Pure $\mathcal{O}$-sequences arising from $2$-dimensional PS ear-decomposable simplicial complexes

We show that the $h$-vector of a $2$-dimensional PS ear-decomposable simplicial complex is a pure $\mathcal{O}$-sequence. This provides a strengthening of Stanley's conjecture for matroid $h$-vectors in rank $3$. Our approach modifies the approach of combinatorial shifting for arbitrary simplicial complexes to the setting of $2$-dimensional PS ear-decomposable complexes, which allows us to greedily construct a corresponding pure multicomplex.


Introduction and Background
In the late 1970s, Stanley [4] conjectured that the h-vector of a matroid simplicial complex is a pure O-sequence. Later, Chari [1] showed that any matroid simplicial complex admits a PS ear-decomposition, which inductively decomposes a simplicial complex into ears whose contribution to the h-vector could correspond to an interval of monomials in the divisibility lattice. As a consequence, it is natural to extend Stanley's conjecture to the family of PS ear-decomposable simplicial complexes. In this paper, we use this approach to show that h-vectors of 2-dimensional PS ear-decomposable simplicial complexes are pure O-sequences. We begin with the relevant background on face enumeration for simplicial complexes, then provide further background on Stanley's conjecture and PS ear-decomposable simplicial complexes.

Simplicial complexes and face numbers
A simplicial complex ∆ on (finite) vertex set V = V (∆) is a collection of subsets F ⊆ V called faces with the property that if F ∈ ∆ and G ⊆ F , then G ∈ ∆ as well. To each abstract simplicial complex, there is an associated geometric object called its geometric realization, ∆ , which contains a geometric simplex for each face F ∈ ∆.
The dimension of a face F ∈ ∆ is dim(F ) = |F | − 1 and the dimension of ∆ is dim(∆) = max{dim(F ) : F ∈ ∆}. We say that ∆ is pure if all of its facets (maximal faces under inclusion) have the same dimension. We will typically assume that ∆ is (d − 1)dimensional and pure, meaning each of its facets contains d vertices.

PS ear-decomposable complexes
Chari [1] defined the family of (d−1)-dimensional PS ear-decomposable simplicial complexes. Before defining PS ear-decomposable simplicial complexes, we need to define the components that are used to build them, which are PS spheres and PS balls.
Definition 1.1. A PS sphere is a triangulated sphere that can be decomposed as a join of simplex boundaries. A PS ball is a triangulated ball that can be decomposed as the join of a simplex and a PS sphere.
Let us illustrate these definitions when d = 3. We will use σ k to denote the k-dimensional simplex and ∂σ k to denote its boundary. For example, when k = 3, σ 3 is a solid tetrahedron and ∂σ 3 is its boundary. When k = 0 we set ∂σ 0 = {∅} by convention.
For d = 3, the possible PS 2-spheres and PS 2-balls are shown in Figure 1. The boundary of each PS ball is highlighted in red.
This leads us to the main object of study in this paper, PS ear-decomposable simplicial complexes.
In other words, a PS ear-decomposable complex can be constructed inductively by starting with a PS sphere, then repeatedly gluing PS balls to the existing complex such that each successive gluing takes place along the boundary of the corresponding ball.

Multicomplexes and O-sequences
The second main objects of study in this paper are multicomplexes, which are multiset analogues of simplicial complexes. A multicomplex M is a (finite) collection of monomials that is closed under divisibility; i.e., if µ ∈ M and ν|µ, then ν ∈ M. A multicomplex is pure if all of its maximal monomials (under divisibility) have the same degree (with deg(x i ) = 1 for all i).
Just as the f -vector of a simplicial complex ∆ counts the faces in ∆ by cardinality, a multicomplex has a corresponding F -vector that enumerates its elements by degree. If M is a multicomplex and d is the maximal degree of a monomial in M, then the F -vector is ≥0 that can be realized as the F -vector of a (pure) multicomplex is called a (pure) O-sequence. Example 1.3. The vector F = (1, 3, 5, 3) is a pure O-sequence, with corresponding multicomplex M = {1, x, y, z, x 2 , xy, xz, y 2 , yz, x 3 , xyz, y 3 }. We see that M is a multicomplex because the divisors of any monomial of M also belong to M. For example, xyz ∈ M, and its divisors -1, x, y, z, xy, xz, yz -also belong to M. Additionally, M is pure because every monomial in M is a divisor of a degree-3 monomial in M.

Matroid h-vectors and Stanley's conjecture
Having defined PS ear-decoposable simplicial complexes, multicomplexes, and O-sequences, we are finally in a position to state the problem we wish to study.
In the late 1970's, Stanley [4] established a deep connection between commutative algebra and the combinatorics of f -and h-vectors of certain families of simplicial complexes in the following result.

