.19. Let M be a graphic matroid. ThenM is isomorphic to
M(G) for some connected graph G.
Proof. AsM is graphic, there is a graphH such that M ∼=M(H). Suppose
H is not connected. Let H1, H2. . . Ht be the connected components of H.
Fori∈[t], let vi∈V(Hi). Let Gbe the graph that identifiesv1, v2. . . vt as
a single vertex. ThenE(G) =E(H),G is connected and clearly the cycles ofG are the same as those of H, so M ∼=M(H)∼=M(G).
Proposition 3.20. Let M be a graphic matroid. ThenM is isomorphic to
M(G) for some connected graph G such that for every vertex v0 ∈ V(G),
the induced graph ofG[V(G)− {v0}]is connected.
Proof. By Proposition 3.19, we have that M = M(H) for some connected graph H. Let v0 ∈ V(H). If H[V(H)− {v0}] is connected we have the
desired result by letting G = H. Suppose now that H[V(H)− {v0}] is
not connected.LetH1, H2. . . Htbe the connected components ofH[V(H)−
{v0}]. Let G1 = H[V(H1)∪ {v0}] and for 1 < i ≤ t, let Gi be the graph
isomorphic to H[V(Hi)∪ {v0}] but instead of having v0 as vertex, let any
vertex of H1 take its place. Let G be the graph formed by the union of
the Gi with i ∈ [t]. Then G is connected, G[V(G) − {v0}] is connected, E(G) =E(H), andGhas the same cycles asHsoM ∼=M(H)∼=M(G).
Proposition 3.21. Let M be a graphic matroid. Then M is linear over every field.
Proof. LetGbe such thatM ∼=M(G). Give any orientation to the graphG, and considerAits incidence matrix. LetF is a set of edges. IfF is a forest, then an edge of F that is incident to a leaf is clearly has its corresponding column in A independent to the rest of the columns corresponding to the rest of the edges of F. If F has a cycle, then the sum of the columns corresponding to that cycle, multiplied by 1 or -1 in order to have an oriented cycle, is 0. Then the corresponding columns ofF inAare linearly dependent. ThenM ∼=M(G)∼=M[A].
3.3
Duals, Minors and Tutte Polynomial
In this section we look at the basic constructions to get new matroids from a given matroid. We do this to understand how to use induction on matroids. We start with the following proposition:
Proposition 3.22. If B is the set of bases of M,B satisfies the following:
(B2’) If B1, B2 ∈ B and x ∈ B2 −B1 then there is y ∈ B1 −B2 such
that(B1− {y})∪ {x} ∈ B.
Note that there is a difference between (B2) and (B2)’ that cannot be solved just by relabeling.
Proof. By proposition 3.10, we have that there is a unique C ⊂ B1∪ {x}.
As B2 is independent, C −B2 =6 ∅, so there exists y ∈ C−B2. Clearly, y∈B1−B2. As (B1− {y})∪ {x} does not containC, which was the only circuit insideB1∪ {x}, we have that (B1− {y})∪ {x} is independent. But
(B1− {y})∪ {x} has the same number of elements thanB1, so it is in fact
a base.
Theorem 3.23. Let M be a matroid with ground setE and set of basesB. ThenB∗={E−B | B ∈ B} is a set of bases for a matroid onE.
Proof. As B is not empty, B∗ is not empty. Now let B1, B2 ∈ B∗ and x∈B1−B2. We have thatE−B1, E−B2 ∈ Bandx∈(E−B2)−(E−B1).
By proposition 3.22 we have that there is an elementy∈(E−B1)−(E−B2) such that ((E−B1)− {y})∪ {x} ∈ B. This means that y ∈B2−B1 and E−(((E−B1)− {y})∪ {x}) = (B1− {x})∪ {y} ∈ B∗. So B∗ satisfy (B1)
and (B2).
The matroid M∗ on E with B∗ as its bases is called the dual of M. It
is easy to see that (M∗)∗ = M, so there is really a sense of duality. Note that in the proof of Theorem 3.23 we showed that (B2) and (B2’) are dually analog, so we have the following corollary.
Corolary 3.24. IfB ⊆ P(E), Bis the set of bases if and only if it satisfies (B1) and (B2’).
We can see, for example, that U∗
r,n = Un−r,n. However, the dual of
a graphic matroid is not necessarily graphic. So we refer to this class of matroids, the dual of a graphic matroid, ascographic matroids. The prefix co is widely used to relate a concept to its dual. For example, a coloop is a
loop of the dual. As a loop is an element that does not belong to any basis, a coloop is an element that belongs to every basis. The following relation will be relevant to us.
Proposition 3.25. LetH be a hyperplane ofM. ThenE−His a cocircuit, a circuit ofM∗.
Proof. H is a hyperplane if and only if it does not contain any basis but for every x ∈E −H, H∪ {x} contains a base. Another way of saying this is thatE−H intersects every basis but for everyx ∈E−H there is a basis
B ∈ B such that E −H∩B = ∅. In terms of the dual this means that
E−His not contained in any element ofB∗ but removing any element of it
makes it a subset of an element ofB∗. This means that H is a hyperplane if and only if E−H is dependent in the dual but every subset of E−H is independent in the dual, which is exactly being a cocircuit.
