Discrete Moments of Half-Open Parallelepipeds

• Discrete moments of half-open parallelepipeds are finite sums of monomials evaluated over lattice points, serving as foundational units in Ehrhart theory and polyhedral enumeration.
• They enable explicit computation of Ehrhart quasi-polynomials through analytic generating functions, Barnes polynomials, and torus flow methodologies.
• Applications extend to lecture hall polytopes and bijective combinatorics, underscoring their significance in both theoretical and computational discrete geometry.

A half-open parallelepiped is a fundamental construct in contemporary combinatorial and discrete geometry, especially within the context of Ehrhart theory and polyhedral enumeration. Discrete moments of such parallelepipeds—finite sums of monomial evaluations over the lattice points they contain—have emerged as central atomic units for explicit formulas regarding Ehrhart quasi-polynomials, generating functions over polytopes, and related combinatorial invariants. This article surveys the rigorous definition, principal results, computational methodology, and applications of discrete moments of half-open parallelepipeds, including their key roles in the modern understanding of lattice-point enumeration and the polyhedral combinatorics of rational polytopes.

1. Definitions and General Framework

Let $\Lambda \subset \mathbb{R}^d$ denote a full-rank lattice, and let $\omega_1, \ldots, \omega_d \in \Lambda$ be linearly independent primitive edge vectors. The associated half-open parallelepiped, based at the origin, is

$$\Pi = \Bigl\{ \lambda_1 \omega_1 + \cdots + \lambda_d \omega_d \mid 0 \leq \lambda_i < 1 \Bigr\} \subset \mathbb{R}^d.$$

For any $v \in \mathbb{R}^d$, the translate $\Pi + v$ is considered. Discrete moments are formulated with respect to a complex direction $z \in \mathbb{C}^d$ and an auxiliary real parameter $t > 0$, defining the $k$-th discrete moment as

$$\mu_k(\Pi, v; t, z) = \sum_{p \in \mathbb{Z}^d - t v\, (\bmod\, \Pi)} \langle p, z \rangle^k,$$

where the sum is over a finite, torus-parametrized subset of $\mathbb{Z}^d$ determined by the lattice flow through the torus $\mathbb{R}^d/\Lambda$ (Robins, 18 Jan 2026). These moments naturally extend to sums of arbitrary polynomials over the integer points in rational polytopes and underpin explicit computations in lattice-point enumeration.

2. Discrete Moments in Ehrhart Theory

Robins (Robins, 18 Jan 2026) demonstrates that the complexity of computing the Ehrhart quasi-polynomial $L_P(t) = |tP \cap \mathbb{Z}^d|$ for a rational simple polytope $P$ can be localized to the evaluation of the corresponding discrete moments of half-open parallelepipeds at the vertices of $P$. Through a combination of Brion's theorem, Barnes polynomials, and an auxiliary complex parameter $z$, the formula for $L_P(t)$ assumes the form: $$L_P(t) = \frac{(-1)^d}{d!} \sum_{v \in \mathrm{Vert}(P)} \sum_{k=0}^d \binom{d}{k} B_k(t\langle v, z \rangle, a_v) \sum_{p \in \mathbb{Z}^d - t v\, (\bmod\, \Pi_v)} \langle p, z \rangle^{d-k},$$ where $B_k$ denotes the $k$-th Barnes polynomial and $\Pi_v$ is the half-open parallelepiped at vertex $v$ (Robins, 18 Jan 2026). The entire periodic and local-geometric structure of Ehrhart quasi-polynomials is thus encapsulated by these discrete moments.

Similarly, discrete moments encode sums of polynomial weights over integer points in dilates of $P$: $$\sum_{p \in t P \cap \mathbb{Z}^d} \langle p, z \rangle^m = (-1)^d m! \sum_{v \in \mathrm{Vert}(P)} \sum_{k=0}^{d+m} \frac{1}{k!(d+m-k)!} B_k(t\langle v, z \rangle, a_v) \sum_{q \in \mathbb{Z}^d - t v\, (\bmod\, \Pi_v)} \langle q, z \rangle^{d+m-k}.$$ Thus, discrete moments serve as the atomic units for all explicitly computable sums of polynomial expressions over lattice polytopes.

3. Torus Flows and Lattice Orbits

The connection between discrete moments and toral dynamics is manifest in the description of the summation index set. For fixed $\Pi_v$, let $\Lambda_v \subset \mathbb{Z}^d$ denote the sublattice generated by $\omega_1(v), \ldots, \omega_d(v)$ so that the real torus $T_v = \mathbb{R}^d / \Lambda_v$ models the geometric structure. The "lattice flow"

$$\mathrm{LatticeFlow}_v(t): x \mapsto x - t v \bmod \Lambda_v$$

generates for each $t$ the intersection of the flow's trajectory with the fundamental domain $\Pi_v$. Discrete moments $\mu_k(\Pi_v, v; t, z)$ sum $\langle p, z \rangle^k$ over points in this moving lattice section; all quasi-periodic phenomena in Ehrhart theory derive from this flow structure (Robins, 18 Jan 2026).