Cohen-Macaulay simplicial complex if and only if it is an O-sequence.
We will not formally define Cohen-Macaulay simplicial complexes at this point as they will not play a direct role in the rest of this paper, however it is worth noting that triangulations of spheres and balls are Cohen-Macaulay simplicial complexes. Another family of interesting simplicial complexes is the family of matroid simplicial complexes. Definition 1.7. A matroid is a nonempty simplicial complex ∆ that satisfies the following additional property: if F and G are faces in ∆ with |F | < |G|, then there exists an element x ∈ G \ F such that F ∪ {x} is also a face of ∆.
The extra structure imposed by this so-called exchange axiom adds tremendous structure to matroid simplicial complexes, as is evidenced by the following theorem of Chari.
Here, the condition that ∆ is coloop-free means that topologically, ∆ is not a cone. As such, there is no harm in considering only coloop-free matroids because cone vertices only append zeros to the end of the h-vector. Further, the following conjecture of Stanley has remained tantalizingly open for the past several decades: Imani et al. [3] established this result for 1-dimensional PS ear-decomposable simplicial complexes. Our goal in this paper is to establish the same result in dimension 2. Our main theorem is the following result.
Before we delve into the technical results leading up to the proof, we will begin with a broad overview. Let ∆ = Σ 0 ∪ Σ 1 ∪ · · · ∪ Σ t be a 2-dimensional PS ear-decomposable simplicial complex. We will transform ∆ into a new PS ear-decomposable simplicial complex with the same number of ears such that h(∆) = h(C(∆)). We call C(∆) the compression of ∆, which will serve as an analogue of classical compression/shifting operators to in the setting of PS ear-decomposable simplicial complexes.
For arbitrary simplicial complexes, compression and shifting operators play an important role in the characterization of f -vectors of simplicial complexes in the Kruskal-Katona Theorem [2,5] and also in Stanley's characterization of h-vectors of Cohen-Macaulay simplicial complexes [4]. The difference in this setting is that C(∆) is not generally a shifted or compressed simplicial complex because these operators do not preserve PS ear-decomposability or even purity of the underlying simplicial complex. Instead, the ears in the PS eardecomposition of C(∆), are added greedily with respect to revlex order (in a certain sense that will be made more precise later), and to each ear we define a corresponding set of monomials, also chosen greedily, that can be used to explicitly construct a corresponding pure multicomplex whose F -vector is h(C(∆)).
The rest of the paper is structured as follows. In Section 2 we study the extremal combinatorics of the underlying graph of a 2-dimensional PS ear-decomposable simplicial complex, proving the main technical results we will use in guaranteeing the existence of the compressed complexes C(∆). In Section 3 we prove the main result. Section 3 has three main subsections corresponding to the three different PS-spheres of dimension 2. The majority of the work goes into handling the case that Σ 0 = ∂σ 3 is the boundary of a tetrahedron. In the case that Σ 0 = ∂σ 1 * ∂σ 2 or Σ 0 = ∂σ 1 * ∂σ 1 * ∂σ 1 is a bipyramid or octahedron, the proof generally reduces to the case of the tetrahedron, with some exceptions for handling small boundary cases. Ultimately, the proof of Theorem 1.11 is given in Theorems 3.2, 3.6, and 3.16.

Acknowledgments
We gratefully acknowledge support from NSF grant DMS-1600048.

Shifting operators and constructible graphs
If ∆ is a 2-dimensional PS ear-decomposable simplicial complex with compression C(∆) = Σ ′ 0 ∪ Σ ′ 1 ∪ · · · ∪ Σ ′ t , our goal is to assign a family of monomials M i to each Σ ′ i so that M 0 ∪· · ·∪M i is a pure multicomplex whose F -vector is the same as the h-vector of Σ ′ 0 ∪· · ·∪Σ ′ i for each i. As we will see, it is relatively easy to describe the families M i when Σ i is one of σ 0 * ∂σ 2 , σ 0 * ∂σ 1 * ∂σ 1 , or σ 1 * ∂σ 0 , but it is more difficult when Σ i = σ 2 fills a missing triangle. Filling a missing triangle contributes (0, 0, 0, 1) to the h-vector, so the challenge in studying missing triangles is to know that there cannot be so many missing triangles to be filled that they would exceed the possible support of degree-2 monomials in the multicomplex. Throughout this section, we will use T (G) to denote the set of of triangles (3-cycles) in a simple graph G and #T (G) to denote the cardinality of that set. In order to better understand triangles in G, we begin by exploring the extremal combinatorics of the graph of a PS ear-decomposable simplicial complex.
Let G = (V, E) be a simple graph with vertices v 1 , v 2 , . . . , v n . For any distinct i, j ∈ [n], define an operator S i,j that acts on the edges of G as follows In other words, S i,j shifts edges incident to vertex v j to become incident to vertex v i whenever possible. We slightly abuse notation and use S i,j (G) to denote the resulting graph.
Proof: We establish an injective map from T (G) to T (S i,j (G)). Let τ = {v k , v ℓ , v m } be a set of vertices that span a triangle in G. If v j / ∈ τ , then τ will remain unaffected by S i,j . Similarly, if v i ∈ τ and v j ∈ τ , then τ also remains unaffected by S i,j . In either case, τ is also a triangle in S i,j (G).
Thus it remains to consider the case that