Proposition 3.26. If M is a matroid with rank function r, then the rank function of its dual isr∗(A) =|A|+r(E−A)−r(E).
LetA⊆E. ConsiderI ⊆Aa maximal co independent subset ofA, that is, a maximal indepndent subset ofAin the dual. Then there is a basis such thatI ⊆E−B. We have thatB is a base such thatE−B∩Ais maximal. So
Bis also a base such thatB∩E−Ais maximal. Thenr(E−A) =|B∩E−A|. We have that|A|=|A∩B|+|A∩E−B|=|B|−|B∩E−A|+|A∩E−B|=
r(E)−r(E−A) +r∗(A). So r∗(A) =|A|+r(E−A)−r(E). Proof.
Definition 3.27. LetM = (E,I) be a matroid. IfA⊆E, andI\A:={I ∈ I | I∩A=∅}, we call the matroidM\A= (E−A,I\A) therestriction of
A.
It is easy to check that M\A is in fact a matroid. There is naturally a dual analog of this operation.
Definition 3.28. LetM = (E,I) be a matroid and A⊆E. We define the contraction of Aas the matroidM/A:= (M∗\A)∗.
Ifx∈E, we denote the restriction and contraction ofxasM\xandM/x, without the{}. Ifxis not a coloop, we have that the bases ofM\xare all of the bases ofM which do not contain x. Ifx is a coloop, it belongs to every basis, so the bases ofM\x are{B− {x} | B ∈ B}whereBare the bases of
Ifx is a loop, it does not belong to any basis so the bases ofM/x are just the same bases ofM. From this we get thatM/x=M\x if and only ifx is a loop or a coloop.
From the definition, it is clear that ifA⊆E−S, we have thatrM\S(A) =
r(A). Now using that and Proposition 3.26 we have that
rM/S(A) = r(M∗\S)∗(A)
= |A|+rM∗\S(E−S−A)−rM∗\S(E−S)
= |A|+rM∗(E−S−A)−rM∗\S(E−S)
= |A|+|E−S−A|+r(S∪A)−r(E)− |E−S| −r(S) +r(E) = r(S∪A)−r(S)
We can use this to prove the following:
Proposition 3.29. Let A1, A2⊆E such that A1∩A2=∅. Then
(M/A1)\A2 = (M\A2)/A1
Proof. LetX ⊆(E−A1)−A2. We have
r(M/A1)\A2(X) = rM/A1(X)
= r(X∪A1)−r(A1)
= rM\A2(X∪A1)−rM\A2(A1)
= r(M\A2)/A1(X)
As the rank functions of both matroids are equal, they are equal.
Any matroid constructed this way, by contracting or restricting elements ofM, is called aminor ofM. In the case of graphic matroids, ifM[G], then
M[G]\e = M[G\e] where G\e is the graph resulting from deleting e from the set of edges. We also have that if e is an edge between the vertices v
and u, then M[G]/e=M[G/e] where G/eis the graph where you identify verticesv and u inG to be the same, and remove e. Figure 3.4 shows the restriction and contraction ofb in graphGfrom figure 3.3.
Now we focus our attention on the Tutte polynomial of a matroid, a bi-variate polynomial that encodes much of the important information of a matroid. We refer the reader to [4] for more information about the Tutte polynomial.
a c d
e
f
a c d
e
f
M\b M/b
Figure 3.4: The restriction and contraction ofb inG.
Definition 3.30. For a matroidM we define the Tutte polynomial as
TM(x, y) =
X
A⊆E
(x−1)r(E)−r(A)(y−1)|A|−r(A)
Proposition 3.31. Let M be a matroid. Then TM(x, y) =TM∗(y, x).
