Numerical semigroups, polyhedra, and posets I: the group cone

Several recent papers have explored families of rational polyhedra whose integer points are in bijection with certain families of numerical semigroups. One such family, first introduced by Kunz, has integer points in bijection with numerical semigroups of fixed multiplicity, and another, introduced by Hellus and Waldi, has integer points corresponding to oversemigroups of numerical semigroups with two generators. In this paper, we provide a combinatorial framework from which to study both families of polyhedra. We introduce a new family of polyhedra called group cones, each constructed from some finite abelian group, from which both of the aforementioned families of polyhedra are directly determined but that are more natural to study from a standpoint of polyhedral geometry. We prove that the faces of group cones are naturally indexed by a family of finite posets, and illustrate how this combinatorial data relates to semigroups living in the corresponding faces of the other two families of polyhedra.


Introduction
Let Z ≥0 denote the set of nonnegative integers. A numerical semigroup S is a subset of Z ≥0 that contains 0, is closed under addition, and has finite complement in Z ≥0 (the final condition is equivalent to requiring the greatest common divisor of the elements of S is 1). We often specify a numerical semigroup via generators, writing S = n 1 , . . . , n k = {a 1 n 1 + · · · + a k n k : a 1 , . . . , a k ∈ Z ≥0 } for the numerical semigroup generated by n 1 , . . . , n k . It is known that any numerical semigroup has a unique generating set that is minimal with respect to containment; its cardinality is known as the embedding dimension of S. The set of gaps of S, denoted G(S), is the finite set Z ≥0 \ S. The cardinality of this set is the genus g(S) of S. The smallest positive element of S is called the multiplicity of S, denoted m(S).
To motivate the contents of this manuscript, we survey two counting problems involving numerical semigroups. Each problem can be realized as an integer point counting problem in some family of rational polyhedra. One of the primary insights of this manuscript is to identify a strong combinatorial connection between these polyhedra.
1.1. Counting by multiplicity and genus. There has been much recent interest in counting numerical semigroups by genus; see the survey article [14] for an overview of problems and results in this area. Let N (g) denote the number of numerical semigroups S with g(S) = g. Bras-Amorós computed the first 50 values of N (g) and made several conjectures about the behavior of this function [5]. Zhai determined the asymptotic growth rate of N (g), thereby proving N (g) ≤ N (g+1) for g sufficiently large [24]. However, the following conjecture remains open, as the smallest upper bound for possible exceptions is well out of range of computation [17,Section 6]. Conjecture 1.1 (Bras-Amóros). For all g ≥ 1, we have N (g) ≤ N (g + 1).
One approach to Conjecture 1.1 is to study a more refined counting problem. Let N m (g) denote the number of numerical semigroups S with m(S) = m and g(S) = g. See [15, Conjecture 7]). For any m ≥ 2 and g ≥ 1, N m (g) ≤ N m (g + 1).
In order to state what is known about N m (g) we introduce some additional notation. A quasipolynomial of degree d is a function f : Z ≥0 → C of the form f (n) = c d (n)n d + c d−1 (n)n d−1 + · · · + c 0 (n) with periodic functions c i having integer periods, c d = 0. The function c d is called the leading coefficient of f . The proof of Theorem 1.3 uses Ehrhart theory and a bijection between numerical semigroups of multiplicity m and certain integer points in a rational polyhedron P m , called the Kunz polyhedron (we defer the formal definition to Section 4). In particular, this yields a geometric interpretation of the leading coefficient of p m (g). Additionally, the face structure of P m was studied in [6] to provide a new approach to a longstanding conjecture of Wilf [23].
1.2. Counting oversemigroups. Let S be a numerical semigroup. An oversemigroup of S is a numerical semigroup T with T ⊇ S. Let o(S) denote the number of oversemigroups of S. Since G(S) is a finite set, and any numerical semigroup T ⊇ S has G(T ) ⊆ G(S) and is determined by its set of gaps, it is clear that o(S) is finite.