Constructible graphs
In this section we define a family of graphs called constructible graphs, which arise as graphs of 2-dimensional PS ear-decomposable simplicial complexes. We will use the following graph theoretical notation: for a vertex v in a graph G, the degree of v will be denoted as Definition 2.2. Let G be a simple graph on vertex set {v 1 , v 2 , . . . , v n } with a subset of edges labeled by elements of [n] ∪ {0}. We say G is constructible if one of the following conditions is satisfied: Viewing this recursive definition as an inductive one, constructible graphs are obtained from the complete graph K 4 with edges labeled 0 through a sequence of three possible operations: adding a new vertex v ℓ of degree three or four, all of whose edges are labeled ℓ, or inserting an unlabeled missing edge. Given a constructible graph, one can roughly see the process through which it was constructed (up to reordering) because the edges labeled ℓ > 0 specify which edges were created at the same time as vertex v ℓ and all other edges were either missing edges that were inserted (unlabeled) or edges that were part of the initial K 4 (labeled 0).
Constructible graphs are relevant to us because the graph of a 2-dimensional PS eardecomposable simplicial complex is constructible. Now we wish to bound the number of triangles in a constructible graph in terms of the types of moves that were used in its construction. For convenience, we will say that an A-move on a constructible graph consists of adding a new vertex of degree 3 with appropriately labeled edges, a B-move adds a new vertex of degree 4 with appropriately labeled edges, and an E-move adds an unlabeled missing edge. For a vertex v ∈ V (G) that is not part of the initial K 4 , we define its type, type(v), to equal 3 or 4 depending on the degree of v when it is created.
Classically, the Kruskal-Katona Theorem tells us that, among all simple graphs with a given number of vertices and edges, the one with the maximal number of triangles is obtained by adding edges reverse lexicographically. Such graphs are known as compressed graphs. Of course, this construction may lead to a graph with a large number of isolated vertices, which is not suitable to our definition of constructible graphs. Therefore, we modify this definition to better suit our needs.
Let G be a constructible graph, and let a, b, and e respectively denote the number of A-, B-, and E-moves used in constructing G. We define the compression C(G) to be the constructible graph on vertex set {v 1 , v 2 , . . . , v 4+a+b } that is built as follows: on n vertices with m edges, the graph on vertex set {v 1 , . . . , v n } whose edges are the first m edges in revlex order has the largest number of triangles. The structure of such a graph can be described simply -we can uniquely express m = p 2 + q with 0 ≤ q < p, and the compressed graph G has the following properties: • G contains a clique on {v 1 , . . . , v p }, Similarly, a compressed constructible graph comes equipped with a vertex order v 1 , . . . , v n such that • the maximal clique spans vertices {v 1 , . . . , v p } for some p ≥ 4, This connection to the classical Kruskal-Katona Theorem will be made precise in Theorem 2.5, which states that C(G) has at least as many triangles as G when G is constructible. The remainder of this section focuses on the proof of Theorem 2.5, which requires a few intermediate lemmas.
• u and v are not adjacent, and and let G ′ be the graph obtained from G by removing k unlabeled edges incident to u and adding k unlabeled edges incident to v whose neighbors lie among {v 1 , . . . , v p }. Then G ′ is constructible and #T (G) ≤ #T (G ′ ).

Proof:
The number deg G (u) − type(u) is the number of unlabeled edges incident to u and the number p − deg G (v) is the number of missing edges between v and vertices among {v 1 , . . . , v p }. So k is the largest number of unlabeled edges that could be moved from u to v. The fact that G ′ is constructible is immediate as we are only changing the unlabeled edges in G.
Because {v 1 , . . . , v p } span a clique, every pair of vertices in the neighborhood of u (respectively, v) form a triangle with u (respectively v). Because of this, and because u and v are not adjacent, we see that