Proof. Using Proposition 3.26, we get thatr∗(E)−r∗(A) =n−r(E)−(|A|+
r(E−A)−r(E)) =n− |A| −r(E−A) =|E−A|+r(E−A). Then
TM(x, y) =
X
A⊆E
(x−1)r(E)−r(A)(y−1)|A|−r(A)
= X
A⊆E
(x−1)r(E)−r(E−A)(y−1)|E−A|−r(E−A)
= X
A⊆E
(x−1)|A|+r∗(A)(y−1)r∗(E)−r∗(A) = TM∗(y, x)
(a) If eis a loop then TM(x, y) =yTM\e(x, y) (b) If eis a coloop then TM(x, y) =xTM\e(x, y)
(c) Ifeis neither a loop nor a coloop thenTM(x, y) =TM\e(x, y)+TM/e(x, y) Proof. If e is a loop, then for every A ⊆ E− {e}, r(A) = r(A∪ {e}). We have
TM(x, y) =
X
A⊆E
(x−1)r(E)−r(A)(y−1)|A|−r(A)
= X
A⊆(E−{e})
(x−1)r(E)−r(A)(y−1)|A|−r(A)
+ X
A⊆(E−{e})
(x−1)r(E)−r(A∪{e})(y−1)|A∪{e}|−r(A∪{e})
= (1 +y−1) X
A⊆(E−{e})
(x−1)r(E)−r(A)(y−1)|A|−r(A) = yTM\e(x, y)
Ifeis a coloop, (b) follow from (a) and Proposition 3.31. Now ifeis neither a loop nor a coloop, we have
TM(x, y) =
X
A⊆(E−{e})
(x−1)r(E)−r(A)(y−1)|A|−r(A)
+ X
A⊆(E−{e})
(x−1)r(E)−r(A∪{e})(y−1)|A∪{e}|−r(A∪{e})
= X
A⊆(E−{e})
(x−1)r((E−{e}))−r(A)(y−1)|A|−r(A)
+ X
A⊆(E−{e})
(x−1)(r(E)−1)−(r(A∪{e})−1)(y−1)|A|−(r(A∪{e})−1)
= TM\e(x, y) +TM/e(x, y)
Note that the last proposition can also be used to define the Tutte poly-nomial recursively. Its is actually its recursive nature that makes it the fun-damental tool for using induction on matroids. Tutte polynomial has many interesting properties when evaluated in specific points and lines. Examples of specific points include:
• TM(1,1) is the number of bases ofM.
• TM(2,1) is the number of independent sets of M.
• TM(2,2) = 2n wheren=|E|
In the next section we see how to relate the Tutte polynomial to theh-vector of a matroid.
3.4
Ordered Matroids and Shellings
We now see shellings in the particular case of matroids. Let<be any total order on the elements of a matroidM. We say that (M, <) is an ordered ma-troid. For a basisB ofM, we writeB = (b1, b2, . . . , bd) ifb1 < b2 <· · ·< bd.
This induces a lexicographic order on the bases of M, where B < C with
C= (c1, c2, . . . cd) if and only if for some e,bi =ci fori < e andbe< ci for
i≥e. This order of the bases ofM is also an order of the facets of I. We will show that this will always be a shelling ofI. This actually characterizes pure simplicial complexes which are matroids. In particular, every matroid complex is shellable.
Proposition 3.33. Let(M, <)an ordered matroid. Then the lexicographical order on the bases ofM is a shelling in the complex I.
Proof. Let B = (b1, b2, . . . , bd) and C = (c1, c2, . . . , cd) be two bases such
thatB < C lexicographically. Then for somee,bi =ci fori < eandbe< ce
fori≥e. By the basis exchange property, there exists an element inC−B, this is ci with i > e, such that A = (C∪ {be})− {ci} is a basis. We have
nowA < C and B∩C ⊆A∩C=C− {ci}. Hence<induces a shelling on
I(M).
Proposition 3.34. Let ∆ be a simplicial complex on vertex set V. Then ∆is a matroid complex if and only if ∆A:={F ∈∆ | F ⊆A} is pure for every subsetA⊂V.
Proof. If ∆ is a matroid complex, with ∆ =I for some matroid M. Then ∆Ais the complex of the restriction ofM toA. This is ∆A=I(M−(V−A)).
Then ∆A is a matroid complex and in particular it is pure. Now suppose
∆A is pure for everyA⊆V. We shall proof the augmentation axiom for ∆.
LetF, G ∈∆, such that |F|<|G|. Consider A= F∪G. ∆A is pure, and
in some facet H of ∆A which is bigger than F. Then there is an element
x∈H−F ⊆G−F and so F∪ {x} ∈∆.
Theorem 3.35. Let ∆ be a pure simplicial complex. Then ∆ is a matroid complex if and only if every order on the vertexesV of ∆induces a shelling of ∆with the lexicographical order of the facets.
Proof. One of the directions is essentially proposition 3.33. Now for the other direction, suppose that every order of ∆ induces a shelling but ∆ is not a matroid complex. By proposition 3.34 we have that there is anA⊂V
such that ∆Ais not pure. LetF be a facet of ∆Aof minimal dimension and
Ga facet of ∆A of dimension larger than F such that |G∩F|is maximal.
Consider an order on V where the elements of F −G come first, then all of the elements of G, and finally all the other vertexes. Let ˜G be the first facet of ∆ to contain G, and ˜F be any facet of ∆ that contains F. We have ˜F < G˜. As the order induces a shelling, there is a facet H <G˜ such that ˜F∩G˜ ⊆H∩G˜ = ˜G− {g} for some g∈ G˜. Then H−G˜ ={h} with
h∈F−G. Ifg∈G˜−G, thenG∪{h} ⊆Hand ash∈F ⊆U,G∪{h} ∈∆A,
which contradicts the fact thatG is a facet in ∆A. Then g ∈ G−F. But
then if G0 = (G− {g})∪ {h} is a subset of both A and H, so G0 ⊆ ∆
A.
Then there is a facetK in ∆A containingGand |K| ≥ |G0|=|G|>|F|and
|K∩F| ≥ |G0∩F|>|G∩F|which is a contradiction of the maximality of
|G∩F|.
