Antichain Simplices

Benjamin Braun and Brian Davis Department of Mathematics University of Kentucky Lexington, KY 40506-0027 USA

                benjamin.braun@uky.edu
                  brian.davis@uky.edu

Abstract

Associated with each lattice simplex Δ is a poset encoding the additive structure of lattice points in the fundamental parallelepiped for Δ. When this poset is an antichain, we say Δ is antichain. For each partition λ of n, we define a lattice simplex Δ_{λ} having one unimodular facet, and we investigate their associated posets. We give a number-theoretic characterization of the relations in these posets, as well as a simplified characterization in the case where each part of λ is relatively prime to n − 1. We use these characterizations to experimentally study Δ_{λ} for all partitions of n with n ≤ 73. Further, we experimentally study the prevalence of the antichain property among simplices with a restricted type of Hermite normal form, suggesting that the antichain property is common among simplices with this restriction. Finally, we explain how this work relates to Poincaré series for the semigroup algebra associated with Δ, and we prove that this series is rational when Δ is antichain.

1 Introduction

Given a lattice simplex Δ, the structure of the lattice points in the fundamental parallelepiped of the cone over Δ reflects a wealth of arithmetic and combinatorial properties of Δ. In this work, we study a partial order on these lattice points that encodes the additive relations among these points. Thus, our main results are primarily arithmetic in nature, and will hopefully be of interest to those working in areas where fundamental parallelepipeds play a role, e.g., Ehrhart theory, partition identities, coding theory, optimization, etc. Our motivation for this investigation comes from questions regarding rationality of Poincaré series for infinite graded resolutions of graded algebras, where it is of particular interest when the partial order associated with Δ has no relations. Thus, after developing our main results using number-theoretic techniques, we explain their algebraic implications.

More precisely, in Section 2 we define the fundamental parallelepiped poset P (Δ) associated with Δ, where we say Δ is antichain if P (Δ) ∖ {0} has no relations. For each partition λ of n, we define a lattice simplex Δ_{λ} having one unimodular facet, and we investigate the posets for these simplices in depth. In Theorem 15 we give a number-theoretic characterization of the relations in P (Δ_{λ}), and in Corollary 16 we give a simplified characterization in the case where each part of λ is relatively prime to n − 1. In Section 3 we use these characterizations to generate empirical data, experimentally studying those Δ_{λ} for all partitions of n with n ≤ 73. These experiments reveal that a substantial fraction of those λ satisfying the relatively prime condition appear to have Δ_{λ} that are antichain. Further, we experimentally study the prevalence of the antichain property among simplices with a restricted type of Hermite normal form, suggesting that the antichain property is common among simplices with this restriction. Finally, in Section 4 we explain the algebraic implications of our work to Poincaré series of semigroup algebras associated with Δ. Specifically, we prove that if Δ is antichain, then the associated Poincaré series is rational.

2 Fundamental parallelepiped posets and their structure

2.1 Lattice simplices and associated posets

Details regarding polytopes, cones, Hilbert bases, etc. as discussed in the following can be found in standard references [3, 13]. For a collection V = {v_{0}, …, v_{m}} of points in ℝ^{d}, we let conv(V) denote the convex hull of V. In the case that m = d and the set V^{∘} := {(v_{1} − v_{0}), …, (v_{d} − v_{0})} is a vector space basis of ℝ^{d}, then we call Δ := conv(V) a d-simplex. We call the v_{i}’s the vertices of Δ, and if each v_{i} is an integer point, i.e., lies in ℤ^{d}, we call Δ a lattice simplex. We define the conical hull of V to be the set

cone(V) := {∑_{i=0}^{m} γ_{i}v_{i} such that 0 ≤ γ_{i}} ⊂ ℝ^{d}.

Notice that the conical hull is unbounded, as in particular it contains the rays ℝ_{≥0} · v_{i} for 0 ≤ i ≤ m.

We are particularly interested in conical hulls of the following kind. Let V = {v_{0}, …, v_{d}} with Δ a lattice d-simplex. Then the cone over Δ is

cone{(1, v_{0}), …, (1, v_{d})} ⊂ ℝ^{d+1} ,

and is denoted cone(Δ). We next recall the fundamental parallelepiped, a distinguished subset of cone(Δ).

Definition 1. For a lattice d-simplex Δ with vertices v_{0} through v_{d}, the fundamental parallelepiped Π_{Δ} is the set

Π_{Δ} := {∑_{i=0}^{d} γ_{i}(1, v_{i}) such that 0 ≤ γ_{i} < 1} ⊂ cone(Δ).

Interest in the fundamental parallelepiped Π_{Δ} arises mainly from the following wellknown fact: every lattice point in cone(Δ) can be written uniquely as a non-negative integer combination of the (1, v_{i})’s and a lattice point in Π_{Δ}. To see this, note that because any element z of cone(Δ) ∩ ℤ^{d+1} lies in cone(Δ), it is a non-negative linear combination of the (1, v_{i})’s, i.e., there exist non-negative real coefficients g_{i} such that

z = ∑_{i=0}^{d} g_{i}(1, v_{i}) = (∑_{i=0}^{d} ⌊g_{i}⌋ (1, v_{i})) + (∑_{i=0}^{d}{g_{i}}(1, v_{i}))

where {g_{i}} means the fractional part of g_{i}. By setting γ_{i} equal to {g_{i}}, we see that any point z may be written as a non-negative integral combination of the (1, v_{i})’s and an integer point in Π_{Δ} ∩ ℤ^{d+1}. In particular, it is well-known that the set cone(Δ) ∩ ℤ^{d+1} has a unique finite minimal additive generating set.

Definition 2. The unique minimal additive generating set of cone(Δ) ∩ ℤ^{d+1} is called the Hilbert basis H of cone(Δ). It consists of the (1, v_{i})’s and some lattice points h_{1} through h_{m} in Π_{Δ} such that

cone(Δ) ∩ ℤ^{d+1} = {(∑_{i=0}^{d} r_{i}(1, v_{i})) + (∑_{j=1}^{m} s_{i} h_{i}) such that r_{i} , s_{j} ∈ ℤ_{≥0}} .

The Hilbert basis consists of the cone generators (1, v_{i}) together with the additively minimal elements h_{j} of Π_{Δ} ∩ ℤ^{d+1}. If the matrix whose columns are given by (1, v_{i}) has determinant ±v, we say that the simplex Δ has normalized volume v. Since v is precisely the index of the sub-lattice generated by (1, v_{0}) through (1, v_{d}), we see that the normalized volume is equal to the number of lattice points in Π_{Δ}. If the normalized volume of Δ is equal to one, then we call Δ a unimodular simplex.