Hellus and Waldi study the function o(S) in the case S = n, q is the smallest semigroup containing n, q and gcd(n, q) = 1. For simplicity, write o(n, q) = o( n, q ). (a) There is a quasipolynomial H n (q) of degree n − 1 taking the value o(n, q) at each positive integer q relatively prime to n. (b) The leading coefficient λ(n) of H n (q) is constant and Like Theorem 1.3, the proof of Theorem 1.4 uses a bijection between oversemigroups of n, q and certain integer points in a rational polyhedral cone O n , which we refer to here as the oversemigroup cone. The leading coefficient λ(n) again has a geometric interpretation, as the volume of a particular cross section of O n , from which the bounds in Theorem 1.4(b) are obtained in [13]. We defer the formal definition of O n to Section 5.

1.3.
Enter the group cone. The goal of this manuscript is to provide a common framework for studying the combinatorial structure of the Kunz polyhedron P m and the oversemigroup cone O n via the introduction of a new family of polyhedra, the group cone C(G) of a finite abelian group G (Definition 3.1), which has also appeared in the context of discrete optimization as the cone of subadditive functions [12,18].
• We give a combinatorial description of the faces of C(G) in terms of certain partially ordered sets. In doing so, we complete the partial description of the faces of the Kunz polyhedron from [6], and provide a previously unknown description of the faces of the oversemigroup cone. In both settings, the poset corresponding to a face F manifests within the divisibility posets of all semigroups lying in F . • We identify particular cross sections of the group cone whose relative volumes equal the leading coefficients of the quasipolynomials in Theorems 1.3 and 1.4. This implies that obtaining a triangulation for the group cone, which is a common method for computing or bounding cross section volumes, simultaneously yields control over the leading coefficients of both quasipolynomials.
Several subsequent papers have already made use of the framework established here. First, one operation that has been studied extensively in the numerical semigroup literature is called gluing, and in [1], the gluing operation is realized geometrically via a collection of combinatorial embeddings of group cones. Second, a major technique for understanding a numerical semigroup S is to study its minimal presentations, which encode minimal algebraic relations among the generators of S, and in [2], a combinatorial connection is given between the face of P m containing S and the minimal presentations of S. Said another way, the face lattice of the group cone gives a stratification of the set of all numerical semigroups, wherein the numerical semigroups within each stratum have minimal presentations of a particular combinatorial type. Third, in forthcoming work, the ideas of this paper are used to develop a combinatorial recipe for specializing free resolutions of numerical semigroup algebras.
After defining the necessary terminology from polyhedral geometry in Section 2, we introduce the group cone C(G) in Section 3 and study the combinatorial data associated to its faces. Sections 4 and 5 contain formal definitions of the Kunz polyhedron P m and the oversemigroup cone O n , respectively, and provide precise connections to the faces of the group cone. Combining results in these sections gives a direct correspondence between the Kunz polyhedron and the oversemigroup cone. In Section 6, we reduce the task of obtaining the precise leading coefficients of the quasipolynomial formulas in the counting problems described above to that of finding a triangulation of the corresponding group cone. We conclude with Section 7, wherein we present an improved algorithm for computing the Apéry set of a numerical semigroup S. We also include an appendix with some computational data related to the quasipolynomial functions introduced above.

Background
In this section, we provide the necessary definitions related to convex rational polyhedra and partially ordered sets. For more thorough introductions, see [25] and [22].
A partially ordered set (or poset, for short) is a set Q equipped with a relation (called a partial order ) that is reflexive, antisymmetric, and transitive. Given q, q ∈ Q, we write q ≺ q whenever q q and q = q . We say q covers q if q ≺ q and there does not exist q ∈ Q with q ≺ q ≺ q. If (Q, ) has a unique minimal element 0 ∈ Q, the atoms of Q are the elements that cover 0. Posets are often depicted using a Hasse diagram in which the elements of Q are drawn so that whenever q covers q , q is drawn above q and an edge is drawn from q down to q . See Figure 2 for examples.
