S-Lecture Hall Simplices Overview

• S-lecture hall simplices are lattice polytopes defined by inequalities based on positive integer sequences that encode lecture hall partitions.
• They establish a geometric framework linking combinatorial partition theory, Ehrhart theory, and inversion sequence statistics through explicit bijections.
• Their study reveals rich algebraic and triangulation properties, including real-rooted h*-polynomials and instances of negative Ehrhart coefficients.

S-Lecture Hall Simplices are a class of lattice polytopes defined by inequalities determined by a positive integer sequence, and serve as the geometric encoding of lecture hall partitions introduced by Eriksson and Bousquet-Mélou (Liu et al., 2012). The paper of these simplices unifies combinatorial partition theory, polyhedral geometry, and algebraic structures, with deep connections to Ehrhart theory, triangulations, permutation statistics, and the arithmetic of lattice point enumeration.

1. Definition, Structure, and Basic Properties

Given a sequence $s = (s_1, s_2, \ldots, s_n)$ of positive integers, the s-lecture hall polytope (simplex) $P_s$ is the convex hull in $\mathbb{R}^n$ of the following vertices:

• $(0, 0, \ldots, 0)$,
• $(0, \ldots, 0, s_n)$,
• $(0, \ldots, s_{n-1}, s_n)$,
• $\ldots$,
• $(s_1, s_2, \ldots, s_n)$.

Equivalently, $P_s$ is the set

$$P_s = \left\{ x \in \mathbb{R}^n: 0 \leq \frac{x_1}{s_1} \leq \frac{x_2}{s_2} \leq \cdots \leq \frac{x_n}{s_n} \leq 1 \right\}.$$

This simplex encodes s-lecture hall partitions—integer vectors $x$ satisfying the above inequalities—with its normalized volume given by $s_1 s_2 \cdots s_{n-1}$ in general, and $n!$ when $s$ is a permutation of $[n]$.

Associated to $P_s$ is the fundamental parallelepiped $\mathsf{Par}_s$, generated by its nonorigin vertices $v_1, \ldots, v_n$, where $v_i = (0, \ldots, 0, s_i, s_{i+1}, \ldots, s_n)$. The set

$$\mathsf{Par}_s = \left\{ \sum_{i=1}^n c_i v_i : 0 \leq c_i < 1 \right\}$$

provides the grading for lattice point enumeration, crucial for the Ehrhart-theoretic analysis.

2. Lattice Point Enumeration and Inversion Sequence Correspondence

A central technical contribution is the explicit description of the integer points in $\mathsf{Par}_s$ (Liu et al., 2012). Every integer point $x \in \mathsf{Par}_s \cap \mathbb{Z}^n$ can be written uniquely in "quotient-remainder" form: $x_i = k_i s_i + r_i,\quad 0 \leq r_i < s_i,$ where the remainder vector $r = (r_1, \ldots, r_n)$ lies in $V_n = [0, s_1-1] \times \cdots \times [0, s_n-1]$. The mapping $\operatorname{REM}_s$ sends each $x$ to its $r$ vector, establishing a bijection (suitably augmented) between integer points in $\mathsf{Par}_s$ and inversion sequences that encode lecture hall partitions. Thus, the geometry of the simplex is tightly connected with inversion sequences and the associated combinatorics.

The grading that determines the h*-vector is tracked by counting the number of certain statistics (ascents/descents) in the inversion sequence, as established by this bijection. This correspondence extends naturally to dual and reversed sequences, with unimodular equivalences constructed between $P_s$ and $P_u$ for the reversed $u = (s_n, \ldots, s_1)$.

3. Ehrhart Theory and h*-Vectors

The Ehrhart series of $P_s$ is written as

$$\sum_{t \geq 0} i(P_s, t) z^t = \frac{d_0 + d_1 z + \cdots + d_n z^n}{(1 - z)^{n+1}},$$

with Ehrhart coefficients $d_i$ forming the h*-vector. Lemma 2.3 (Liu et al., 2012) shows that

$d_{P_s, i} = \#\{\text{lattice points in %%%%31%%%% of grading %%%%32%%%%}\},$

which can be interpreted directly in terms of inversion sequences and their ascent/descent statistics, generalizing the Eulerian numbers in classical cases.

For specific $s$ (e.g., $(1, 2, ..., n)$), the h*-vector components $d_{P_s, i}$ match Eulerian numbers $A(n, i+1)$, demonstrating the geometric-combinatorial link between $P_s$ and permutation statistics. The paper proves that the h*-vector encodes ascent/descent distributions for arbitrary $s$, and in many cases, $\mathrm{h}^*$ is symmetric or unimodal due to underlying real-rootedness results.

4. s-Ascents, s-Descents, and Combinatorial Statistics

For a remainder vector $r = (r_1, ..., r_n)$, an index $i$ is defined as an s-descent if $r_i/s_i > r_{i+1}/s_{i+1}$, and as an s-ascent if $r_i/s_i < r_{i+1}/s_{i+1}$. These generalizations of classical ascent/descent statistics provide fine combinatorial stratifications of the simplex.