The set of lattice points ℤ^{d+1} ∩ Π_{Δ} can be equipped with the following partial order, inherited from a well-known partial order on the lattice points in ℤ^{d+1} ∩ cone(Δ).

Definition 3. The set ℤ^{d+1} ∩ Π_{Δ} is partially ordered by letting σ ≺ μ if and only if μ − σ is an element of ℤ^{d+1} ∩ Π_{Δ}. We call this poset the fundamental parallelepiped poset P(Δ).

Observe that the zero element of ℤ^{d+1} ∩ Π_{Δ} is below every other element of P(Δ), and that the minimal elements of P(Δ) ∖ {0} are precisely the elements h_{1}, …, h_{m} of the Hilbert basis of cone(Δ). Our interest is in the case where the Hilbert basis contains all the elements of P(Δ) ∖ {0}, leading to the following definition.

Definition 4. If Δ is a simplex such that P (Δ)∖{0} is an antichain, we call Δ an antichain simplex and say Δ is antichain.

Example 5. Recall that an empty simplex is one whose only lattice points are its vertices. It is known that any 2-dimensional empty simplex is unimodularly equivalent to the convex hull of the origin and the standard basis vectors; this is easily verified to be antichain. Let Δ be a 3-dimensional empty simplex, for which a classification of such simplices was given by G. K. White [20]. Since Δ is empty, no lattice points of Π_{Δ} have 0-th coordinate equal to one, and thus the only possible 0-th coordinates of lattice points in Π_{Δ} are 2 or 3. However, sums of pairs of such lattice points have 0-th coordinate equal to 4, 5, or 6, and hence P(Δ) has no relations. Thus, every 3-dimensional empty simplex is antichain.

When attempting to determine whether or not Δ is antichain, the first problem encountered is to enumerate the elements of Π_{Δ} ∩ ℤ^{d+1}. As an initial attempt in this direction, let the matrix A have columns given by {(1, v_{i})}_{0≤i≤d}, where the v_{i}’s are the vertices of Δ. Recall that the normalized volume v, the number of elements of Π_{Δ}, may be computed by v = | det A|. Recall also that the set Π_{Δ} is the image of [0, 1)^{d+1} under the linear transformation A, so that the preimage of a lattice point of Π_{Δ} must be a rational point of [0, 1)^{d+1} with denominator v. We may therefore compute the set of points in Π_{Δ} ∩ ℤ^{d+1} by considering each element of the form

\{A · (\frac{b_{0}}{v} , ⋯, \frac{b_{d}}{v} )^{T} such that 0 ≤ b_{i} < v\} ,

throwing out the ones which are not integer points. Unfortunately, this test set grows as v^{d+1}, and there is no easy way to describe the lattice points among them.

The software Normaliz [7] gives a more efficient implementation based on the fact that (possibly after a lattice translation) the matrix A has a representation A = UH where U is a unimodular matrix and H is in Hermite normal form. Bruns et al. [8] show that, for {c_{i,i}}_{0≤i≤d} given by the diagonal entries of the matrix H, lattice points in

[0, c_{0,0}) × ⋯ × [0, c_{d,d})

are representatives of the quotient classes (in ℤ^{d+1} modulo the (1, v_{i})’s) of the elements of Π_{Δ} ∩ ℤ^{d+1}. It is then sufficient to consider the image under A of the elements (A^{−1} · x) mod ℤ^{d+1} for x ∈ [0, c_{0,0}) × ⋯ × [0, c_{d,d}). This modular arithmetic is implemented in a computer easily enough, but introduces number theory to any analysis of the poset P (Δ).

2.2 Lattice simplices with a unimodular facet and their posets

In this work, we study a restricted class of simplices in order to avoid both of the methods described above for determining Π_{Δ} ∩ ℤ^{d+1}.

Definition 6. We say that a lattice d-simplex has a unimodular facet if there exists a permutation π in 𝔖_{d+1} such that conv({v_{π_{1}}, …, v_{π_{d}}}) is a unimodular lattice (d−1)-simplex.

One way to represent these simplices is through their Hermite normal form, which is of the form

[ \begin{matrix}0 & 1 & 0 & ⋯ & 0 & a_{1} \\ 0 & 0 & 1 & ⋯ & 0 & a_{2} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & 0 & ⋯ & 1 & a_{d−1} \\ 0 & 0 & 0 & ⋯ & 0 & n \end{matrix} ]

where for i = 1, …, d − 1 we have 0 ≤ a_{i} < n. Simplices with Hermite normal form having only a single non-trivial column, such as the ones given above, were previously considered by Hibi, Higashitani, and Li [11, Section 3] in the context of Ehrhart theory.

Alternatively, if Δ has a unimodular facet, then we may define a lattice preserving transformation taking Δ to conv(e_{1}, …, e_{d}, z) where the e_{i} are the standard basis vectors of ℝ^{d} and z is a lattice point in ℤ^{d}. This is the general approach we will take in this paper, and we will focus our analysis on those Δ having z with strictly positive entries. Our goal in this section is to find a description of the relations in P (Δ) in terms of the coordinates of the point z for such simplices.

Note that in the following definition, we can assume that λ is a partition of n, as permuting the entries of λ corresponds to a unimodular transformation of the Δ_{λ} that is defined. This provides us with a unique lattice simplex for each integer partition, which is our motivation for this restriction. Definition 7. Let λ = (λ_{1}, …, λ_{d}) be a lattice point in ℕ^{d} such that ∑_{i=1}^{d} λ_{i} = n. We define Δ_{λ} := conv(e_{1}, …, e_{d}, λ) ⊂ ℝ^{d} and use the shortened notation Π_{λ} := Π_{Δ_{λ}} and P (λ) := P (Δ_{λ}). Remark 8. The simplices Δ_{λ} are defined in a similar manner to the simplices Δ_{(1,q)} that have recently been studied by multiple authors [4, 5, 6, 9, 14, 16, 17]. However, these are not the same families of simplices. Specifically, the matrix giving the Hermite normal form of Δ_{λ} (after translating Δ_{λ} by −e_{1}) is

[ \begin{matrix}0 & 1 & 0 & ⋯ & 0 & λ_{2} \\ 0 & 0 & 1 & ⋯ & 0 & λ_{3} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & 0 & ⋯ & 1 & λ_{d} \\ 0 & 0 & 0 & ⋯ & 0 & −1 + ∑_{i} λ_{i} \end{matrix} ] .

Note that ∑_{i=2}^{d} λ_{i} ≤ −1 + ∑_{i=1}^{d} λ_{i}.