A rational polyhedron P ⊂ R d is the set of solutions to a finite list of linear inequalities with rational coefficients, that is, for some matrix A and vector b. If none of the inequalities can be omitted without altering P , we call this list the H-description or facet description of P (such a list of inequalities is unique up to reording and multiplying both sides by a positive constant). The inequalities appearing in the H-description of P are called facet inequalities of P .
Given a facet inequality a 1 x 1 + · · · + a d x d ≤ b of P , the intersection of P with the hyperplane a 1 x 1 + · · · + a d x d = b is called a facet of P . A face F of P is a subset of P equal to the intersection of some collection of facets of P . The set of facets containing F is called the H-description or facet description of F . The dimension of a face F is the dimension dim(F ) of the affine linear span of F . We say F is a vertex if dim(F ) = 0, an edge if dim(F ) = 1 and F is bounded, a ray (or an extremal ray) if dim(F ) = 1 and F is unbounded, and a ridge if dim(F ) = d − 2.
If the origin is the unique point lying in the intersection of all facets of P (or, equivalently, if b = 0 in the H-description of P ), then we call P a pointed cone. Separately, we say P is a polytope if P is bounded. If P is a pointed cone, then any face F equals the nonnegative span of the rays of P it contains, and if P is a polytope, then any face F equals the convex hull of the set of vertices of P it contains; in each case, we call this the V-description of F .
The set of faces of a polyhedron P forms a poset under containment that is a lattice (i.e., any two faces have a unique greatest common descendant and a unique least common ancestor) and is graded by dimension (i.e., whenever F covers F , we have dim(F ) = dim(F ) + 1). If P is a polytope, then every face of P equals the convex hull of some collection of vertices of P and the intersection of some collection of facets of P , meaning the face lattice is both atomic and coatomic.

The group cone
We begin by defining the group cone of a finite abelian group.
where the coordinates are indexed by the nonzero elements of G. Proof. The first claim follows from the fact that for each nonzero a ∈ G, the vector v with v a = 2 and v b = 1 for b = a lies in C(G), since R m−1 is spanned by these vectors. For the second claim, suppose m ≥ 3. We first verify that for each a ∈ G \ {0}, the inequality x a ≥ 0 is redundant. Let k = |a|. The key is for any x ∈ C(G), for any positive integer c < k. As such, if k ≥ 3, then while if k = 2, then there exists some b ∈ G \ {0, a}, so In either case, we obtain x a ≥ 0. Lastly, given a, b ∈ G \ {0} with a + b = 0, we see the The remainder of this section is dedicated to a combinatorial interpretation of the face lattice of the group cone in terms of Kunz-balanced posets.
Throughout the rest of this section, when we have a finite abelian group G and a subgroup H ⊂ G, we write x for the image of x in G/H.
is a subgroup of G and is the Kunz-balanced poset on G/H with minimal element 0 and the property that Next, define a reflexive relation on G/H with unique minimal element 0 such that for each nonzero a, b ∈ G with a + b = 0, we have a a + b whenever x a + x b = x a+b for all x ∈ F (note that is well defined by the last sentence of the preceding paragraph).
so is transitive and thus a partial order. Since is Kunzbalanced by construction, the proof is complete.   Figure 1 depicts the Kunz-balanced poset corresponding to each of these eight proper nontrivial faces. Notice that whenever two faces F, F ⊂ C(G) satisfy F ⊂ F , the Kunz subgroup of F contains the Kunz subgroup of F , and if these subgroups coincide, then the Kunz poset of F refines the Kunz poset of F . In particular, the lower right ray has nontrivial Kunz subgroup since the posets of its facets have contradictory orderings (indeed, 1 3 in one while 3 1 in the other).
The remaining 8 rays each have their Kunz poset on Z 6 as depicted in Figure 2. Each of the first 6 posets appears next to the poset lying in the same orbit under the action of the automorphism group of Z 6 discussed in Remark 3.8, and the last 2 are fixed by both automorphisms. This is also visually evident in the Hasse diagrams of Figure 2, where the last 2 are again the ones fixed under both automorphisms.
Next, suppose b ∈ G is nonzero and not an atom, so that b covers some other element a ∈ G. By part (a), b = a + m for some m ∈ M , and x b = x a + x m , so part (b) now follows from induction on the height of b in .