Proof:
We prove the claim by induction on the number of edges in G| W . The claim is trivial when there are no edges.
First suppose that there exists an unlabeled edge e ∈ G| W . By the inductive hypothesis, To obtain C(G) from C(G−e), we add the revlex smallest missing edge. If the revlex smallest missing edge is {v i , v j } and j ≤ |W |, then holds trivially. Thus, we are left to handle the case that G| W contains only labeled edges. Order the vertices in W as v i 1 , v i 2 , . . . , v i |W | in order of their creation. In G, there is at most one edge whose largest vertex is v i 2 , at most two edges whose largest vertex is v i 3 , at most three edges whose largest vertex is v i 4 , and at most type(v i j ) edges whose largest vertex is v i j for j > 4. The same is true in C(G), and because the vertices of C(G) are ordered so that vertices of type 4 come before vertices of type 3, it must be the case that Proof: We prove the claim by induction on the number of vertices in G. When #V (G) = 4, the claim is clear as G and C(G) are both the complete graph. Therefore, we may assume #V (G) = n + 1 > 4.
If possible, pick a vertex v ∈ V (G) with type(v) = 4. Otherwise, all vertices in G that are not part of the original K 4 have type 3; pick one of those vertices arbitrarily. Let δ = deg G (v). Note that the number of triangles in G that contain v is equal to the number of edges in G| N (v) . Therefore, by the inductive hypothesis and Lemma 2.4, This label will be temporary, as we need to determine the position where v should be inserted into the given order on the vertices of C(G − v). By construction, Moreover, for all i > 4, the labeled edges incident to v i are the those incident to the first By Eqs (2.1) and (2.2), it follows that #T (G) ≤ #T (G ′ ), so we need only show that #T (G ′ ) ≤ #T (C(G)). We examine several cases.  As above, let δ = deg(v) and insert v into the given vertex order so that v 4 < v < v 5 . Having done this, the graph G ′ fails to be compressed because there are unlabeled edges of the form {v, v i } for p + 2 ≤ i ≤ δ that are not the revlex smallest edges that could have been added. However, it is still the case that deg Therefore, we may repeatedly apply Lemma 2.3, first to the pair of vertices {v p+1 , v δ }, then {v p+1 , v δ−1 }, and so on, each time moving the unlabeled edge {v, v k } for k > p + 1 to the revlex smallest missing edge incident to v p+1 . Each of these moves weakly increases the number of triangles in the graph, and hence #T (G ′ ) ≤ #T (C(G)) as desired. We note that at some point in this process, it could be the case that v p+1 becomes adjacent to all vertices among {v, v 5 , . . . , v p }. In this case, we simply start moving unlabeled edges so that they form the revlex smallest missing edge incident to v p+2 instead (and, if necessary, then to v p+3 , v p+4 , etc.).

Case 3: deg(v) < p
This is the most complicated of the cases and requires a few sub-cases.
Then we apply Lemma 2.3 to the pair of vertices {v, v p+1 } to obtain C(G) and see that #T (G ′ ) ≤ #T (C(G)).
If type(v) = type(v p+1 ), then by our choice of v it must be the case that type(v) = 4 and type(v p+1 ) = 3. If there exists an index 5 ≤ i ≤ p such that type(v i ) = 3, we can pick the smallest such i and perform an operation that declares type(v i ) = 4 and type(v) = 3. Since 4 < i ≤ p, we know that {v 4 , v i } ∈ G ′ , so we can give that edge the label i and remove the label v from the edge {v 4 , v}. This does not change the number of trees in G ′ , but now we have arranged for type(v) = 3 = type(v p+1 ) and we can apply the argument used in the previous case.
Thus we need only examine the case that type(v p ) = type(v) = 4, but type(v p+1 ) = 3.
This preserves the condition that all vertices of type 4 come before the vertices of type 3 in the vertex order, and then we can apply Lemma 2.3 to the pair of vertices {v, v p+1 } to obtain C(G) from G ′ . On the other hand, if deg(v) < deg(v p+1 ), we first perform the above operation to swap edge labels so that type(v) = 3 and type(v p+1 ) = 4. Then we insert v into the vertex order on C(G − v) so that v p+1 < v < v p+2 and then apply Lemma 2.3.
Our goal in this section is to prove that the h-vector of a PS ear-decomposable simplicial complex is a pure O-sequence. The proof breaks into three main cases for the three possible base spheres Σ 0 , along with several subcases. We begin with a broad overview of the main cases and subcases in the the remainder of the paper.
1. The case that Σ 0 = ∂σ 3 is the boundary of a tetrahedron is the simplest case given Theorem 2.5. Algorithm 3.1 defines the compressed complex C(∆) and corresponding pure multicomplex M(∆). The fact that this algorithm terminates and produces a pure multicomplex is proved in Theorem 3.2.
(a) When η F > 0, we can reduce the problem to the case that Σ 0 = ∂σ 3 in Theorem 3.3.
(b) When η F = 0, Algorithm 3.4 defines the compressed complex C(∆) and corresponding pure multicomplex M(∆). The fact that this algorithm terminates and produces a pure multicomplex is proved in Theorem 3.5.
(a) When η F = 0, Algorithm 3.7 defines the compressed complex C(∆) and corresponding pure multicomplex M(∆). The fact that this algorithm terminates and produces a pure multicomplex is proved in Theorem 3.8.
(b) When η E = 0, Algorithm 3.10 defines the compressed complex C(∆) and corresponding pure multicomplex M(∆). The fact that this algorithm terminates and produces a pure multicomplex is proved in Theorem 3.11.
i. When {v 1 , v 4 } ∈ ∆, we reduce the problem to the case that Σ 0 = ∂σ 1 * ∂σ 2 is a bipyramid in Proposition 3.12. ii. When {v 1 , v 4 } / ∈ ∆, we use shifting operators S i,j to reduce to the case that Σ 0 = ∂σ 3 is a tetrahedron, but with two extra monomials of degree-3 in Theorem 3.14. We then show that two monomials of degree-3 can be removed in Theorem 3.15, which completes the proof.
Throughout the remainder of the paper, we will label the vertices of the tetrahedron, bipyramid, and octahedron as shown in Figure 2.