Setting Q = 1 + ∑_{i} q_{i}, the Hermite normal form for Δ_{(1,q)} is

[ \begin{matrix}0 & 1 & 0 & ⋯ & 0 & Q − q_{2} \\ 0 & 0 & 1 & ⋯ & 0 & Q − q_{3} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & 0 & ⋯ & 1 & Q − q_{d} \\ 0 & 0 & 0 & ⋯ & 0 & Q \end{matrix} ] .

Note that for d ≥ 3, we have ∑_{i=2}^{d} Q − q_{i} > Q. Thus, these are distinct classes of simplices.

The following is a straightforward determinant calculation that also follows from the Hermite normal form given in Remark 8.

Proposition 9. The number of lattice points in Π_{λ}, which is equal to the normalized volume of Δ_{λ}, is ∑_{i=1}^{d} λ_{i} − 1 = n − 1.

We can now describe the integer points in Π_{λ} using only the entries of λ.

Proposition 10. For each integer b with 0 ≤ b < n − 1, there is a unique lattice point p(b) in Π_{λ} given by

p(b) = ((∑_{i=1}^{d} ⌈\frac{bλ_{i}}{n − 1}⌉) − b , ⌈\frac{bλ_{1}}{n − 1}⌉ , …, ⌈\frac{bλ_{d}}{n − 1}⌉) .  (1)

Every integer point in Π_{λ} arises in this manner, and thus we identify the integer b with the lattice point p(b).

Proof. For an element ∑_{i=1}^{d} γ_{i}(1, e_{i}) + γ_{d+1}(1, λ) ∈ Π_{λ} ∩ ℤ^{d+1}, we have

((∑_{i=1}^{d+1} γ_{i}) , (γ_{1} + γ_{d+1}λ_{1}) , …, (γ_{d} + γ_{d+1}λ_{d})) ∈ ℤ^{d+1} .

Because of the condition that each γ_{i} is strictly less than one, for each i we have

γ_{i} = ⌈γ_{d+1}λ_{i}⌉ − γ_{d+1}λ_{i} ,

thus

(γ_{d+1} + ∑_{i=1}^{d} (⌈γ_{d+1}λ_{i}⌉ − γ_{d+1}λ_{i}) , ⌈γ_{d+1}λ_{1}⌉, …, ⌈γ_{d+1}λ_{d}⌉)

= (γ_{d+1} (1 − ∑_{i=1}^{d} λ_{i}) + ∑_{i=1}^{d}⌈γ_{d+1}λ_{i}⌉, ⌈γ_{d+1}λ_{1}⌉, …, ⌈γ_{d+1}λ_{d}⌉) .

Observe that the first coordinate of this vector is an integer, hence

γ_{d+1} (1 − ∑_{i=1}^{d} λ_{i}) = γ_{d+1}(1 − n) ∈ ℤ .

It follows that γ_{d+1} is a rational number of the form b/(n − 1), and every lattice point arises in this manner and is of the form

((∑_{i=1}^{d} ⌈\frac{bλ_{i}}{n − 1}⌉) − b , ⌈\frac{bλ_{1}}{n − 1}⌉ , …, ⌈\frac{bλ_{d}}{n − 1}⌉) .

Since there are n − 1 lattice points in Π_{λ} by Proposition 9, there must be one unique lattice point for each 0 ≤ b < n − 1.

Using the notation from (1), for 0 ≤ b < n − 1 we have that the zeroth coordinate of p(b) is

p(b)_{0} := (∑_{i=1}^{d} ⌈\frac{bλ_{i}}{n − 1}⌉) − b .

Recall that we freely identify the integer b with the lattice point p(b). The following lemma provides a connection between the parameterization of the integer points in Π_{λ} and the order in P (λ).

Lemma 11. For i, j ∈ P (λ) with i ≠ j, we have i ≺ j if and only if i < j and p(i)+p(j−i) = p(j).

Proof. For the forward direction, if i ≺ j, then by Proposition 10 there exists a point p(ℓ) ∈ P (λ) such that p(i) + p(ℓ) = p(j). Note that ℓ > 0 since p(0) = 0. It follows that for all 1 ≤ t ≤ d, we have

⌈\frac{iλ_{t}}{n − 1}⌉ + ⌈\frac{ℓλ_{t}}{n − 1}⌉ = ⌈\frac{jλ_{t}}{n − 1}⌉ .

Given this, we have that p(i)_{1} + p(ℓ)_{1} = p(j)_{1} reduces to i + ℓ = j, forcing ℓ = j − i > 0, as desired.

For the reverse direction, if i < j and p(i) + p(j − i) = p(j), then we have i ≺ j by definition.

We now give two propositions demonstrating how Lemma 11 can be used in practice.

Proposition 12. If 0 ≠ i ≺ j in P (λ), then also j − i ≺ j in P (λ).

Proof. By Lemma 11, we have 0 ≠ i ≺ j if and only if 0 ≠ i < j and p(i) + p(j − i) = p(j) if and only if 0 ≠ j − i < j and p(i) + p(j − i) = p(j) if and only if 0 ≠ j − i ≺ j.

Proposition 13. Let λ = (n − 2, 2). Then P (n − 2, 2) is equal to the following poset on the elements \{1, 2, …, n − 2\}: The minimal elements of P (n − 2, 2) are \{1, 2, …, ⌊\frac{n−1}{2}⌋\} and the maximal elements are \{⌊\frac{n−1}{2}⌋ + 1, …, n − 2\}. The cover relations are that the maximal element ⌊\frac{n−1}{2}⌋ + j covers \{j, j + 1, …, ⌊\frac{n−1}{2}⌋\}.

5 6 7 8 1 2 3 4

Figure 1: The poset P (6, 2).

Proof. By Lemma 11, we see that i ≺ j if and only if i < j and the following hold:

⌈\frac{2i}{n − 1}⌉ + ⌈\frac{2(j − i)}{n − 1}⌉ = ⌈\frac{2j}{n − 1}⌉  (2)
⌈\frac{i(n − 2)}{n − 1}⌉ + ⌈\frac{(j − i)(n − 2)}{n − 1}⌉ = ⌈\frac{j(n − 2)}{n − 1}⌉  (3)

It is straightforward to verify that these equations hold for the values claimed in the proposition statement.

To show that no other pairs i < j lead to relations i ≺ j, suppose that 1 ≤ i < j ≤ ⌊\frac{n−1}{2}⌋. Then in (2), we obtain 1 + 1 = 1, which is false. Similarly, if ⌊\frac{n−1}{2}⌋ + 1 ≤ i < j ≤ n − 2, then in (2) we obtain 2 + 2 = 2, which is again false.