Now we consider how to compute the shelling polynomial of a matroid complex and hence its h-vector. Let (M, <) be an ordered matroid with
M = (E,I). Let B be a basis of M. We say that an element x of B is internally passive if there is an element y in E −B such that y < x and
B−{x}∪{y}is a basis ofM. We say thatx∈Bisinternally activeif it is not internally passive. Now for an elementy∈E−Bwe say thatyisexternally passive if there is an element x∈B such thaty < xandB− {x} ∪ {y}is a basis ofM. We say thaty ∈E−B isexternally active ifyis not externally passive. Note thatx ∈B is internally passive (or internally active) if and only if x is externally passive (externally active) with respect to E−B in
M∗. We write IP(B),IA(B),EP(B) andEA(B) for the sets of internally
passive, internally active, externally passive and externally active elements of B respectively. This is a partition of E. Note that for b∈B,B− {b} is contained in a basis preceding B if and only if b ∈IP(B). In other words R(B) =IP(B). Hence IA(B) =B−R(B). Using equation 2.2 we get
Proposition 3.36. Let(M, <)be an ordered matroid withBits set of bases. Then
hM(x) =
X
B∈B
x|IP(B)|
We have that the number of bases with i internally passive elements is
hi, so this does not depend on the order of M.
1
2
3
4
5
6
v
0v
1v
2v
3Figure 3.5: Graph G
Example 5. Let G be the graph of figure (5) and consider its matroid
M(G). We have for the natural order that
Base Int. Active Ext. Active
123 123
124 12 3
126 12
135 13
136 1 2
145 1 3
146 1 23
156 1
235 3 1
245 13
256 1
356 12
We get that h(M(G)) =h(M(G)∗) = (1,3,5,4)
Note that ifB is the first basis ande /∈B,eis always internally passive in every basis Be where e belongs (by using (B2) with B and Be). Then
IP(Be) ={e} if and only ife /∈B andBeis the first basis whereeappears.
From this we get
h1(M) =n−r (3.1)
Note also that both loops and coloops are never internally passive. Hence they do not affect the non-zero entries of the h-vector (coloops add 0’s at the end).
Proposition 3.37. Let (M, <) be an ordered matroid. Let B be a basis of
M. Then
EA(B)⊆cl(IP(B))
Proof. Considery an externally active element in B. As {y} ∪B is depen-dent, then there is a circuitC that contains y. Moreover, C is unique. Let
x∈B∩C. We have that (B− {x})∪ {y}is a basis, asyis externally active we havey < x. Thenxis internally passive. Then C− {y} ⊆IP(B), hence
x∈cl(C)⊆cl(IP(B)).
Proposition 3.38. The boolean intervals[IP(B), E−EP(B)]partition the boolean algebra of subsets ofE.
Proof. We want to show that for eachA⊆Ethere is a unique basisBAsuch
thatIP(BA)⊆BA⊆EP(BA). Recall that by proposition 2.8 the intervals
[IP(B), B] partition the set of independent sets of M. Consider MA, the
restriction ofM toA. LetIAbe the greatest basis ofMAin lexicographical
order. We have thatXAis an independent set ofM and so there is a unique
basisBA such that IP(BA) ⊆XA⊆BA. Now suppose there is an element
a ∈ A−BA that is externally passive. Then there is a b ∈ BA such that
b < a and (BA− {b})∪ {a} is a basis of M. If b /∈A, we would have that
XA∪ {a} is independent, but that contradicts the fact that XA is a basis
of MA. Then b ∈ A, then b ∈ A∪BA = XA and (XA− {b})∪ {a} is a
basis ofMA. But then this basis is bigger in lexicographical order thanXA,
which contradicts its maximality. Hence A−BA ⊆ EP(BA) and we have
IP(BA)⊆A⊆E−EP(BA).
Now to proof uniqueness, suppose there is aBsuch thatA∈[IP(B), E− EP(B)]. By proposition 3.37, we have that EA(B) ⊆ cl(IP(B)), so A−
cl(IP(B)) ⊆ B. Then X = A∩B is a basis of MA. Suppose X 6= XA.
If X = (x1, x2, . . . xd) and XA = (y1, y2, . . . yd), asXA is the greatest basis
there is anesuch thatyi =xifori < eandye< xi fori≥e. Then by basis
exchange axiom we have that there is ani≥e such that (X− {xi})∪ {ye}
is a basis. But thenye is an externally passive element of X and hence also
of B, which is a contradiction as A ⊆E−EP(B). Then X =XA and we
have uniqueness.
