arXiv subjects

Benjamin Braun

Publications and source records attributed to Benjamin Braun.

At least 37 records · Page 2Linked to original sources

arXiv2023

Facets of Symmetric Edge Polytopes for Graphs with Few Edges

Symmetric edge polytopes, also called adjacency polytopes, are lattice polytopes determined by simple undirected graphs. We introduce the integer array $\mathrm{maxf}(n,m)$ giving the maximum number of facets of a symmetric edge polytope for a connected graph having $n$ vertices and $m$ edges, and the corresponding sequence $\mathrm{minf}(n,m)$ of minimal values. We establish formulas for the number of facets obtained in several classes of sparse graphs and provide partial progress toward conjectures that identify facet-maximizing graphs in these classes. These formulas are combinatorial in nature and lead to independently interesting observations and conjectures regarding integer sequences defined by sums of products of binomial coefficients.

math.CO↗

arXiv2022

Ehrhart Limits

We introduce the definition of an Ehrhart limit, that is, a formal power series with integer coefficients that is the limit in the ring of formal power series of a sequence of Ehrhart $h^*$-polynomials. We identify a variety of examples of sequences of polytopes that yield Ehrhart limits, with a focus on reflexive polytopes and simplices.

math.CO↗

arXiv2022

The Integer Decomposition Property and Weighted Projective Space Simplices

Reflexive lattice polytopes play a key role in combinatorics, algebraic geometry, physics, and other areas. One important class of lattice polytopes are lattice simplices defining weighted projective spaces. We investigate the question of when a reflexive weighted projective space simplex has the integer decomposition property. We provide a complete classification of reflexive weighted projective space simplices having the integer decomposition property for the case when there are at most three distinct non-unit weights, and conjecture a general classification for an arbitrary number of distinct non-unit weights. Further, for any weighted projective space simplex and $m\geq 1$, we define the $m$-th reflexive stabilization, a reflexive weighted projective space simplex. We prove that when $m$ is $2$ or greater, reflexive stabilizations do not have the integer decomposition property. We also prove that the Ehrhart $h^\ast$-polynomial of any sufficiently large reflexive stabilization is not unimodal and has only $1$ and $2$ as coefficients. We use this construction to generate interesting examples of reflexive weighted projective space simplices that are near the boundary of both $h^*$-unimodality and the integer decomposition property.

math.CO↗

arXiv2021

Triangulations, order polytopes, and generalized snake posets

This work regards the order polytopes arising from the class of generalized snake posets and their posets of meet-irreducible elements. Among generalized snake posets of the same rank, we characterize those whose order polytopes have minimal and maximal volume. We give a combinatorial characterization of the circuits in these order polytopes and then conclude that every triangulation is unimodular. For a generalized snake word, we count the number of flips for the canonical triangulation of these order polytopes. We determine that the flip graph of the order polytope of the poset whose lattice of filters comes from a ladder is the Cayley graph of a symmetric group. Lastly, we introduce an operation on triangulations called twists and prove that twists preserve regular triangulations.

math.CO↗

arXiv2021

Decompositions of Ehrhart $h^*$-polynomials for rational polytopes

The Ehrhart quasipolynomial of a rational polytope $P$ encodes the number of integer lattice points in dilates of $P$, and the $h^*$-polynomial of $P$ is the numerator of the accompanying generating function. We provide two decomposition formulas for the $h^*$-polynomial of a rational polytope. The first decomposition generalizes a theorem of Betke and McMullen for lattice polytopes. We use our rational Betke--McMullen formula to provide a novel proof of Stanley's Monotonicity Theorem for the $h^*$-polynomial of a rational polytope. The second decomposition generalizes a result of Stapledon, which we use to provide rational extensions of the Stanley and Hibi inequalities satisfied by the coefficients of the $h^*$-polynomial for lattice polytopes. Lastly, we apply our results to rational polytopes containing the origin whose duals are lattice polytopes.

math.CO↗

arXiv2020

Rationality of Poincaré Series for a Family of Lattice Simplices

We investigate multi-graded Gorenstein semigroup algebras associated with an infinite family of reflexive lattice simplices. For each of these algebras, we prove that their multigraded Poincaré series is rational. Our method of proof is to produce for each algebra an explicit minimal free resolution of the ground field, in which the resolution reflects the recursive structure encoded in the denominator of the finely-graded Poincaré series. Using this resolution, we show that these algebras are not Koszul, and therefore rationality is non-trivial. Our results demonstrate how interactions between multivariate and univariate rational generating functions can create subtle complications when attempting to use rational Poincaré series to inform the construction of minimal resolutions.