Remark 3.11. The inequality on dim F in Proposition 3.10(b) can be strict, as demonstrated by 6 of the 8 posets in Example 3.9. Also, the number of maximal elements of a Kunz poset need not bound the dimension of its corresponding face in the group cone in either direction. Indeed, the poset corresponding to the ray (1, 2, 1, 2, 1) of C(Z 6 ) in Example 3.9 has 2 maximal elements, and the 2-dimensional facet in C(Z 4 ) with defining equation x 3 = x 1 + x 2 has only 1 maximal element.
The following two propositions demonstrate that not every Kunz-balanced poset on an abelian group G corresponds to a face of C(G). They also provide evidence that characterizing precisely the set of Kunz-balanced posets that correspond to faces is likely difficult in general; see Remark 3.16. Proof. Under the given assumptions, a + b and a + b + c are nonzero since neither is minimal under . Additionally, a + c must be nonzero, since otherwise This implies x a + x c = x a+c and x a+c + x b = x a+b+c , as desired.
Remark 3.13. Proposition 3.12 is a kind of "diamond property" that is a reflection of the commutativity of G; see the left graphic in Figure 3 for a depiction.
Proposition 3.14. Fix a face F ⊂ C(G) with Kunz poset and trivial Kunz subgroup, a subgroup G ⊂ G with |G | odd, and a ∈ G \ G with 2a / ∈ G . The following are equivalent: Proof. The condition 2a / ∈ G ensures every element of G in both statements is nonzero. For any b ∈ G , if g, g ∈ G satisfy 2(a + g) + b = 2(a + g ) + b, then the order of g − g in G divides 2, and since |G | is odd, we can conclude g = g . From this, we obtain for all x ∈ C(G) by applying each inequality x i + x i+b ≥ x 2i+b exactly once for each i ∈ a + G . As such, if (a) holds for some b ∈ G , then equality holds in (3.2), so (a) must hold for all b ∈ G , from which (b) follows. Since (b) clearly implies (a), this completes the proof.  ≺ c for c ∈ 1 + G and c ∈ 2 + G to hold once the relations 1 ≺ 2, 4 ≺ 8, and 7 ≺ 5 hold. Intuitively, the sums of elements of 1 + G are "evenly distributed" in 2 + G , so once sufficiently many relations between them are included, the rest must also appear.
Remark 3.16. Propositions 3.12 and 3.14 are not the only restrictions on Kunz posets. For example, when G = Z 8 , intersecting the facets x 1 + x 5 = x 6 and x 3 + x 7 = x 2 yields a face whose poset is depicted on the right in Figure 3. This is particularly noteworthy since it is an example of two facets with no variables in common whose intersection is not a ridge (a face of codimension 2), a phenomenon that does not occur in C(Z m ) for m ≤ 7. In fact, the face of C(Z 8 ) corresponding to this poset only has dimension 2.

The Kunz polyhedron
We begin this section by defining the Kunz polyhedron P m and explaining the bijection between its integer points and numerical semigroups containing m. Although many of the results in this section have appeared elsewhere, we state them here using the language of Section 3. Doing so answers [6, Problem 3.14] by providing a complete combinatorial characterization of the faces of P m (Theorem 4.7).
One of the primary new insights of this section is Corollary 4.8, which identifies a correspondence between integer points in C(Z m ) and numerical semigroups containing m. This correspondence first identifies these points of C(Z m ) with integer points in P m . This allows one to move freely between the inequalities defining P m and those defining C(Z m ). This is often helpful when working with particular families of semigroups, since the homogeneity of the inequalities defining C(Z m ) makes them easier to work with. It is easy to see that not every vector (z 1 , . . . , z m−1 ) ∈ Z m−1 ≥0 is the Kunz coordinate vector of a numerical semigroup containing m. The following set of linear inequalities that determine the image of KV m in Z m−1 ≥0 can be found in [16,19].
and the strict Kunz polyhedron P m ⊂ R m−1 is given by P m = P m ∩ R m−1 ≥1 .