Theorem 3.39. Let (M, <) be an ordered matroid with B its set of bases. If TM(x, y) is the Tutte polynomial of M we have
TM(x, y) =
X
B∈B
x|IA(B)|y|EA(B)|
Proof. Let A ⊆ E and B basis such that A ∈ [IP(B), E −EP(B)]. By proposition 3.37, we have thatr(A) =|B∩A|. Thenr(E)−r(A) =|B−A|. Also,|A| −r(A) =|A−B|. So we have
TM(x, y) =
X
A⊆E
(x−1)r(E)−r(A)(y−1)|A|−r(A)
= X
B∈B
X
A∈[IP(B),E−EP(B)]
(x−1)|B−A|(y−1)|A−B|
= X B∈B d X i,j=0
B−IP(B)
i
E−EP(B)
j
(x−1)i(y−1)j
= X B∈B d X i,j=0
IA(B)
i
EA(B)
j
(x−1)i(y−1)j
= X
B∈B
(x−1 + 1)|IA(B)|(y−1 + 1)|EA(B)|
= X
B∈B
x|IA(B)|y|EA(B)|
Theorem 3.40. Let M be a martroid wtih tutte polynomialTM(x, y) andh polynomial hM(x). Then
h∆(x) =xrTM(
1
Proof. By theorem 3.39 and 3.36 we get that
xdTM(
1
x,1) = x
r X
B∈B
1
x|IA(B)|
= X
B∈B
xr−|IA(B)|
= X
B∈B
x|IP(B)|
Chapter 4
Order Ideals and Stanley’s
Conjecture
Definition 4.1. A (monomial) order ideal O is a finite collection of mono-mials such that ifm∈O and m0|m, thenm0 ∈O. If the maximal elements ofO are all of the same degree we say thatO is pure
Consider Λ(O) to be the maximal monomials ofO. We have that O is completely determined by Λ(O), as O ={m | ∃n∈ Λ(O) m|n}. If O has as variables x = (x1, . . . xr), and V(O) be the minimal monomials in k[x]
that are not inO, we also get thatOis uniquely determined by the elements ofV(O). Moreover,O is a base fork[x]/ < V(O)>as a vector space overk
where< V(O)>is the ideal generated by the elements of V(O). Note that
< V(O)> is a monomial ideal.
Order ideals are in a natural bijection with multicomplexes, that is, a simplicial complex ∆ where the elements of ∆ can be multisets. Suppose that [r] is the vertex set of ∆. Then a multiset Aover [r] can be associated to a monomialmA=xα11xα22. . . xαrr where for eachi∈[r], αi is the number
of times iappears in A. That way, the condition that for each A ∈∆ and
B ⊆ A then B ∈ ∆ is equivalent that for each mA ∈ O and mB|mA then
mB ∈O. That way multicomplexes and order ideals are basically the same.
The order ideal is pure if and only if its analog multicomplex is pure.
In the same way simplicial complexes have an f-vector, one can de-fine the f-vector of an order ideal O as the f-vector of its multicomplex. However, to differentiate them from usual simplicial complexes, we denote the f-vector with capital F. That is, for a given order ideal O, Fi(O) =
|{m ∈ O | deg(m) = i}| and we have that the F-vector of O is F(O) = (F0, F1. . . Fr).
Definition 4.2. Let a = (a0, a1. . . at) be a sequence of non negative
inte-gers. If there is an order ideal O such that F(O) = a we say that a is an O-sequence. If there is a pure order idealO such thatF(O) =awe say that
ais apure O-sequence.
Much of the theory known for pure O-sequences can be found in [13]. Order ideals are often represented bystaircase diagrams. If O ⊆k[x] is an order ideal, to draw its staircase diagram one must go to the postive part of
Rr, that is, where all coordinates are non-negative. For each monomial xa
wherea= (a1. . . ar), we associate that monomial to the unitary hypercube
{(b1· · ·r) ∈Rr | ∀i∈[r] ai ≤ bi < ai+ 1}. The staircase diagram of O is
the union of all of those cubes.
Example 6. LetO be the order ideal generated by Λ(O) ={x6, x5y, xy2y4}
and U the order ideal generated by Λ(U) = {x2, y2, z2, xyz}. Figure 4.1
shows the 2-dimensional and 3-dimensional staircase diagrams ofO and U
respectively. The ◦ represents the elements of V(O) and V(U) and the × represents the elements of Λ(O) and Λ(U).
x
y
x
y
z
O
U
Figure 4.1: Staircase diagrams forO and U
The following is a conjecture given by Stanley in the 70’s in [20] and has been object of much study in recent years.
Conjecture 1. (Stanley’s Matroid Conjecture) The h-vector of a matroid is a pure O-sequence.
In [20], Stanley proves that O-sequences are equivalent to the h-vectors of shellable complexes. So it became a natural question whether theh-vector of matroids, more specific than shellable complexes, were pure O sequences, more specific than O sequences. The conjecture in general remains open but many partial results have been obtained. It has been shown that Conjecture 1 holds for matroids of rank 2 [21], rank 3 matroids [8, 7], rank 4 matroids [10], corank at most 2 [7], matroids with at most 9 elements [7] lattice path matroids [18] and more generally cotransversal matroids [15], cographic matroids [11], paving matroids [12], positroids [16] and matroids of Cohen-Macaulay type at most 5 [6].