The case that Σ 0 = ∂σ 3
We now present the main algorithm that will be used to prove that h-vectors of 2-dimensional PS ear-decomposable complexes are pure O-sequences when Σ 0 = ∂σ 3 . Step 0: Step 1: • For 5 ≤ i ≤ η B + 4: -Introduce vertex v i to C(∆) through a B-move whose boundary is glued along the -Introduce vertex v i to C(∆) through an A-move whose boundary is glued along the Step 2: • For 1 ≤ ℓ ≤ η E : -Let {v i , v j } be the revlex smallest missing edge in C(∆).
-Add the edge {v i , v j } to C(∆) through an E-move whose boundary is the cycle Step 3: • For 1 ≤ ℓ ≤ η F : -Let G be the revlex smallest missing 2-face in C(∆). Add G to C(∆).

-Let µ be the revlex smallest degree-3 monomial not belonging to M(∆) whose proper divisors all belong to M(∆). Add µ to M(∆).
Note first that upon the completion of Step 1, M(∆) is a pure multicomplex containing x 2 j for all 4 ≤ j ≤ η A + η B + 4. The degree-2 monomials that do not belong to M(∆) at this point are those of the form x 4 x j for which vertex v j was introduced through an A-move, along with all of those of the form x i x j with 5 ≤ i < j ≤ η A + η B + 4. Thus the number of such monomials is η A + η A +η B 2 . But this is exactly the same as the number of missing edges in C(∆) after the completion of Step 1, meaning that Step 2 of the Algorithm will terminate and the resulting multicomplex M(∆) is still a pure multicomplex. Now we turn our attention to Step 3, which is slightly more complicated. Again we count the number of missing 2-faces in C(∆) and the number of degree-3 monomials not belonging to M(∆) upon the completion of Step 2.
Let G be the underlying graph of ∆ and G ′ the underlying graph of C(∆). By the construction of C(∆) in Steps 0, 1, and 2, G ′ = C(G) is a compressed constructible graph. Let p be the size of the largest clique in G ′ , and let q denote the degree of v p+1 . Furthermore, let a ′ be the number of vertices among {v 5 , . . . , v p+1 } that were introduced through a type-A move and let a ′′ be the number of vertices among {v p+2 , . . . , v n } that were introduced through a type-A move. Define b ′ and b ′′ similarly for vertices introduced through a type-B move.
Observe that p + 1 = 4 + a ′ + b ′ , so Further, because there are p 2 + q edges among the first p + 1 vertices of G ′ , but 3a ′ + 4b ′ were introduced as part of the A-or B-moves used to create vertices and 6 were part of the initial PS sphere Σ 0 .
Next, we can directly count that However, 3η A + 4η B + 2η E + 4 of those triangles span triangular faces in C(∆) after the completion of Step 2 because A-, B-, and E-moves respectively introduce 3, 4, and 2 triangular faces, and there are 4 triangular faces in Σ 0 . Finally, by Theorem 2.5, the graph of ∆ cannot have more triangles than G ′ , and ∆ also has 3η A + 4η B + 2η E triangular faces that are introduced through A-, B-, and E-moves, and 4 additional triangular faces in Σ 0 . Therefore, On the other hand, upon the completion of Step 2, the degree-2 monomials in M(∆) support the following degree-3 monomials: • the a ′′ monomials x 3 j corresponding to vertices v j that were introduced through an A-move for j > p + 1, and • the 3b ′′ monomials of the form {x 2 4 x j , x 4 x 2 j , x 3 j } corresponding to vertices v j that were introduced through a B-move for j > p + 1.
Use m = p−1 3 + q−1 2 + a ′′ + 3b ′′ denote the number of available monomials. Upon the completion of Step 2, 1 + η A + η B + η E of these monomials have been added to M(∆), and hence the number of available degree-3 monomials that can be added to M(∆) is This tells us that η F is bounded by the number of degree-3 monomials that can be added to M(∆) upon the completion of Step 2 while still preserving the property that M(∆) is a multicomplex. Consequently, Step 3 will terminate and upon its completion, M(∆) will be a pure multicomplex. This proves the following theorem.