4. Explicit Computation and Complexity Considerations

Though the reduction of Ehrhart formulas to sums over parallelepipeds is algorithmically clarifying, the corresponding half-open parallelepipeds $\Pi_v$ may contain exponentially many integer points in $d$ relative to the bit-lengths of generators, precluding naive polynomial-time summation. Nonetheless, it is conjectured (Robins, Conj. 6.1) that for fixed $d$ and $k$, the discrete moments $\mu_k(\Pi)$ can be computed in polynomial time in the encoding size of $\Pi$ (Robins, 18 Jan 2026). Structural vanishing identities relate lower and higher moments, offering recursive avenues for efficient computation: $$0 = \sum_{k=0}^m \binom{m}{k}\sum_{v \in V} B_k(t\langle v, z \rangle, a_v) \sum_{p \in \Pi_v \cap \mathbb{Z}^d} \langle p, z \rangle^{m-k}$$ for $0 \leq m \leq d-1$ and integral polytopes. The use of generic $z$ values streamlines intermediate expressions, although final answers are independent of $z$.

5. The Case of Lecture Hall Parallelepipeds and Bijective Combinatorics

For the specific case of the $s$-lecture-hall simplex $P_s$ and its associated half-open parallelepiped $\mathrm{Par}_s$ (Liu et al., 2012), the integer points admit a complete combinatorial parametrization via s-descents. Explicitly, disregarding the origin, vertices $v_i$ are defined by $$v_1 = (s_1, ..., s_n), v_2 = (0, s_2, ..., s_n), ..., v_n = (0, ..., 0, s_n)$$. The half-open parallelepiped is

$$\mathrm{Par}_s = \left\{ c_1 v_1 + \cdots + c_n v_n: 0 \leq c_i < 1 \right\},$$

and $V_n = \prod_{i=1}^n \{0, ..., s_i-1\}$ parametrizes all its integer points. The s-descent set and the explicit bijection $h_s: V_n \to \mathrm{Par}_s \cap \mathbb{Z}^n$ yield closed-form expressions for all discrete moments: $$M_{k_1, ..., k_n}(s) = \sum_{x \in \mathrm{Par}_s \cap \mathbb{Z}^n} x_1^{k_1} \cdots x_n^{k_n} = \sum_{r \in V_n} \prod_{i=1}^n ( \operatorname{dess}_s^{<i}(r) s_i + r_i )^{k_i}$$ where $\operatorname{dess}_s^{<i}(r)$ counts certain descent statistics of $r$. This extends to explicit formulas for the lecture hall simplex's Ehrhart $\delta$-vector, with combinatorics fully encoding geometric data (Liu et al., 2012).

6. Low-Dimensional Examples

For $d = 1$, the parallelepiped is trivial, and the classic quasi-polynomial formulas for lattice point enumeration on intervals are recovered, with moments easily computed and linked to fractional parts of rational endpoints (Robins, 18 Jan 2026). For $d = 2$, e.g., lattice polygons, moments such as $\mu_1(\Pi_v; z)$ can be written directly in terms of the geometric centroid and the parity of the midpoint, with area terms explicitly isolatable: $$L_P(t) = \mathrm{Area}(P)t^2 + \left( \frac{1}{2} \sum_{v \in V} \langle v, z \rangle \left( \frac{1}{\langle w_1(v), z \rangle} + \frac{1}{\langle w_2(v), z \rangle} \right) (1 - \mathbf{1}_{\mathbb{Z}^2}(c_v)) \right)t + 1.$$ These computations exhibit the full range of combinatorial and analytic intricacy found in higher-dimensional situations.

7. Applications and Directions

Discrete moments of half-open parallelepipeds unify combinatorial enumeration, analytic generating function methods, and geometric decompositions—acting as the core arithmetic objects underlying closed-form Ehrhart and weighted lattice-point formulas (Robins, 18 Jan 2026). Through the explicit combinatorial models available in the lecture hall and similar settings (Liu et al., 2012), these moments also bridge polyhedral geometry and symmetric function theory, with ramifications for combinatorial representation theory and algebraic combinatorics. Efficient computation, algorithmic complexity, and the development of further vanishing identities remain active avenues for further research.

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References:

• “Ehrhart quasi-polynomials via Barnes polynomials and discrete moments of parallelepipeds” (Robins, 18 Jan 2026)
• “The Lecture Hall Parallelepiped” (Liu et al., 2012)