Remark 4.3. The terminology used for P m and P m varies across the literature. It has often been called the "Kunz polytope," although this conflicts with the conventions of polyhedral geometry, where "polytopes" are bounded polyhedra. This was corrected in [6], wherein P m and P m were called the "relaxed Kunz polyhedron" and "Kunz polyhedron" respectively. We believe the names in Definition 4.2 are the most appropriate, as (i) nonnegativity and positivity inequalities are frequently viewed as implicit or extra in the lattice point and integer optimization literature, and (ii) we will see below that the relationship between numerical semigroups and the faces of P m is more direct than the connection to faces of P m , as P m has several additional faces that come from the inequalities z i ≥ 1.  Notation 4.5. Given a numerical semigroup S and a face F ⊂ P m , we write S ∈ F and say "S is in the face F " to mean the Kunz coordinates of S lie in F , that is, In what follows, we show that the Kunz polyhedron P m is a translation of the group cone C(Z m ), inducing a correspondence between their faces.  For the second claim, note that if a face F ⊂ C(Z m ) has nontrivial Kunz subgroup H ⊂ Z m , then some coordinate of the corresponding face F of P m must be negative throughout F . As such, any face F containing semigroups has trivial Kunz subgroup, and so the result now follows from [6, Theorem 3.10].
The following gives a method to identify semigroups in the group cone C(Z m ) directly. Proof. This follows immediately upon checking the appropriate facet equations. The action of Aut(G) on C(G) given by coordinate permutation induces an action on the face lattice of C(G), and consequently on the face lattice of P m . The following result implies the property "has a numerical semigroup" is preserved by this action. Proof. Fix g, h ∈ Z ≥1 with gh ≡ 1 mod m. Suppose S is a numerical semigroup in F with Apéry set Ap(S; m) = {0, a 1 , . . . , a m−1 }. By Corollary 4.8, a = (a 1 , . . . , a m−1 ) lies in the corresponding face of C(Z m ). Acting on a by g ∈ Aut(Z m ) yields the point a = (a h , a 2h , . . . , a (m−1)h ), and scaling a by g yields (ga h , ga 2h , . . . , ga (m−1)h ) in the same face as a , where all of the subscripts are taken modulo m. Moreover, we see  Proof. This follows from Theorem 4.4, as the vector difference of KV n (S) and the vertex of P n must have all positive entries, so adding a multiple of this difference to KV n (S) yields an integer point with all positive entries.
Remark 4.12. There are two reasons why a face F ⊂ P m may fail to contain any points corresponding to numerical semigroups. The first is that some coordinates are negative throughout F ; such faces are fully characterized by Theorem 3.4. The second is that F contains positive rational points but no integer points, a property that is also reflected in the corresponding Kunz poset; Example 4.13 demonstrates this. Example 4.13. Let m = 6. The Kunz polyhedron P 6 is obtained by translating C(Z 6 ) by the vector v = (− 1 6 , − 2 6 , − 3 6 , − 4 6 , − 5 6 ), so Example 3.9 implies P 6 has 11 extremal rays. Of these rays, only the 2 with vector directions (1,2,3,4,5) and (5, 4, 3, 2, 1) contain integer points. These correspond to the rays of C(Z 6 ) whose Kunz posets are total orderings; see Figure 2.
The 3 rays of P 6 that correspond to rays of C(Z 6 ) that are not listed in Figure 2 each have a coordinate that is always negative. For example, in the face v + rR ≥0 with vector direction r = (1, 0, 1, 0, 1), every vector has second coordinate − 2 6 . The remaining 6 rays of P 6 contain points with all positive entries but still do not contain integer points. This can be verified numerically from the coordinates of the vector direction of each ray, but can also be verified by proving that the corresponding posets in Figure 2 cannot occur as the Apéry poset of a numerical semigroup.
Since we have a 2 = 2a 1 = a 3 + a 5 , either a 3 < a 1 < a 5 or a 5 < a 1 < a 3 . In either case, it is impossible to have a 4 = 2a 5 = a 1 + a 3 . This gives a contradiction. Corollary 4.10 implies the remaining 2 rays of P 6 also contain no integer points.