The inverse of $\operatorname{REM}_s$ is described explicitly: $$x_i = (\#\,\text{s-descents among } r_1, ..., r_i)\cdot s_i + r_i$$ [(Liu et al., 2012), Thm 3.6]. Therefore, the total number of s-descents in a remainder vector translates directly into the grading of the related lattice point in $\mathsf{Par}_s$. The i-th entry of the h*-vector is the number of $r \in V_n$ with exactly $i$ s-descents. For sequences with $s_1 = 1$, grading corresponds to ordinary ascent counts, recovering classical results.

This framework is further extended by considering reversed sequences and duality: lattice points in the parallelepiped for $P_s$ correspond bijectively to those in $P_u$. As a result, descent and ascent statistics are intertwined under reversal, supporting dual interpretations of the h*-vector.

5. Generalizations and Connections to Triangulation, Symmetry, and Algebraic Properties

The paper provides several generalizations:

• Multiple bijections ($\operatorname{REM}_s$, $\operatorname{REM}_+$, etc.) between lattice points and combinatorial objects (inversion sequences, lecture hall partitions).
• Explicit recovery of known Ehrhart polynomials, e.g., for $s = (1,2,\dots,n)$ or its reversal, the Ehrhart polynomial matches that of the unit cube.
• Affine and unimodular equivalences between polytopes for different $s$-sequences, with explicit dualities established (see Lemma 5.1 (Liu et al., 2012)).
• Triangulation results for special $s$ (e.g., those with monotonicity or small differences), including flag, regular, and unimodular triangulations in cases with first order $s$-differences in $\{0,1\}$ (Brändén et al., 2019); explicit one-point extension triangulations for broader classes (Caicedo et al., 26 Aug 2025).
• In special cases (e.g. $s_1 = 1$), simpler bijections between lattice points and inversion sequences yield direct enumerative formulas.

Structural properties such as IDP (integer decomposition property) are verified for monotone $s$-sequences (Hibi et al., 2016, Brändén et al., 2019), and algebraic properties (existence of quadratic, square-free Gröbner bases for associated toric ideals) are characterized for certain $s$ (Brändén et al., 2019). Gorenstein and level properties are classified explicitly in terms of arithmetic and combinatorial properties of $s$ and its inversion sequences (Kohl et al., 2017).

6. Real-Rootedness, Unimodality, and Ehrhart Nonpositivity Phenomena

It is established for all $s$ that the h*-polynomial $\sum_{e} z^{\operatorname{asc}(e)}$ (aggregate over $s$-inversion sequences) is real-rooted (Savage, 2016, Corteel et al., 2018, Gustafsson et al., 2018, Olsen, 2018), implying unimodality and log-concavity of Ehrhart coefficients. Local h*-polynomials, or box polynomials, generalize classical derangement polynomials and are also real-rooted and unimodal (Gustafsson et al., 2018, Olsen, 2018). This supports conjectures relating to subdivision theory and algebraic topology.

Ehrhart positivity—nonnegativity of coefficients—does not always hold for s-lecture hall simplices. As shown in (Caicedo et al., 26 Aug 2025), for $s = (a, ..., a, a+1)$ with large $a$ and $n\geq 5$, the coefficient of $t^{n-4}$ in the Ehrhart polynomial is negative: $$[t^{n-4}]L_p(t) \sim - \frac{1}{720 (n-4)!} a^{n-1},$$ demonstrating that proximity to standard simplex structure does not guarantee positivity.

7. Explicit Triangulations and Computational Implications

Regular, flag, and unimodular triangulations exist for wide classes of s-lecture hall simplices. For sequences with $s_{i+1} - s_{i} \in \{0, 1\}$, such triangulations are constructed via alcoved polytope techniques and quadratic square-free Gröbner bases (Brändén et al., 2019). In (Caicedo et al., 26 Aug 2025), explicit inductive constructions via one-point extensions enable flag, regular, unimodular triangulations even in families with negative Ehrhart coefficients, answering conjectures and broadening available triangulation methods well beyond prior Gröbner basis-dependent approaches.

The existence of explicit triangulations supports efficient computation of h*-vectors, provides combinatorial control over semigroup algebras, and links algebraic and geometric paper via toric ideals and Stanley–Reisner theory.

---

In sum, S-lecture hall simplices provide a unified geometric-combinatorial framework encapsulating lecture hall partitions, inversion sequences, and permutation statistics. Their lattice point enumeration, Ehrhart theory, symmetry and duality properties, Gorenstein and level classification, and explicit triangulation constructions not only recover and extend classical results (such as Eulerian numbers and derangement polynomial behavior) but also reveal subtle phenomena such as Ehrhart non-positivity and the coexistence of "nice" triangulations with negative Ehrhart coefficients. Ongoing research (cf. (Caicedo et al., 26 Aug 2025)) continues to expand our understanding of both their geometric structure and combinatorial algebraic implications.