The case that
The case that Σ 0 is a bipyramid is very similar to the case that Σ 0 is the boundary of a tetrahedron. We can start by labeling the vertices of the bipyramid as in Figure 2. Now we examine two cases. First suppose η F > 0. If at some point the missing face {v 1 , v 2 , v 3 } is filled through an F-move, then we can view ∆ as a PS ear-decomposable complex with Σ 0 = ∂σ 3 by starting with the boundary of the tetrahedron on {v 1 , v 2 , v 3 , v 4 }, then adding vertex v 5 through an A-move, and then attaching the remaining ears in order to construct ∆. On the other hand, if {v 1 , v 2 , v 3 } is not filled through an F-move, pick any 2-face filled through an F-move, remove it from ∆, and fill {v 1 , v 2 , v 3 } instead. This creates a new PS ear-decomposable complex ∆ ′ with h(∆ ′ ) = h(∆). In either of these cases, we can then apply Theorem 3.2 to see that h(∆) is a pure O-sequence. This proves the following theorem.
On the other hand, if η F = 0, we can use the following algorithm to construct a pure multicomplex whose F -vector is the same as h(∆).
Step 1: • For 6 ≤ i ≤ η B + 5: -Introduce vertex v i to C(∆) through a B-move whose boundary is glued along the • For η B + 6 ≤ i ≤ η A + η B + 5: -Introduce vertex v i to C(∆) through an A-move whose boundary is glued along the Step 2: . The proof that this algorithm terminates and produces a pure multicomplex is identical to the proof that Steps 0, 1, and 2 in Algorithm 3.4 terminate and produce a pure multicomplex. Upon the completion of Step 1, the number of missing edges in C(∆) is η A +η B 2 +2η A +η B +1. Indeed, none of the newly introduced vertices are adjacent, giving η A +η B 2 missing edges; each vertex v j with j > 5 is part of a missing edge with v 5 , and it is part of a missing edge with v 4 if it was introduced through an A-move; finally, the edge {v 4 , v 5 } is missing. On the other hand, the number of missing degree-2 monomials in M(∆) upon the completion of Step 1 is Once again, this guarantees that Step 2 will terminate. Moreover, {v 4 , v 5 } is the revlex smallest missing edge in C(∆) after the completion of Step 1. This means that if η E > 0, then the first missing edge that is inserted also adds x 2 5 to M(∆), at which point x 2 j ∈ M(∆) for all j. This guarantees that the monomials added to M(∆) in subsequent iterations of Step 2 preserve the property that M(∆) is a pure multicomplex. Thus, we have proved the following theorem.  Theorem 3.6. Let ∆ = Σ 0 ∪Σ 1 ∪· · ·∪Σ t be a 2-dimensional PS ear-decomposable simplicial complex with Σ 0 = ∂σ 1 * ∂σ 2 . Then h(∆) is a pure O-sequence.