2.3 Characterizing the relations in P (λ)

While Lemma 11 is a reasonable first tool, as Propositions 12 and 13 illustrate, in general it is difficult to compute these relations directly. Thus, we need to create a more sophisticated mechanism through which to study P (λ). In this section, we establish in Theorem 15 a number-theoretic characterization of the relations in P (λ). Further, Corollary 16 provides a particularly simple characterization in the case where each part of λ is relatively prime to n − 1.

For 0 ≤ i < n − 1, define the non-negative integers r_{t,i} and 0 ≤ s_{t,i} < n − 1 by

iλ_{t} = r_{t,i}(n − 1) + s_{t,i} .  (4)

Lemma 14. We have i ≺ j in P (λ) if and only if i < j and for every t ∈ {1, …, d} we have

\frac{s_{t,i} + s_{t,j−i} − s_{t,j}}{n − 1} = ⌈\frac{s_{t,i}}{n − 1}⌉ + ⌈\frac{s_{t,j−i}}{n − 1}⌉ − ⌈\frac{s_{t,j}}{n − 1}⌉  (5)

Proof. After adding and subtracting (4) for the values i, j − i, and j, we obtain

r_{t,i} + r_{t,j−i} − r_{t,j} = \frac{−s_{t,i} − s_{t,j−i} + s_{t,j}}{n − 1} .  (6)

By dividing both sides of (4) by n − 1 and taking the ceiling of both sides, we see that

⌈\frac{ℓλ_{t}}{n − 1}⌉ = r_{t,ℓ} + ⌈\frac{s_{t,ℓ}}{n − 1}⌉ .  (7)

Adding (7) with itself for ℓ equal to i and j − i, then subtracting the equation with ℓ = j, and further applying (6), we obtain

⌈\frac{iλ_{t}}{n − 1}⌉ + ⌈\frac{(j − i)λ_{t}}{n − 1} ⌉ − ⌈\frac{jλ_{t}}{n − 1}⌉

= r_{t,i} + r_{t,j−i} − r_{t,j} + ⌈\frac{s_{t,i}}{n − 1}⌉ + ⌈\frac{s_{t,j−i}}{n − 1}⌉ − ⌈\frac{s_{t,j}}{n − 1}⌉
= \frac{−s_{t,i} − s_{t,j−i} + s_{t,j}}{n − 1} + ⌈\frac{s_{t,i}}{n − 1}⌉ + ⌈\frac{s_{t,j−i}}{n − 1}⌉ − ⌈\frac{s_{t,j}}{n − 1}⌉ .

Recall that i ≺ j in P (λ) if and only if p(i) + p(j − i) = p(j) if and only if for all t, we have that

⌈\frac{iλ_{t}}{n − 1}⌉ + ⌈\frac{(j − i)λ_{t}}{n − 1} ⌉ − ⌈\frac{jλ_{t}}{n − 1}⌉ = 0 ,

which by our computation above holds if and only if

\frac{s_{t,i} + s_{t,j−i} − s_{t,j}}{n − 1} = ⌈\frac{s_{t,i}}{n − 1}⌉ + ⌈\frac{s_{t,j−i}}{n − 1}⌉ − ⌈\frac{s_{t,j}}{n − 1}⌉ .

Theorem 15. Let λ be a partition of n. We have i ≺ j in P (λ) if and only if i < j and for each t ∈ {1, …, d}, one of the following holds:

1. s_{t,i} > s_{t,j} > 0,

2. s_{t,i} = 0 and s_{t,j} = s_{t,j−i}, or

3. s_{t,j} = s_{t,i} > 0 and s_{j−i} = 0. Proof. Forward implication: Suppose that i ≺ j in P (λ), and thus by Lemma 14 the s-values satisfy (5). We consider five cases:

• s_{t,i} = 0

• s_{t,i} > s_{t,j} = 0

• s_{t,i} = s_{t,j} > 0

• s_{t,i} > s_{t,j} > 0

• s_{t,j} > s_{t,i} > 0

Case 1: s_{t,i} = 0. If s_{t,i} = 0, then by (5) we have that

\frac{s_{t,j−i} − s_{t,j}}{n − 1} = ⌈\frac{s_{t,j−i}}{n − 1}⌉ − ⌈\frac{s_{t,j}}{n − 1}⌉ .

Thus \frac{s_{t,j−i} − s_{t,j}}{n − 1} is equal to an integer, and the fact that 0 ≤ s_{t,ℓ} < n − 1 implies that s_{t,j−i} − s_{t,j} = 0. Thus, we must have s_{t,j−i} = s_{t,j}. This establishes the second condition in the theorem statement.

Case 2: s_{t,i} > s_{t,j} = 0. In this case, (5) implies

\frac{s_{t,i} + s_{t,j−i}}{n − 1} = ⌈\frac{s_{t,i}}{n − 1}⌉ + ⌈\frac{s_{t,j−i}}{n − 1}⌉ .

Thus \frac{s_{t,i} + s_{t,j−i}}{n − 1} is an integer, and again since 0 ≤ s_{t,ℓ} < n − 1 and 0 < s_{t,i} < n − 1 we have that Thus, it is impossible to have s_{t,i} > s_{t,j} = 0.

Case 3: s_{t,i} = s_{t,j} > 0. In this case, (5) implies

\frac{s_{t,j−i}}{n − 1} = ⌈\frac{s_{t,j−i}}{n − 1}⌉ .

This forces s_{t,j−i} = 0, resulting in the third condition in the theorem statement.

Case 4: s_{t,i} > s_{t,j} > 0. In this case, (5) implies \frac{s_{t,i} + s_{t,j−i} − s_{t,j}}{n − 1} is equal to an integer, and the fact that every 0 ≤ s_{t,ℓ} < n−1 implies this integer is 0 or 1. Since n−1 > s_{t,i}−s_{t,j} > 0,

we must have \frac{s_{t,i} + s_{t,j−i} − s_{t,j}}{n − 1} = 1, and also the right-hand side of (5) is equal to 1. Thus, the first condition in the theorem statement is possible if i ≺ j.

Case 5: s_{t,j} > s_{t,i} > 0. Following the same logic as in the previous case, we must have

\frac{s_{t,i} + s_{t,j−i} − s_{t,j}}{n − 1} = 0 and thus s_{t,j−i} ≠ 0. But then the right-hand side of (5) is equal to 0 while the right-hand side is equal to 1, a contradiction.

Reverse implication: We verify that each of the three conditions listed in the theorem statement imply that (5) is valid.