The conjecture became an even more natural question when bothh-vectors and pureO-sequences were both proved to satisfy the same inequalities,
Theorem 4.3. [3, 5, 9] Let h = (h0. . . hr) a matroid h-vector or a pure O-sequence with hr 6= 0. Then
1. h0 ≤h1 ≤ · · · ≤hbd
2c
2. For all 0≤i≤ bd
2c,hi ≤hd−i
3. For every 0≤k≤d andα≥1 we have k
X
i=0
(−α)k−ih i ≥0
The last inequality is well known as the Brown-Colbourne inequality [3]. Now we show the proof given in [7] for corank 2 matroids.
Theorem 4.4. Let M be a matroid of rank 2. Then h(M∗) is a pure O-sequence.
Proof. Suppose M is loopless. Fix an ordering E1, E2. . . Et on the
paral-lelism classes of M. Now order the elements of M such that if i < j then for every a ∈ Ei and b ∈ Ej we have a < b. Rename the elements of
M such that E(M) = [n] with the usual order. For any basis B = {a, b}
of M with a < b let α1(B) be the number of elements c of M such that a < c < b and {c, b} is a basis and let α2(B) be the number of elements
elements of B smaller than b and α2(B) counts the externally passive
ele-ments bigger than b. Let µB := xα1yα2. Now note that b = n−α2(B).
If b∈ Ej then the elements c that are externally passive in B and smaller
thanb are exactly those which belong to a parallelism class Ek with k < j
and c > a. So a = | S
k<j
Ek| −α1(B). Then if for two bases B and B0 of M we have µB =µB0 thenB =B0. Recall that the h vector ofM∗ orders
the basis of M by external passivity and α1(b) +α2(b) = e(B). Then if
O := {µB | B ∈ B(M)}, we have f(O) = h(M∗). We now need to show
thatO is a pure order ideal.
Letxα1yα2 ∈ O. Then there is a basis B={a, b}such that α1 =α1(B)
and α2 =α2(B). If α1 >0 then there is at least one element bigger than asuch that it is independent with b. That means thata+ 1∈/ cl(b), which means that B0 ={a+ 1, b} is a basis and µB0 =xα1−1yα2. Now if α2 >0
then b+ 1 ∈ E(M). If b+ 1 ∈ cl(b) then B00 = {a, b+ 1} is a basis and
µB00=xα1yα2−1. Ifb+ 1∈/ cl(b) then a≤b− |cl(b)|soa+|cl(b)| ≤b. Then B000 ={a+|cl(b)|, b+1}is a basis andµB000 =xα1yα2−1. We have thatOis an
order ideal. To show that it is pure consider ˜b=min(cl(b)) and ˜B :={1,˜b}
we have thatα1 ≤α1( ˜B), α2 ≤α2( ˜B) andα1( ˜B) +α2( ˜B) = n−2. So all
Chapter 5
Cographic Matroids
In this chapter we recursively build order ideals whose F vector is the h
vector of a given matroid. We start with M = (E,I) a matroid of rank r
and we will attempt to build an order ideal forM∗. We want to do induction
onr, that is, building the order ideal from order ideals that represent theh
vector of matroids of corank r−1 (that is, order ideals whose F vector is the same as theh-vector of that matroid). To do so, we use the following:
Proposition 5.1. Let M be a matroid and e∈E(M) such that eis not a coloop or a loop. Then
hM∗(x) =h(M/e)∗(x) +xh(M\e)∗(x)
Proof. By theorem 3.40 we have that ifeis not a loop or a coloop,
hM∗(x) = xr−nTM(1,1 x)
= xr−n(TM/e(1, x) +TM\e(1, x))
= h(M/e)∗(x) +xh(M\e)∗(x)
asr((M\e)∗) =r(M∗)−1.
Lemma 5.2. LetM be a loopless matroid andA={a1, a2. . . ak}be a subset of E such thatr(E−A) =r(M). Then
hM∗(x) =
k
X
i=1
xi−1h(M/ai\{a1...ai−1})∗(x) +x
kh
Proof. As r(E−A) = r(M), we have that each ai is neither a loop nor a
coloop. We get the desired result by applying proposition 5.1 for eachai in
the matroidM\{a1. . . ai−1}.
Lemma 5.3. Let A={a1, a2. . . ak} be a subset ofM such that E−A is a hyperplane, then
hM∗(x) =
k
X
i=1
xi−1h(M/ai\{a1...ai−1})∗(x)
Proof. Note that ifeis a coloop we simply have thathM∗(x) =h(M/e)∗(x).
We have thatr(E−(A− {ak})) =r(M), so by applying the Lemma 5.2 to
A− {ak} we have that
hM∗(x) =
k−1
X
i=1 xi−1h
(M/ai\{a1...ai−1})∗(x) +x
kh
(M\(A−{ak}))∗(x)
Note that ifa∈A,E−Ais still a hyperplane inM\eand so inM\(A−{ak}).
Thenakis a coloop inM\{a1. . . ak−1}and we have thath(M\(A−{ak}))∗(x) =
h(M\{a1...ai−1}/ak)∗(x). By replacing in the above equation we get the desired
result.