The case that
The case that Σ 0 = ∂σ 1 * ∂σ 1 * ∂σ 1 is the boundary complex of the octahedron is more complicated than the previous two cases and cannot immediately be reduced to the case that Σ 0 = ∂σ 3 . This case requires special handling of base cases for small values of η A , η B , η E , and η F . We can start by labeling the vertices of the octahedron as in Figure 2. 3.3.1 The case that η F = 0 Algorithm 3.7. Let ∆ be a 2-dimensional PS ear-decomposable simplicial complex with Σ 0 = ∂σ 1 * ∂σ 1 * ∂σ 1 and η F = 0. We define a new PS ear-decomposable complex C(∆) and a set of monomials M(∆) in the variables {x 4 , x 5 , x 6 , . . . , x η A +η B +6 } inductively as follows.
Step 1: • For η B + 7 ≤ i ≤ η A + η B + 6: -Introduce vertex v i to C(∆) through an A-move whose boundary is glued along the cycle v 1 v 2 v 3 .
Step 2: -Let {v i , v j } be the revlex smallest missing edge in C(∆). - Step 0 and Step 1 in Algorithm 3.7 proceed as they have in all other cases, and it is clear that M(∆) is a pure multicomplex upon the completion of Step 1. In Step 2, the revlex smallest missing edges {v 1 , v 4 }, {v 2 , v 5 }, and {v 3 , v 6 } correspond to the values ℓ = 1, ℓ = 2, and ℓ = 3 respectively. When ℓ ≥ 4, we see that x 2 i ∈ M(∆) for all 4 ≤ i ≤ η A + η B + 6 and moreover that the revlex smallest missing edge {v i , v j } satisfies j ≥ 7 and i ≥ 4. Therefore the contribution of the monomials {x i x j , x 2 i x j } preserves the property that M(∆) is a pure multicomplex. This proves the following theorem.
Proof: Attaching a PS ball by an A-move does not change the number of missing triangles in a PS ear-decomposable simplicial complex. Therefore, since Σ 0 does not contain any missing triangles, neither will ∆ if η B = 0. Now suppose η B > 0. When a new vertex is introduced through a type-B move, there is a potential for two missing triangles to be introduced to ∆. Specifically, if vertex v is introduced through a B-move whose boundary vertices are labeled as shown below, then the new missing triangles that are created are {v, x, z} (as long as {x, z} was already an edge) and {v, w, y} (as long as {w, y} was already an edge).
v w x y z Therefore, η F ≤ 2η B whenever η B > 0. However, if η A = 0, then the first ear attached through a B-move cannot contribute two missing triangles because the graph of the octahedron does not contain an induced K 4 subgraph (meaning either {x, z} or {w, y} will be missing). This means that when η A = 0 and η B > 0, it must be the case that η F ≤ 2η B − 1.
Step 0: Step 1: • If η A = 0 and η B > 0: -Introduce vertex v 7 to C(∆) through a B-move whose boundary is glued along the -For 7 ≤ i ≤ η A + 6: * Introduce vertex v i to C(∆) through an A-move whose boundary is glued along Step 2: • If η A = 0: -Add the first η F monomials in S (under revlex order) to M(∆).
-Add the first η F faces in F (under revlex order) to C(∆).
-Add the first η F faces in F (under revlex order) to C(∆).
This algorithm requires more case analysis because of the bound on η F when η A = 0 and η B > 0 in Lemma 3.9. In Step 1, we introduce all vertices through A-and B-moves. In the case that η A > 0, we break with our previous strategy and introduce all type-A vertices first out of convenience. As noted in the proof of Lemma 3.9, we do this so that there will be a K 4 subgraph on the vertices {v 1 , v 2 , v 3 , v 7 } where the boundaries of the type-B balls can be glued. In Step 2, when η A = 0, there are 2η B − 1 faces in set F , which correspond to the maximum number of missing triangles in a PS ear-decomposable complex with η E = 0 and η A = 0 by Lemma 3.9. There are 2η B − 1 corresponding monomials in the set S that can be added to M(∆). Similarly, when η A > 0, there are 2η B faces in set F and 2η B monomials in set S. Therefore, Algorithm 3.10 terminates and produces a pure O-sequence. This proves the following theorem. Our goal in the case that η E > 0 and η F > 0 is to reduce to the cases already established in which Σ 0 is the boundary of a tetrahedron or a bipyramid. Proposition 3.12. Let ∆ = Σ 0 ∪ Σ 1 ∪ · · · ∪ Σ t be a 2-dimensional PS ear-decomposable simplicial complex with Σ 0 = ∂σ 1 * ∂σ 1 * ∂σ 1 . Assume the vertices of Σ 0 are labeled as in Figure 2. If {v 1 , v 4 } ∈ ∆, then there exists a PS ear-decomposable simplicial complex Proof: Let Σ i be the ear in which edge {v 1 , v 4 } is introduced. The ear Σ i must be of type-E, so there are vertices u and w such that Σ i has triangular faces {v 1 , v 4 , u} and {v 1 , v 4 , w}.
The proof proceeds in two steps. First we handle the case that {u, w} = {v 2 , v 5 }. Next we reduce to that case.