math.CO↗

arXiv2020

A regular unimodular triangulation of reflexive 2-supported weighted projective space simplices

For each integer partition $\mathbf{q}$ with $d$ parts, we denote by $Δ_{(1,\mathbf{q})}$ the lattice simplex obtained as the convex hull in $\mathbb{R}^d$ of the standard basis vectors along with the vector $-\mathbf{q}$. For $\mathbf{q}$ with two distinct parts such that $Δ_{(1,\mathbf{q})}$ is reflexive and has the integer decomposition property, we establish a characterization of the lattice points contained in $Δ_{(1,\mathbf{q})}$. We then construct a Gröbner basis with a squarefree initial ideal of the toric ideal defined by these simplices. This establishes the existence of a regular unimodular triangulation for reflexive 2-supported $Δ_{(1,\mathbf{q})}$ having the integer decomposition property.

math.CO↗

arXiv2020

Phase transitions and control measures for network epidemics caused by infections with presymptomatic, asymptomatic,and symptomatic stages

We investigate phase transitions associated with three control methods for epidemics on small world networks. Motivated by the behavior of SARS-CoV-2, we construct a theoretical SIR model of a virus that exhibits presymptomatic, asymptomatic, and symptomatic stages in two possible pathways. Using agent-based simulations on small world networks, we observe phase transitions for epidemic spread related to: 1) Global social distancing with a fixed probability of adherence. 2) Individually initiated social isolation when a threshold number of contacts are infected. 3) Viral shedding rate. The primary driver of total number of infections is the viral shedding rate, with probability of social distancing being the next critical factor. Individually initiated social isolation was effective when initiated in response to a single infected contact. For each of these control measures, the total number of infections exhibits a sharp phase transition as the strength of the measure is varied.

physics.soc-ph↗

arXiv2020

The TRaCaR Ratio: Selecting the Right Storage Technology for Active Dataset-Serving Databases

Main memory database systems aim to provide users with low latency and high throughput access to data. Most data resides in secondary storage, which is limited by the access speed of the technology. For hot content, data resides in DRAM, which has become increasingly expensive as datasets grow in size and access demand. With the emergence of low-latency storage solutions such as Flash and Intel's 3D XPoint (3DXP), there is an opportunity for these systems to give users high Quality-of-Service while reducing the cost for providers. To achieve high performance, providers must provision the server hosts for these datasets with the proper amount of DRAM and secondary storage, as well as selecting a storage technology. The growth of capacity and transaction load overtime makes it expensive to flip back-and-forth between different storage technologies and memory-storage combinations. Servers set up for one storage technology must now be reconfigured, repartitioned, and potentially replaced altogether. As more low-latency storage solutions become available, how does one decide on the right memory-storage combination, as well as selecting a storage technology, given a predicted trend in dataset growth and offered load? In this paper, we describe and make the case for using the TRaCaR ratio - the transaction rate divided by the storage capacity needed for a workload - for allowing providers to choose the most cost-effective memory-storage combination and storage technology given their predicted dataset trend and load requirement. We explore how the TRaCaR ratio can be used with 3DXP and Flash with a highly-zipfian b-tree database, and discuss potential research directions that can leverage the ratio.

cs.DC↗

arXiv2019

Antichain Simplices

To each lattice simplex $Δ$ we associate 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. To each partition $λ$ of $n$, we associate 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 $n\leq 73$. We also investigate the structure of these posets when $λ$ has only one or two distinct parts. Finally, we explain how this work relates to Poincaré series for the semigroup algebra associated to $Δ$, and we prove that this series is rational when $Δ$ is antichain.

math.CO↗

arXiv2018

Homomorphism Complexes and Maximal Chains in Graded Posets

We apply the homomorphism complex construction to partially ordered sets, introducing a new topological construction based on the set of maximal chains in a graded poset. Our primary objects of study are distributive lattices, with special emphasis on finite products of chains. For the special case of a Boolean algebra, we observe that the corresponding homomorphism complex is isomorphic to the subcomplex of cubical cells in a permutahedron. Thus, this work can be interpreted as a generalization of the study of these complexes. We provide a detailed investigation when our poset is a product of chains, in which case we find an optimal discrete Morse matching and prove that the corresponding complex is torsion-free.