First, by equation (6) we have \frac{s_{t,i} + s_{t,j−i} − s_{t,j}}{n − 1} ∈ ℤ. Combining n − 1 > s_{t,i} > s_{t,j} > 0

and the general bounds 0 ≤ s_{t,ℓ} < n − 1 for all ℓ, it follows that \frac{s_{t,i} + s_{t,j−i} − s_{t,j}}{n − 1} = 1. Thus,

s_{t,i} + s_{t,j−i} − s_{t,j} = n − 1. Since n − 1 > s_{t,i} − s_{t,j} > 0, we have s_{t,j−i} = n − 1 − (s_{t,i} − s_{t,j}) > 0, and thus

⌈\frac{s_{t,i}}{n − 1}⌉ + ⌈\frac{s_{t,j−i}}{n − 1}⌉ − ⌈\frac{s_{t,j−i}}{n − 1}⌉ = 1 .

We conclude that equation (5) holds.

Second, if s_{t,i} = 0 and s_{t,j} = s_{t,j−i}, then it is immediate that (5) holds.

Finally, if s_{t,j} = s_{t,i} > 0 and s_{j−i} = 0, then again it is immediate that (5) holds.

The following corollary illustrates a special case of Theorem 15 that we will focus on in the remainder of this paper.

Corollary 16. Let λ be a partition of n where each coordinate is coprime to n − 1, i.e., gcd(n − 1, λ_{t}) = 1. Then i ≺ j in P (λ) if and only if s_{t,i} > s_{t,j} > 0 for every t.

Proof. If gcd(n − 1, λ_{t}) = 1, then s_{t,i} ≠ 0 for all i. Thus, the second and third conditions in Theorem 15 do not apply.

Remark 17. Ehrhart-theoretic properties of simplices Δ_{λ} that satisfy the relatively prime condition in Corollary 16 have been previously studied by Hibi, Higashitani, and Li [11, Section 3].

We can use Corollary 16 to prove the following structural result regarding P (λ) in the case where each part of λ is coprime to n − 1.

Theorem 18. Let λ be a partition of n such that each λ_{t} is coprime to n − 1. Then P (λ) is self-dual.

Proof. We claim that ϕ : x → n−1−x for x ∈ [n−2] is an order-reversing poset isomorphism. It is clear that ϕ is a bijection. To see that ϕ is order-reversing, observe that by Corollary 16, we have that i ≺ j if and only if

s_{t,i} > s_{t,j} for all t .  (8)

Due to the fact that gcd(n − 1, λ_{t}) = 1, we have that s_{t,i} + s_{t,n−i−i} = n − 1 for all i and t, and thus (8) holds if and only if

s_{t,n−1−j} > s_{t,n−1−i} for all t .  (9)

This final condition holds if and only if n − 1 − j ≺ n − 1 − i, as desired.

2.4 Partitions with one distinct part

As an example of how these results can be applied, we consider the case where λ = (x, x, …, x) has v occurrences of x. In this case, it is immediate that x is coprime to n − 1 = vx − 1. The following theorem shows that P (λ) has a direct interpretation as a subposet of ℤ^{2}.

Theorem 19. For λ = (x, x, …, x) with v occurrences of x, we have that P (λ) is isomorphic to the poset with elements

{(r, p) : 0 ≤ r < x, 0 ≤ p < v} \ {(0, 0), (x − 1, v − 1)}

and order relation (r, p) ≺ (r′, p′) if both p > p′ and r′ > r.

Proof. For 1 ≤ i ≤ vx − 2, write

i = r_{i}v + p_{i}

where 0 ≤ r_{i} < x and 0 ≤ p_{i} < v, but we do not have simultaneously r_{i} = x − 1 and p_{i} = v − 1. Then

s_{i} = ix − ⌊\frac{ix}{xv − 1}⌋ (xv − 1)

= x(r_{i}v + p_{i}) − ⌊\frac{(r_{i}v + p_{i})x}{xv − 1}⌋ (xv − 1)

= xr_{i}v + xp_{i} − ⌊\frac{xr_{i}v − r_{i} + r_{i} + p_{i}x}{xv − 1}⌋ (xv − 1)

= xr_{i}v + xp_{i} − (r_{i} + ⌊\frac{r_{i} + p_{i}x}{xv − 1}⌋) (xv − 1)

= r_{i} + xp_{i} − ⌊\frac{r_{i} + p_{i}x}{xv − 1}⌋ (xv − 1)

= r_{i} + xp_{i},

where the final equality is a result of the bounds on r_{i} and p_{i} forcing the floor function to be zero. Thus, if i = r_{i}v + p_{i} and j = r_{j}v + p_{j}, then we have i ≺ j in P (λ) if and only if i < j and s_{i} > s_{j}, which happens if and only if the following two conditions simultaneously occur:

• p_{i} > p_{j} or p_{i} = p_{j} with r_{i} > r_{j}

• r_{j} > r_{i} or r_{j} = r_{i} with p_{j} > p_{i}

The only way for both conditions to simultaneously occur is to have p_{i} > p_{j} and r_{j} > r_{i}, and thus our proof is complete.

The following corollary follows immediately.

Corollary 20. The posets for λ = (x, x, …, x) where x occurs v times and λ′ = (v, v, …, v) where v occurs x times are isomorphic.

Corollary 20 is interesting because the two lattice simplices corresponding to λ and λ′ are in different dimensions. As an aside, we remark that the order on the lattice points within a rectangular grid given in Theorem 19 corresponds to the reflexive closure of the direct product of two strict total orders.

Example 21. Figures 2 and 3 show the Hasse diagrams of the posets P (4, 4, 4, 4, 4, 4) and P (6, 6, 6, 6), respectively, embedded in ℤ^{2} as described in Theorem 19. This illustrates the isomorphism obtained by switching the roles of x and v.

Figure 2: P (4, 4, 4, 4, 4, 4).

Figure 3: P (6, 6, 6, 6).

3 Experimental results

3.1 Exhaustive search of Δ_{λ} over all partitions of n

The results in Section 2, particularly Theorem 15 and Corollary 16, provide explicit tools for studying the relations in P (λ). Also, Corollary 16 and Theorem 18 demonstrate that the condition that the parts of λ be relatively prime to ∑_{i} λ_{i} − 1 imposes additional structure on P (λ), leading us to the following definition.

Definition 22. Given a partition λ of n, if gcd(λ_{i}, n − 1) = 1 for all i, we say λ satisfies the relatively prime condition. Let Part(n) denote the set of partitions of n, and let part(n) := |Part(n)|. Let Relprime(n) denote the set of partitions of n that satisfy the relatively prime condition, and set relprime(n) := |Relprime(n)|. Finally, let Rpac(n) denote the subset of Relprime(n) for which Δ_{λ} is an antichain simplex, and set rpac(n) := |Rpac(n)|.