This way, the h vector of a matroid can be seen as sum of h vectors of matroids of lower corank. We want to construct an order ideal that representsM∗. As h1(M∗) =r by equation 3.1 the number of variables of the desired order ideal is going to be r. Then if we use order ideals that represent matroids of lower corank, we would have that these order ideals are on r−1 variables. On the other hand, we want a pure order ideal of degree n−r. When contracting e from M we get that the rank of (M/e)∗ is stilln−r but theh vector may have 0’s at the end if (M/e)∗ has coloops.
As the coloops of (M/e)∗ are precisely the elements of cl(e)− {e}, we have that the pure order ideal representingM/eis of degreen−r−cl(e) + 1. So we want to multiply this order ideal by the remaining variable in order to make it of the desired degree. This motivates the following construction of order ideals that represent cographic matroids. This would prove Stanley’s conjecture for cographic matroids, which was first proved by Merino in [11]. LetGbe a graph. We assumeGis connected by proposition 3.19. More-over, we assumeGis loopless, as loops do not affect theh-vector. However, we allow parallel edges as parallel elements do affect theh-vector of the ma-troid. Letv0 be any vertex of G. We can also assume thatG[V − {v0}] is
connected by proposition 3.20. Ifr is the rank of M = M(G), then G has
r+ 1 vertices which we are going to label v0, v1. . . vr. Letnbe the number
of edges of G(the number of elements ofM(G)). We define recursively the wrapWG(v0,x) as a set of monomials ink[x]. Ifr= 0 thenWG(v0,x) ={1}.
Now for eachvi ∈N(v0) letei be an edge betweenvi and v0 and define the
piece ofvi as
Pi := WG/cl(ei)(v0,x−xi)[xi]|cl(ai)|
= {mxki | m∈WG/cl(ei)(v0,x−xi) k <|cl(ei)|}
Let J(v0) ={i | vi ∈N(v0)}. We define WG(v0,x) := S i∈J(v0)
Pi. Note
that each vertex (different from v0) is being associated to a variable. So when contracting the graph bycl(ei), we let all vertices to keep their label
except for vi which would become v0. Without loss of generality, assume J(v0) = [t] for some positive integer t. For notation purposes, for i < t
we let ci = |cl(ei)|, that is, the number of edges between v0 and vi. We
also do the following abbreviations: W := WG(v0,x), if A ⊆ [r] then GA
is the subgraph generated byv0 and all vi such that i∈A as vertices. Let
WA:=WG/E(GA)(v0,{xi ∈x | i /∈A}). With this notation Pi =Wi[xi]ci.
Proposition 5.4. WG(v0,x) is a pure order ideal whose maximal elements are of degreen−r.
Proof. If r = 0, then as we assumed G to be loopless, n = 0 and in fact
{1} is a pure order ideal of degree 0. Now assume the statement holds for graphs wtih at mostr vertices. Then for eachi < t,Wi is a pure order ideal
of degreen−ci−r+ 1. Now by multiplying Wi by [xi]ci, we get that the
degree of Pi is the degree of Wi plus ci−1. This means that Pi is a pure
order ideal of degree n−r. As W is the union of pure order ideals of the same degree,W is a pure order ideal of that same degree, n−r.
The idea of this definition is that the wrap of a graph will be an order ideal that represents M(G)∗. Note that the set of edges coming out of v0
is a cut set, which means it is the compliment of a hyperplane. So we want to use lemma 5.3 on that set. If we assume that Wi represents M/ei, we
have thatFPi(x) =h(M/ei)∗(x)[x]ci. So the idea of multiplyingWi by [xi]ci
is both to make sureW is pure and to take into account the ci summands
W
1W
3W
2P
1P
3P
2W
x1
x2
x3 x1
x1
x2 x2
x3
x3
Example 7. Recall graphGof example 5. By using the method for corank 2 matroids (for each {i, j, k} = {1,2,3} if j < k then in M/i one must order j first and k last) or by doing the recursive process we get that
W1 = {1, x2, x22, x3, x2x3, x22x3, x23, x2x23}, W2 = {1, x1, x3, x1x3, x23, x1, x23}
and W3 = {1, x1, x2, x1x2, x22}. We have that c1 = c2 = 1 and c3 = 2 so P1 =w1,P2=W2 andP3=W3{1, x3}. W results as the union ofP1P2and P3. Figure (7) shows the 2 dimensional staircase diagrams for W1,W2 and W3 and the 3 dimensional staircase diagrams forP1,P2,P3 andW. We can
see that for i∈ {1,2,3},Wi determines the shape ofPi while ci determines
its width. Note thatF(W) =h(M(G)∗) = (1,3,5,4).
Now for any monomial collection O, we define Rji(O) :={m ∈O | xji 6 |m} and Lji(O) := {m | xi 6 |m ∧mxji−1 ∈ O}. So if O is an order ideal
ofr variables Rji(O) is an order ideal on r variables that can be seen as the truncation of O when ignoring all monomials with degree in xi bigger or
equal thanj. Lji(O) is an order ideal on r−1 variables that can be seen as thelevel ofO where the degree ofxi is fixed to bej.