If {u, w} = {v 2 , v 5 }, let Σ ′ 0 be the bipyramid and let Σ ′ 1 be the PS ear shown in Figure 3. Figure 3: The new PS sphere Σ ′ 0 (left) and PS ball Σ ′ 1 (right). If τ ∈ ∆, then τ is added to ∆ through an F-move because {v 1 , v 5 } and {v 4 , v 5 } are edges in Σ 0 and {v 1 , v 4 } is introduced through Σ i , and neither Σ 0 nor Σ i contains τ . Therefore, there is some j > i such that Σ j is an F-move introducing face τ . We can replace Σ i with the PS ear Σ ′ i , which is an E-move introducing edge {v 1 , v 4 } using the triangles {v 1 , v 2 , v 4 } and {v 1 , v 4 , v 5 }, and replace Σ j with the PS ear Σ ′ j , which is an F-move introducing the face {v 1 , v 4 , w}. This is an alternate PS ear decomposition of ∆ that satisfies the claim.
On the other hand, if τ / ∈ ∆, let ∆ ′ be the complex obtained from ∆ by removing the face {v 1 , v 4 , w} and adding the face τ . This complex is PS ear-decomposable because we can replace Σ i with the ear Σ ′ i , which is an E-move with triangles {v 1 , v 2 , v 4 } and {v 1 , v 4 , v 5 }. Clearly h(∆ ′ ) = h(∆), so ∆ ′ satisfies the claim.
Now we turn our attention to the case that {v 1 , v 4 } / ∈ ∆. We begin with a lemma whose proof is essentially Algorithm 3.1.
Lemma 3.13. Let G be a constructible graph and C(G) its compression. There exists a 2-dimensional PS ear-decomposable simplicial complex ∆ whose underlying graph is C(G).
Proof: We begin with the boundary of a tetrahedron on vertices {v 1 , v 2 , v 3 , v 4 }. For each vertex v i of type-B in C(G), we attach an ear of type-B whose boundary is the cycle v 1 v 2 v 3 v 4 , and for each vertex of type-A, we attach an ear of type-A whose boundary is the cycle v 1 v 2 v 3 . If {v i , v j } is an edge added to C(G), then v 1 and v 2 are common neighbors of v i and v j by construction, so we attach an ear of type-E whose boundary is the cycle v 1 v i v 2 v j . The underlying graph of the resulting complex is C(G).
For the remainder of the paper, we assume ∆ is a 2-dimensional PS ear-decomposable simplicial complex with Σ 0 = ∂σ 1 * ∂σ 1 * ∂σ 1 such that {v 1 , v 4 } / ∈ ∆. Let G be the underlying graph of ∆. Let G ′ be the graph obtained from G by applying the shifting operator S 4,5 . We know G is constructible and claim that G ′ is also constructible. First, S 4,5 acts on the graph of Σ 0 by removing edge {v 1 , v 5 } and adding edge {v 1 , v 4 }.
The resulting graph is constructible, starting with the complete graph on {v 1 , v 2 , v 3 , v 4 }, then adding edges {v 1 , v 6 }, {v 3 , v 6 }, and {v 4 , v 6 } through an A-move that introduces vertex v 6 , and adding edges {v 2 , v 5 }, {v 4 , v 5 }, and {v 5 , v 6 } through an A-move that introduces vertex v 5 . We now proceed inductively. For each vertex v ℓ with ℓ > 6, consider the ℓ-labeled edges incident to v ℓ in G. Let N denote the set of vertices incident to v ℓ by such edges. If v 5 / ∈ N or if {v 4 , v 5 } ⊂ N, add ℓ-labeled edges {v i , v ℓ } to G ′ for all i ∈ G. Otherwise, if v 5 ∈ N but v 4 / ∈ N add ℓ-labeled edges {v 4 , v ℓ }, and {v i , v n } for i ∈ N \ {v 5 } to G ′ .
Having done this, we move on to consider the unlabeled edges in G. Suppose {v i , v j } is such an edge and i < j. If i > 5, add the unlabeled edge {v i , v j } to G ′ as well. If i = 4, it is possible that {v 4 , v j } already belongs to G ′ as a j-labeled edge. If this happens, then {v 4 , v j } is an unlabeled edge in G and {v 5 , v j } is a j-labeled edge in G. This means {v 5 , v j } / ∈ G ′ , so we can add the edge {v 5 , v j } to G ′ . Finally, if i = 5 and {v 4 , v j } is not an edge in G ′ , add the unlabeled edge {v 4 , v j } to G ′ . Otherwise, add the unlabeled edge {v 5 , v j } to G ′ . This proves that G ′ is also constructible.
Next, by Lemma 2.1 we know #T (G) ≤ #T (G ′ ), and because G ′ is constructible, Theorem 2.5 tells us #T (G ′ ) ≤ #T (C(G ′ )). Finally, by Lemma 3.13, there is a PS eardecomposable simplicial complex ∆ ′ whose underlying graph is C(G ′ ). Use η ′ A , η ′ B , η ′ E , and η ′ F to denote the number of ears of type A, B, E, and F respectively in the construction of ∆ ′ . It follows from the proof of Lemma 3.13 that the PS sphere used in the construction of ∆ ′ is ∂σ 3 and that η ′ A = η A + 2, η ′ B = η B , η ′ E = η E , and η ′ F = 0. Finally, since #T (G) ≤ #T (C(G ′ )), there are at least η F missing triangles in ∆ ′ that can be filled. This proves the following theorem.

Now we can observe that M \ {x
x 5 x 6 1 Now we can observe that M \ {x 3 4 , x 4 x 2 5 } is a pure multicomplex whose F -vector is h(∆).