Using SageMath [18] via CoCalc.com [12], we computed part(n), relprime(n), and rpac(n) for all 1 ≤ n ≤ 73; the results are given in Table 1. Figure 4 plots relprime(n)/part(n) for these values, and Figure 5 plots the ratio rpac(n)/relprime(n).

Figure 4: The ratio relprime(n)/part(n) for 1 ≤ n ≤ 73.

Figure 5: The ratio rpac(n)/relprime(n) for 1 ≤ n ≤ 73.

What follows are some observations regarding this experimental data.

1. Figure 5 shows that regardless of the total value of relprime(n), rpac(n)/relprime(n) is generally above 0.8 and as n grows it is clustering between 0.85 and 0.95. Thus, these experiments suggest that when λ satisfies the relatively prime condition, it is likely that Δ_{λ} is antichain.

2. Figure 4 shows that when n − 1 is not prime or the square of a prime, the ratio relprime(n)/part(n) appears to be small, and thus our consideration of the relatively prime condition does not broadly apply to partitions in this case. However, it is immediate that when n − 1 is prime, every partition of n except for 1 + (n − 1) satisfies the relatively prime condition, and thus rpac(n)/relprime(n) = rpac(n)/(part(n) − 1). Thus, it appears that one likely source of antichain simplices are those Δ_{λ} for which n − 1 is prime.

3. In Figure 5, the values of rpac(n)/relprime(n) for n ≥ 13 that lie on the upper hull of the data plot arise from n in {13, 19, 31, 43, 61, 67, 73}. These are all prime numbers, and an OEIS [19] search finds that these values arise in three known sequences, including the sequence \underline{A040047} of those primes p such that x^{3} = 6 has no solution mod p.

4. Again in Figure 5, the values of rpac(n)/relprime(n) for 70 ≥ n ≥ 20 that lie on the lower hull of the data plot arise from n in {20, 26, 32, 38, 44, 50, 62, 68}. These values are all of the form 6k + 2, though whether due to coincidence or mathematics it is not clear.

5. When n − 1 is a superabundant \underline{A004394} or highly composite \underline{A002182} number, one might expect to see particularly low numbers of relatively prime antichain simplices, which is supported by the data given in Table 1.

At this time, the authors do not have an explanation for why the ratio rpac(n)/relprime(n) appears to be clustering as n grows, leading to the following problems.

Problem 23. Determine if there is a limiting value to which the sequence rpac(n)/relprime(n) converges as n increases. Alternatively, determine if there are any connections between a liminf or limsup value for rpac(n)/relprime(n) and the values of n corresponding to subsequences achieving those values, as hinted at in the observations above.

3.2 Random sampling of simplices with one non-trivial column in Hermite normal form

It is worthwhile to compare the results for our restricted Δ_{λ} simplices to arbitrary simplices with Hermite normal form given by

[ \begin{matrix}0 & 1 & 0 & ⋯ & 0 & a_{1} \\ 0 & 0 & 1 & ⋯ & 0 & a_{2} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & 0 & ⋯ & 1 & a_{d−1} \\ 0 & 0 & 0 & ⋯ & 0 & n \end{matrix} ]

where for i = 1, …, d − 1 we have 0 ≤ a_{i} < n. We will call a simplex of this form a onecolumn (n,d) Hermite normal form simplex. Let OCH(n, d) denote the family of one-column (n, d) Hermite normal form simplices, and let

OCH^{+}(n, d) := {A ∈ OCH(n, d) : 1 ≤ a_{i} < n for all i} .

n rpac(n) relprime(n) part(n)
1 1  1  1
2 2  2  2
3 2  2  3
4 3  4  5
5 3  3  7
6 7  10  11
7 3  3  15
8 15  21  22
9 7  8  30
10 17  22  42
11 8  8  56
12 58  76  77
13 7  7  101
14 103  134  135
15 18  21  176
16 45  56  231
17 33  38  297
18 316  384  385
19 15  16  490
20 513  626  627
21 36  41  792
22 180  215  1002
23 78  89  1255
24 1317  1574  1575
25 31  34  1958
26 1169  1414  2436
27 148  170  3010
28 750  874  3718
29 143  162  4565
30 4779  5603  5604
31 26  28  6842
32 7050  8348  8349
33 392  448  10143
34 1675  1951  12310
35 478  539  14883
36 4850  5625  17977
37 115  126  21637
38 22109 26014  26015
39 816  918  31185
40 4410  5047  37338

n rpac(n) relprime(n) part(n)
41 433  481  44583
42 45819  53173  53174
43 104  112  63261
44 64731  75174  75175
45 1362  1522  89134
46 4192  4747  105558
47 2202  2468  124754
48 129242 147272  147273
49 365 399 173525
50 106948 123165  204226
51 1233  1362  239943
52 24641  27874  281589
53 3597  3986  329931
54 339300 386154  386155
55 623 679 451276
56 128590 145176  526823
57 3426  3781  614154
58 54230  60927  715220
59 8575  9496  831820
60 864231 966466  966467
61 302 324 1121505
62 1146930 1300155  1300156
63 13151  14458  1505499
64 55541  61850  1741630
65 16496  18200  2012558
66 522255 586074  2323520
67 1012 1091 2679689
68 2761384 3087734  3087735
69 20580 22503 3554345
70 234794 261034  4087968
71 3040 3287 4697205
72 4875893 5392782  5392783
73 2715  2931  6185689

Table 1: Experimental data Thus, OCH^{+}(n, d) contains those simplices in OCH(n, d) that are not obviously arising as lattice pyramids over simplices of smaller dimension.

There are (n − 1)^{d−1} simplices in OCH^{+}(n, d). For 67 random choices (uniform without replacement) of (n, d) ∈ {3, …, 20} × {3, …, 20}, we selected n^{3} random samples from OCH^{+}(n, d) and computed the resulting fraction f(n, d) of antichain simplices in this sample. We plotted the points (n/d, f(n, d)) in Figure 6. It is particularly interesting that when d is large relative to n, the percentage of antichain simplices among those sampled appears to be close to 1, leading to the following problem.

Problem 24. Fix n ≥ 2. Is it true that the fraction of antichain simplices in OCH^{+}(n, d)

goes to 1 as d → ∞? Alternatively, let ac_{n}(d) denote the fraction of ⋃_{j=3}^{d} OCH^{+}(n, j) that are

antichain simplices; what is the liminf of ac_{n}(d) as d → ∞?