Example 8. LetU be the order ideal generated by{x21, x22, x23, x1x2x3}. We
have that R21(U) = {1, x2, x22, x3, x2x3, x23, x1, x1x2, x1x3, x1x2x3}, L11(U) =
{1, x2, x22, x3, x2x3, x23}andL21(U) ={1, x2, x3, x2x3}. Recall figure 4.1 where
the staircase diagram of U was drawn for clarity.
Lemma 5.5. Let i∈[r]. Then Rci
i (W) =Pi. Proof. Clearly Rci
i (W) ⊇Pi. Now we will prove the other containment by
induction on r. Letm ∈Rci
i (W). Then m ∈Pj for somej. Suppose j 6=i
(otherwisem∈Pi trivially). Thenm∈Rcii(Pj). But
Rci
i (Pj) = R ci
i (Wj[xj]cj)
= Rci
i (Wj)[xj]cj
= Wi,j[xi]ci[xj]cj
where the last equality comes from the induction hypothesis. But the last expression is symmetric with respect to i and j, so Rci
i (Pj) = R cj
j (Pi), so
m∈Rcj
j (Pi)⊆Pi.
As a corollary we get
Theorem 5.6. Let G be a connected loopless graph and v0 a vertex of G
such that G[V(G)− {v0}] is connected. Then if W = WG(v0,x) we have F(W) =h(M∗)
Proof. We do induction on r. If r= 0 we have thatW ={1} butGwould be a loopless graph of one vertex, soE =∅and h(M∗) = (1) =F(W). Now assumer >0 and the theorem holds for every graph withror fewer vertices. LetD={d1, d2. . . dk}be the set of edges that come out of v0 ordered in a
way such that if i < j, each edge that connects v0 with vi occurs earlier in
D than every edge that connects v0 with vj. Moreover, for i < t let ei be
the first edge that connects v0 with vi, so that ei =dc1+...ci−1+1. Now for i≤kletMi =M\{dj | j < i}/di. We have that thatDis the complement
of a hyperplane. Letsi = i
P
j=1
cj, in other words,dsi is the last element in D
of its parallelism class. For notation simplicity sets0 = 0. LetUi=Sj≤iPi,
soUt=W.
We are going to proof by induction on ithat
FUi(x) =
si
X
j=1
xj−1hM∗
j(x) (5.2)
Note that we are doing 2 inductions, one on r and inside it, we do in-duction on i. We assume equation (5.2) holds whenever r0 < r or r0 = r
and i0 < i. Replacing iby t in equation (5.2) and using Lemma 5.3 we get
thath(M∗) =F(W). Also note that forr= 1 equation (5.2) is exactly the conclusion of this theorem, so it serves as again as first case for supposing equation (5.2) as induction hypothesis.
Now to proof equation (5.2) consider first i = 1. As dj is parallel to
e1 for every j < s1, we have that Mj is M/e1 with j−1 less loops. As
coloops do not affect the non zero entries of theh vector, Mj has the same
h vector (up to non zero entries) for everyj ≤s1. Now using the induction
hypothesis (of the induction onr) we have thatF(W1) =h(M/e1)∗. Then as P1 =W1[x1]c1, we get
FU1(x) = FP1(x)
=
s1
X
j=1
xj−1FW1(x)
=
s1
X
j=1
xj−1h(M/e1)∗(x)
=
s1
X
j=1
xj−1h(Mj)∗(x)
This proves (5.2) for i= 1. Now suppose i >1 and the equation holds for i−1. Again, hMj∗ is the same for si−1 + 1 ≤ j ≤ si. We have that
F(Ui) = F(Ui−1 ∪Pi) = F(Ui−1) +F(Pi)−F(Ui−1∩Pi). So we need to
proof that
FPi(x)−FUi−1∩Pi(x) =
si
X
j=si−1+1
xj−1hMj∗(x)
AsPi =Wi[xi]ci we have thatFPi(x) =h(M/ei)∗(x)[x]ci by the induction
hypothesis onr. Now using equation (5.1) we getUi−1∩Pi = S j<i
Pj∩Pi =
S
j<i
Wj,i[xj]cj[xi]ci. Again we have by induction on r that equation (5.2)
holds for M/ei. So if we define Ui0−1 :=
S
j<i
Wj,i[xj]cj then FUi0−1(x) =
si−1
P
j=1 xj−1h
(Mj/ei)∗(x). Then as Ui−1∩Pi=U
0
i−1[xi]ci, we have
FUi−1∩Pi(x) = FUi0−1(x)[x]ci
=
si−1
X
j=1
xj−1h(Mj/ei)∗(x)
[x]ci
FPi(x)−FUi−1∩Pi(x) =
h(M/ei)∗(x)−
si−1
X
j=1
xj−1h(Mj/ei)∗(x)
[x]ci
Now as G[V − {v0}] is connected, there is an edge from vi to some vj