Faces of the oversemigroup cone
In this section, we give a definition of the oversemigroup cone O n ⊂ R n and explain the correspondence between integer points of O n and numerical semigroups containing n. We then give a connection between O n and the group cone C(Z n ). Combining this with the results of the previous section gives a correspondence between O n and the Kunz polyhedron P n .
Definition 5.1. For n ≥ 2, the oversemigroup cone O n ⊂ R n is the set of points (y 1 , . . . , y n ) satisfying y i + y j ≤ y i+j , for all 1 ≤ i ≤ j ≤ n − 1 with i + j < n, and y i + y j ≤ y i+j−n + y n , for all 1 ≤ i ≤ j ≤ n − 1 with i + j > n.
We also set notation for a particular face of the cone O n , defining O n ⊂ R n by Note that we can view O n as a cone in R n−1 . Unlike the Kunz polyhedron, any numerical semigroup S with n ∈ S corresponds to infinitely many points in O n , one for each q ∈ S with gcd(n, q) = 1. Some of this redundancy is handled by the following proposition. Proposition 5.3. Fix n ≥ 2. Every y ∈ O n can be written uniquely in the form y = y + y 1 (1, 2, . . . , n) where y ∈ O n and y 1 = 0. Moreover, if y corresponds to a numerical semigroup S, then y also corresponds to S, and y n ∈ Ap(S; n) if and only if y 1 = 0.
This proves the first claim. For the second claim, since gcd(y n , n) = gcd(y n − y 1 n, n) = gcd(y n , n) = 1 we know y must correspond to some numerical semigroup S under Proposition 5.2. Moreover, for each i = 1, . . . , n, we have iy n − ny i = i(y n − ny 1 ) − n(y i − iy 1 ) = iy n − ny i so Ap(S ; n) = Ap(S; n) and thus S = S . The final claim follows from Proposition 5.2, since y n − y 1 n ∈ Ap(S; n) implies that y n ∈ Ap(S; n) if and only if y 1 = 0. By Proposition 5.3, (1, 2, . . . , n) ∈ O n is the only ray with positive first coordinate. Every face of O n that is not contained in O n is simply the Minkowski sum of some face of O n with the ray (1, 2, . . . , n). This implies that in order to characterize the face lattice of O n it suffices to characterize the face lattice of O n . In the theorem below, we choose to think of O n as a cone in R n−1 .
Theorem 5.4. For each n ≥ 2, the linear map (y 2 , . . . , y n ) → (x 1 , . . . , x n−1 ) given by . , x n−1 = n−1 n y n − y n−1 maps O n bijectively onto C(Z n ). Additionally, if a numerical semigroup S corresponds to a point y in the relative interior of some face F ⊂ O n , then applying the action of the automorphism σ of Z n with σ(1) = y n to each element of the ground set of A(S; n) yields the Kunz-balanced poset of the face of C(Z n ) corresponding to F .
Proof. For clarity of notation we set y 1 = 0 for the rest of the proof. We prove the first part of the statement by comparing the defining hyperplanes for O n and for C(Z n ). For each inequality x i + x j ≥ x i+j of C(Z n ) with 1 ≤ i ≤ j ≤ n − 1, there are 2 cases: n y n − y i+j , which simplifies to y i + y j ≤ y i+j ; and (ii) if i + j > n, then i+j n y n − y i − y j = x i + x j ≥ x i+j = x i+j−n = i+j−n n y n − y i+j−n , which simplifies to y i + y j ≤ y i+j−n + y n .
Since C(Z n ) and O n are both full dimensional cones in R n−1 , this proves the first claim.
Much of the structure highlighted in Example 5.6 would not be readily clear without the explicit bijection in Theorem 5.4, as the Apéry set construction in Proposition 5.2 breaks down for points whose last coordinate is not coprime to n.