Figure 6: For 67 random choices (uniform without replacement) of (n, d) ∈ {3, …, 20} × {3, …, 20}, we plot the point (n/d, f(n, d)) where f(n, d) is the fraction of antichain simplices among n^{3} random samples from OCH^{+}(n, d).

4 Algebraic implications

In this section, we discuss the algebraic implications of our analysis of the fundamental parallelepiped poset. Our main result is Theorem 32 showing that the Poincaré series for the semigroup algebra associated with an antichain simplex is rational. Unlike previous work of the authors [4] establishing rationality of Poincaré series for lattice simplex semigroup algebras, our proof technique in this work involves the bar resolution.

4.1 A review of resolutions and Poincaré series

For all background regarding graded resolutions of algebras, see the book by Peeva [15]. Recall that the semigroup (Λ, +) associated with a d-simplex Δ is the intersection Λ := cone(Δ) ∩ ℤ^{d+1} with + given by the usual coordinate-wise addition on ℤ^{d+1}. The semigroup algebra K[Λ] associated with a semigroup Λ ⊂ ℤ^{d+1} is the K-vector space with basis {e_{α}}_{α∈Λ} equipped with the product e_{α} · e_{β} = e_{α+β}. For K a field, a K-algebra R is called graded with respect to ℤ^{n} if it can be written as a direct sum

R = ⨁_{α∈ℤ^{n}} R_{α},

where for x ∈ R_{α} and y ∈ R_{β}, we have that x · y ∈ R_{α+β}. It is immediate that K[Λ] is a ℤ^{d+1}-graded K-algebra. It is common to “coarsen” the grading of K[Λ] by considering it to be a ℤ-graded algebra with grading given by the zeroth coordinate of its ℤ^{d+1}-grading.

In this context, the seemingly arbitrary definition of the cone over a simplex Δ is shown to be natural and helpful by the following observation. For a point x = (x_{0}, x_{1}, …, x_{d}) in ℝ^{d+1}, we define the height of x to be height(x) = x_{0}. Letting X_{n} denote the collection of points x ∈ ℝ^{d+1} with height equal to n, we have the set equality

X_{n} ∩ cone(Δ) = {(n, n · x) ∈ ℝ^{d+1} such that x ∈ Δ}.

Observe that the set ℤ^{d+1} ∩ X_{n} ∩ cone(Δ) is in bijection with the set of lattice points of nΔ (by dropping the zeroth coordinate). Thus, the coarsened grading of K[Λ] corresponds to the height function in the cone.

We need to consider complexes of vector spaces in order to define free resolutions of K-algebras. Given a collection of vector spaces {F_{i}}_{i∈ℤ_{≥0}}, together with linear maps ∂_{i} from F_{i} to F_{i−1}, we call the sequence

F : F_{0} ⟵^{∂_{1}} F_{1} ⟵^{∂_{2}} ⋯ ⟵^{∂_{i}} F_{i} ⟵^{∂_{i+i}} F_{i+1} ⟵^{∂_{i+2}} ⋯

a complex of vector spaces if the image of ∂_{i+1} is contained in the kernel of ∂_{i} for all i ≥ 1. The ith homology of the complex F is the quotient vector space H_{i}(F) := ker ∂_{i}/im∂_{i+1}. Let M be a finitely generated graded module over R, F_{i} be a free R-module and ∂_{i} be a graded R-module homomorphism such that the image of ∂_{i+1} is equal to the kernel of ∂_{i} for all i ≥ 1. Then the complex F is a free resolution of M over R if M ≅ F_{0}/im∂_{1}. Because it is graded, we may split the free resolution F into a direct sum of K-vector space complexes by writing each F_{i} as a direct sum ⨁_{α∈ℤ^{n}} F_{i,α}.

For (F, ∂) a complex of free R-modules, we can define a tensor complex (M ⊗ F, Id ⊗ ∂). If F is a graded free resolution of M, the Betti number β_{i,α}^{R}(M) of a graded R-module M is the vector space dimension of the ith homology of the graded component of K ⊗ F of degree α. This leads to our primary object of interest.

Definition 25. The Poincaré series P_{R}^{M}(z; t) is the ordinary generating function for the Betti numbers of the R-module M, i.e.,

P_{R}^{M}(z; t) = ∑_{α∈ℤ^{n}} ∑_{i≥0} β_{i,α}^{R}(M)z^{i}t^{α}.

In the case that R is a polynomial ring in n variables, the Hilbert Syzygy Theorem says that the Poincaré series P_{R}^{M}(z; t) is a polynomial for any finitely generated R-module M. However, when R is not a polynomial ring, the growth of the Betti numbers is not so simple.

4.2 Rational Poincaré series

We call a ℤ^{n}-graded algebra R connected if R_{0} ≅ K (as in the case of a semigroup ring K[Λ] associated with a lattice simplex Δ). By a slight abuse of notation, we write

𝔪 := ⨁_{α∈Λ\0} R_{α} and K ≅ R/𝔪

as R-modules. It has been shown [10] that if the Poincaré series for the ground field K as an R-module is rational for all R, then the Poincaré series is rational for any finitely generated module, leading to the question of Serre-Kaplansky:

Question 26. Is the Poincaré series of the ground field K over R rational for all K-algebras R?

This question was answered in the negative by Anick [1], and much subsequent work has focused on determining the properties of R that lead to rationality or irrationality. Our interest is in the rationality of the Poincaré series for K[Λ], which leads us to define a related algebra as follows.

Because K[Λ] is finitely generated (by its Hilbert basis H given in Definition 2) it has a presentation

0 → ker φ → K[V_{0}, …, V_{d}, x_{1}, …, x_{m}] ⟶^{φ} K[Λ] → 0,  (10)

where the map φ is defined by the image of variables: the image of V_{i} is the vector space basis element e_{(1,v_{i})} associated with the Hilbert basis element (1, v_{i}) in Λ, and the image of x_{i} is e_{h_{i}} where the h_{i} are the remaining elements of the Hilbert basis. This defines a surjective degree map deg(·) from the set of monomials of K[V_{1}, …, V_{d+1}, x_{1}, …, x_{m}] onto Λ by

deg (∏ V_{i}^{s_{i}} · ∏ x_{j}^{r_{j}}) = ∑ s_{i}(1, v_{i}) + ∑ r_{j}h_{j} .

Extending deg(·) K-linearly, we see that ker φ is the toric ideal I generated by all binomials

V^{u_{V}} x^{u_{x}} − V^{w_{V}} x^{w_{x}}

such that deg (V^{u_{V}} x^{u_{x}}) = deg(V^{w_{V}} x^{w_{x}}).

