s-Lecture Hall Partitions, Self-Reciprocal Polynomials, and Gorenstein Cones

In 1997, Bousquet-Melou and Eriksson initiated the study of lecture hall partitions, a fascinating family of partitions that yield a finite version of Euler's celebrated odd/distinct partition theorem. In subsequent work on s-lecture hall partitions, they considered the self-reciprocal property for various associated generating functions, with the goal of characterizing those sequences s that give rise to generating functions of the form $((1-q^{e_1})(1-q^{e_2})...(1-q^{e_n}))^{-1}$. We continue this line of investigation, connecting their work to the more general context of Gorenstein cones. We focus on the Gorenstein condition for s-lecture hall cones when s is a positive integer sequence generated by a second-order homogeneous linear recurrence with initial values 0 and 1. Among such sequences s, we prove that the n-dimensional s-lecture hall cone is Gorenstein for all n greater than or equal to 1 if and only if s is an l-sequence. One consequence is that among such sequences s, unless s is an l-sequence, the generating function for the s-lecture hall partitions can have the form $((1-q^{e_1})(1-q^{e_2})...(1-q^{e_n}))^{-1}$ for at most finitely many n. We also apply the results to establish several conjectures by Pensyl and Savage regarding the symmetry of h*-vectors for s-lecture hall polytopes. We end with open questions and directions for further research

Level algebras and s\boldsymbol{s}-lecture hall polytopes

Given a family of lattice polytopes, a common endeavor in Ehrhart theory is the classification of those polytopes in the family that are Gorenstein, or more generally level. In this article, we consider these questions for $\boldsymbol{s}$-lecture hall polytopes, which are a family of simplices arising from $\boldsymbol{s}$-lecture hall partitions. In particular, we provide concrete classifications for both of these properties purely in terms of $\boldsymbol{s}$-inversion sequences. Moreover, for a large subfamily of $\boldsymbol{s}$-lecture hall polytopes, we provide a more geometric classification of the Gorenstein property in terms of its tangent cones. We then show how one can use the classification of level $\boldsymbol{s}$-lecture hall polytopes to construct infinite families of level $\boldsymbol{s}$-lecture hall polytopes, and to describe level $\boldsymbol{s}$-lecture hall polytopes in small dimensions.Comment: Final version, to appear in Electronic Journal of Combinatoric

Hilbert Bases and Lecture Hall Partitions

In the interest of finding the minimum additive generating set for the set of $\boldsymbol{s}$-lecture hall partitions, we compute the Hilbert bases for the $\boldsymbol{s}$-lecture hall cones in certain cases. In particular, we compute the Hilbert bases for two well-studied families of sequences, namely the $1\mod k$ sequences and the $\ell$-sequences. Additionally, we provide a characterization of the Hilbert bases for $\boldsymbol{u}$-generated Gorenstein $\boldsymbol{s}$-lecture hall cones in low dimensions.Comment: Final Version. To appear in Ramanujan

Lecture hall partitions and the affine hyperoctahedral group

In 1997 Bousquet-M\'elou and Eriksson introduced lecture hall partitions as the inversion vectors of elements of the parabolic quotient $\widetilde{C}/C$. We provide a new view of their correspondence that allows results in one domain to be translated into the other. We determine the equivalence between combinatorial statistics in each domain and use this correspondence to translate certain generating function formulas on lecture hall partitions to new observations about $\widetilde{C}/C$.Comment: 15 page

A partition inequality involving products of two qq-Pochhammer symbols

We use an injection method to prove a new class of partition inequalities involving certain $q$-products with two to four finitization parameters. Our new theorems are a substantial generalization of work by Andrews and of previous work by Berkovich and Grizzell. We also briefly discuss how our products might relate to lecture hall partitions.Comment: 14 pages, 5 table

Generating functions and triangulations for lecture hall cones

We investigate the arithmetic-geometric structure of the lecture hall cone $$L_n \ := \ \left\{\lambda\in \mathbb{R}^n: \, 0\leq \frac{\lambda_1}{1}\leq \frac{\lambda_2}{2}\leq \frac{\lambda_3}{3}\leq \cdots \leq \frac{\lambda_n}{n}\right\} .$$ We show that $L_n$ is isomorphic to the cone over the lattice pyramid of a reflexive simplex whose Ehrhart $h^*$-polynomial is given by the $(n-1)$st Eulerian polynomial, and prove that lecture hall cones admit regular, flag, unimodular triangulations. After explicitly describing the Hilbert basis for $L_n$, we conclude with observations and a conjecture regarding the structure of unimodular triangulations of $L_n$, including connections between enumerative and algebraic properties of $L_n$ and cones over unit cubes

Mahonian Partition Identities Via Polyhedral Geometry

In a series of papers, George Andrews and various coauthors successfully revitalized seemingly forgotten, powerful machinery based on MacMahon's $\Omega$ operator to systematically compute generating functions \sum_{\la \in P} z_1^{\la_1}...z_n^{\la_n} for some set $P$ of integer partitions \la = (\la_1,..., \la_n). Our goal is to geometrically prove and extend many of the Andrews et al theorems, by realizing a given family of partitions as the set of integer lattice points in a certain polyhedron.Comment: 10 page

Relaxed complete partitions: an error-correcting Bachet's problem

Motivated by an error-correcting generalization of Bachet's weights problem, we define and classify relaxed complete partitions. We show that these partitions enjoy a succinct description in terms of lattice points in polyhedra, with adjustments in the error being commensurate with translations in the defining hyperplanes. Our main result is that the enumeration of the minimal such partitions (those with fewest possible parts) is achieved via Brion's formula. This generalizes work of Park on classifying complete partitions and that of R{\o}dseth on enumerating minimal complete partitions.Comment: 14 pages, 1 figure. Major revision - main result is now shown using Brion's formula for lattice point enumeration in polyhedr

Coefficients of the Inflated Eulerian Polynomial

It follows from work of Chung and Graham that for a certain family of polynomials $T_{n}(x)$, derived from the descent statistic on permutations, the coefficient sequence of $T_{n-1}(x)$ coincides with that of the polynomial $T_{n}(x)/\left(1+x+\cdots+x^{n-1}\right)$. We observed computationally that the inflated $\mathbf{s}$-Eulerian polynomial $Q_{n}^{(\mathbf{s})}(x)$, which satisfies $Q_{n}^{(\mathbf{s})}(x) = T_{n}(x)$ when $\mathbf{s}=(1,2,\ldots,n)$, also satisfies this property for many sequences $\mathbf{s}$. In this work we characterize those sequences $\mathbf{s}$ for which the coefficient sequence of $Q_{n-1}^{(\mathbf{s})}(x)$ coincides with that of the polynomial $Q_{n}^{(\mathbf{s})}(x)/\left(1+x+\cdots+x^{s_{n}-1}\right)$. In particular, we show that all nondecreasing sequences satisfy this property. We also settle a conjecture of Pensyl and Savage by showing that the inflated $\mathbf{s}$-Eulerian polynomials are unimodal for all choices of positive integer sequences ${\bf s}$. In addition, we determine when these polynomials are palindromic and show our characterization is equivalent to another of Beck, Braun, K\"oppe, Savage, and Zafeirakopoulos.Comment: New results and new collaborator added. 26 pages, 4 figure

Odd number and Trapezoidal number

In this paper, we give a bijective proof of the reduced lecture hall partition theorem. It is possible to extend this bijection in lecture hall partition theorem. And refined versions of each theorems are also presented