Leading coefficients of Ehrhart quasipolynomials
Recall that N m (g) equals the number of numerical semigroups with multiplicity m and genus g, and o(n, q) equals the number of numerical semigroups containing two relatively prime integers n and q. These functions coincide with quasipolynomials p m (g) (for g 0) and H n (q) by Theorems 1.3 and 1.4, respectively. The main result of this section is Theorem 6.1, which expresses the leading coefficients in the quasipolynomials p m (g) and H m (q) in terms of an arbitrary triangulation of the group cone C(Z m ). Finding an explicit triangulation is still open (Problem 6.3), and will likely require first characterizing the extremal rays of C(Z m ).
Let γ(m) denote the leading coefficient of p m (g). In what follows, given a subset P of Euclidean space whose affine linear span has dimension d, the relative volume of P , denoted vol(P ), is the d-dimensional Euclidean volume of P normalized with respect to the sublattice of the affine span of P .
For each m ≥ 2, [3, Theorem 4] and Proposition 5.2 imply the leading cofficients of p m (g) and H m (q) are given by respectively, where vol(−) denotes relative volume and · 1 denotes the 1 -norm. As stated previously, both (6.1) and (6.2) use Ehrhart's theorem [11]. For a different perspective on the function o(n, q) that exploits a bijection between oversemigroups of n, q and integer points in a different polyhedron, see [9].
Fix m ∈ Z ≥2 and a triangulation T of C(Z m ). For each simplicial cone T ∈ T , write The final observation is that for each T ∈ T , we have Note that, as a consequence of Theorem 3.4, every nonzero vector on a ray of C(Z m ) has nonzero first coordinate. This proves the following. As such, the leading coefficient of p 4 (g) is and the leading coefficient of the H 4 (q) is 1 192 (48V (T 1 ) + 16V (T 2 )) = 1 72 , which agree with the computations in [3] and [13], respectively. Problem 6.3. For each finite abelian group G, find a triangulation of C(G).

Computing the Apéry set of a numerical semigroup
In the process of writing this paper, an improved implementation of the Apéry set function was written for the GAP package numericalsgps [10]. The original implementation, based on the circle-of-lights algorithm proposed by Wilf [23], enumerates each positive integer, beginning at the multiplicity m, and stops when all m − 1 positive elements a 1 , . . . , a m−1 of the Apéry set have been obtained. The key idea is that one only needs to enumerate within equivalence classes modulo m for which a i has not yet been found, and checking if a given integer n ≡ i mod m lies in Ap(S, m) can be done by checking if n = a j + a i−j for some previously obtained elements a j , a j−i ∈ Ap(S, m). In this sense, the circle-of-lights algorithm is in fact computing the Apéry poset of S.
Algorithm 7.1, in contrast, walks up the Kunz poset instead of the Apéry poset. The Apéry set elements are obtained starting with the bottom of the Kunz poset and using Proposition 3.10(b) to check potential cover relations above each new element. New elements are traversed in the order in which they are encountered, using a queue (first-in-first-out) data structure, and only the smallest element in each equivalence class modulo m is retained (denoted as a(0), . . . , a(m − 1) in the algorithm).
The resulting implementation is particularly effective for numerical semigroups with (i) "small" embedding dimension, or (ii) some generators that are much larger than the multiplicity (i.e., those represented by "large points" in the Kunz polyhedron), as such semigroups can have long sequences of successive integers outside of Ap(S, m). We ran calculations for large numbers of randomly chosen numerical semigroups and include a representative sample comparing the runtimes of Algorithm 7.1 and the original implementation in Table 1.
coefficient of L G (g) with G = Z m equals the leading coefficient of p m (g). We also include in the latter table the initial value N of g for which p m (g) coincides with N m (g) for all g ≥ N .
On the other hand, H m (q) is a quasipolynomial of degree m − 1 whose constant leading coefficient is described geometrically in Theorem 6.1. If gcd(m, q) = 1, H m (q) coincides with o(m, q), the number of oversemigroups of m, q . Table 4 gives the top two coefficients and period of H m (g). It is important to note that the period of H m (q) (on all values) may be strictly larger than that of o(m, q) (considered only when gcd(m, q) = 1).
Examining the data in Tables 2 and 3 suggests the following.
Conjecture 7.3. The following hold.