Definition 27. The Fundamental Parallelepiped Algebra FPA(Δ) associated with the simplex Δ may be constructed in two ways; firstly as the quotient

K[V_{0}, …, V_{d}, x_{1}, …, x_{m}] / ker φ + (V_{0}, …, V_{d}),

and secondly as the algebra with K-vector space basis

{e_{σ} such that σ ∈ ℤ^{d+1} ∩ Π_{Δ}}

and with multiplication given by

e_{σ} · e_{μ} = \begin{cases}e_{σ+μ}, & if σ + μ ∈ ℤ^{d+1} ∩ Π_{Δ}; \\ 0, & otherwise. \end{cases}

One inspiration for defining this algebra is the fact that, due to an argument presented earlier, every element of Λ may be written uniquely as a non-negative sum of points (1, v_{i}) and a single point in Π_{Δ}. Because the generators e_{(1,v_{i})} form a linear system of parameters for K[Λ], we have the following result [2, Prop. 3.3.5].

Theorem 28. For the ℤ-graded algebra K[Λ], we have the following equality:

P_{K[Λ]}^{K}(z; t) = ∏_{i} (1 + zt^{(1,v_{i})}) · P_{FPA(Δ)}^{K}(z; t)
= P_{K[V_{0},...,V_{d}]}^{K}(z; t) · P_{FPA(Δ)}^{K}(z; t).

4.3 Bar resolutions and antichain simplices

We will use the Bar resolution of K, with K as a module over a graded K-algebra, which is a standard construction. We will define the Bar resolution in the special case of a fundamental parallelepiped algebra. In the following, we use the bar symbol | to mean a tensor over K, and reserve the tensor symbol ⊗ to mean a tensor over the ring under consideration.

Definition 29. Consider the ℤ^{n}-graded K-algebra FPA(Δ). The Bar resolution 𝔹 of the module K over FPA(Δ) is a sum of graded components, where the component [𝔹_{i}]_{α} has a formal vector space basis given by all e_{δ_{0}} | e_{δ_{1}} | ⋯ | e_{δ_{i}} such that

• δ_{0} is in Π_{Δ},

• δ_{j} is in Π_{Δ} ∖ {0} for j ≥ 1, and

• ∑_{j=0}^{i} δ_{j} = α.

The differential map ∂_{i} acts by sending e_{δ_{0}} | ⋯ | e_{δ_{i}} ∈ 𝔹_{i} to the sum

∑_{j=0}^{i−1}(−1)^{j} e_{δ_{0}} | ⋯ | e_{δ_{j−1}} | e_{δ_{j}}e_{δ_{j+1}} | e_{δ_{j+2}} | ⋯ | e_{δ_{i}}

in 𝔹_{i−1}.

Example 30. For a unimodular simplex Δ, the FPA(Δ) is one-dimensional as a K-vector space, and has basis e_{0}. Consequently, [B_{i}]_{α} has empty basis (and dimension zero) unless α is equal to zero in Λ and i = 0. It follows that the complex B is given by

0 ← K ← 0 ← 0 ← ⋯

and that

β_{i,α} = \begin{cases}1 & if i = 0 and α = 0, \\ 0 & otherwise. \end{cases}

Thus, P_{FPA(Δ)}^{K}(z; t) = 1. The result is consistent with the fact that K[Λ] is a polynomial ring in the case that Δ is unimodular.

Example 31. Consider Δ_{λ} for λ = (2, 2, 3, 5). It can be checked using Theorem 15 that P (Δ_{λ}) \ {0} has 10 elements and no relations. Thus, for any pair of non-zero elements δ_{i}, δ_{j} ∈ Π_{Δ_{λ}}, we have δ_{i} + δ_{j} ∉ Π_{Δ_{λ}}, hence e_{δ_{i}}e_{δ_{j}} = 0. Thus, in the Bar resolution for FPA(Δ_{λ}), the differential maps are uniformly equal to zero.

Recall from the paragraph prior to Definition 25 that in order to compute the Betti number β_{i,α}, we must compute homology in the tensor complex B := K ⊗ 𝔹. Because we identify K with the quotient R_{0} ≅ R/𝔪 with basis e_{0}, we see that [B_{i}]_{α} is generated as a vector space by the collection {e_{0} ⊗ e_{δ_{0}} | e_{δ_{1}} | ⋯ | e_{δ_{i}}}. Observe that unless δ_{0} is equal to the point 0 ∈ Λ, the product e_{0} ⊗ δ_{0} is equal to zero, since for σ not equal to zero, e_{0} · e_{σ} is equal to zero in the module K, and hence

e_{0} ⊗ e_{σ} = e_{0} · e_{σ} ⊗ e_{0} = 0 ⊗ e_{0} = 0 .

Consequently, for i ≥ 1, [B_{i}]_{α} has a vector space basis in bijection with the collection of e_{δ_{1}} | ⋯ | e_{δ_{i}} such that each δ_{j} ∈ Π_{Δ} ∖ {0} and ∑_{j=1}^{i} δ_{j} = α. We further have that [B_{0}]_{α} is the trivial vector space unless α is zero in Λ, and that [B_{0}]_{0} is isomorphic to K.

We are now ready to prove our final result.

Theorem 32. For an antichain simplex Δ, we have

P_{FPA(Δ)}^{K}(z; t) = ( 1 − ∑_{\begin{matrix}σ∈P(Δ) \\ σ≠0 \end{matrix}} zt^{σ})^{−1} ,

and thus the Poincaré series is rational.

Proof. As illustrated by Example 31, for an antichain simplex the differential map is uniformly zero, since e_{δ_{j}} · e_{δ_{j+1}} equals zero for all j. Therefore, β_{i,α} is equal to the dimension of [B_{i}]_{α}. From the description of the basis of [B_{i}]_{α} in the Bar resolution, it follows that there is a recurrence

dim_{K}[B_{i}]_{α} = ∑_{\begin{matrix}σ∈P(Δ) \\ σ≠0 \end{matrix}} dim_{K}[B_{i}]_{α−σ} ,

from which our result follows.

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[19] N. J. A. Sloane et al., The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org, 2019.

[20] G. K. White, Lattice tetrahedra, Canad. J. Math. 16 (1964), 389–396.

2010 Mathematics Subject Classification: Primary 05E40, Secondary 52B20, 13D02, 11A05. Keywords: simplex, poset, resolution, Poincaré series.

(Concerned with sequences \underline{A002182}, \underline{A004394}, \underline{A040047}, \underline{A323256}, and \underline{A323257}.)

Received January 10 2019; revised version received May 16 2019; November 12 2019. Published in Journal of Integer Sequences, December 28 2019.

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