math.CO↗

arXiv2018

Hajós-Type Constructions and Neighborhood Complexes

Any graph $G$ with chromatic number $k$ can be constructed by iteratively performing certain graph operations on a sequence of graphs starting with $K_k$, resulting in a variety of Hajós-type constructions for $G$. Finding such constructions for a given graph or family of graphs is a challenging task. We show that the basic steps in these Hajós-type constructions frequently result in the presence of an $S^1$-wedge summand in the neighborhood complex of the resulting graph. Our results imply that for a graph $G$ with a highly-connected neighborhood complex, the end behavior of the construction sequence is quite restricted, and we investigate these restrictions in detail. We also introduce two graph construction algorithms based on different Hajós-type constructions and conduct computational experiments using these.

math.CO↗

arXiv2018

A Brief Survey on Lattice Zonotopes

Zonotopes are a rich and fascinating family of polytopes, with connections to many areas of mathematics. In this article we provide a brief survey of classical and recent results related to lattice zonotopes. Our emphasis is on connections to combinatorics, both in the sense of enumeration (e.g. Ehrhart theory) and combinatorial structures (e.g. graphs and permutations).

math.CO↗

arXiv2018

$h^*$-Polynomials With Roots on the Unit Circle

For an $n$-dimensional lattice simplex $Δ_{(1,\mathbf{q})}$ with vertices given by the standard basis vectors and $-\mathbf{q}$ where $\mathbf{q}$ has positive entries, we investigate when the Ehrhart $h^*$-polynomial for $Δ_{(1,\mathbf{q})}$ factors as a product of geometric series in powers of $z$. Our motivation is a theorem of Rodriguez-Villegas implying that when the $h^*$-polynomial of a lattice polytope $P$ has all roots on the unit circle, then the Ehrhart polynomial of $P$ has positive coefficients. We focus on those $Δ_{(1,\mathbf{q})}$ for which $\mathbf{q}$ has only two or three distinct entries, providing both theoretical results and conjectures/questions motivated by experimental evidence.

math.CO↗

arXiv2018

Detecting the Integer Decomposition Property and Ehrhart Unimodality in Reflexive Simplices

A long-standing open conjecture in combinatorics asserts that a Gorenstein lattice polytope with the integer decomposition property (IDP) has a unimodal (Ehrhart) $h^\ast$-polynomial. This conjecture can be viewed as a strengthening of a previously disproved conjecture which stated that any Gorenstein lattice polytope has a unimodal $h^\ast$-polynomial. The first counterexamples to unimodality for Gorenstein lattice polytopes were given in even dimensions greater than five by Musta{ţ}{ǎ} and Payne, and this was extended to all dimensions greater than five by Payne. While there exist numerous examples in support of the conjecture that IDP reflexives are $h^\ast$-unimodal, its validity has not yet been considered for families of reflexive lattice simplices that closely generalize Payne's counterexamples. The main purpose of this work is to prove that the former conjecture does indeed hold for a natural generalization of Payne's examples. The second purpose of this work is to extend this investigation to a broader class of lattice simplices, for which we present new results and open problems.

math.CO↗

arXiv2017

Unimodality Problems in Ehrhart Theory

Ehrhart theory is the study of sequences recording the number of integer points in non-negative integral dilates of rational polytopes. For a given lattice polytope, this sequence is encoded in a finite vector called the Ehrhart $h^*$-vector. Ehrhart $h^*$-vectors have connections to many areas of mathematics, including commutative algebra and enumerative combinatorics. In this survey we discuss what is known about unimodality for Ehrhart $h^*$-vectors and highlight open questions and problems.

math.CO↗

arXiv2017

Generating functions and triangulations for lecture hall cones

We investigate the arithmetic-geometric structure of the lecture hall cone \[ L_n \ := \ \left\{λ\in \mathbb{R}^n: \, 0\leq \frac{λ_1}{1}\leq \frac{λ_2}{2}\leq \frac{λ_3}{3}\leq \cdots \leq \frac{λ_n}{n}\right\} . \] 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.

math.CO↗

arXiv2017

Euler-Mahonian statistics and descent bases for semigroup algebras

We consider quotients of the unit cube semigroup algebra by particular $\mathbb{Z}_r\wr S_n$-invariant ideals. Using Gröbner basis methods, we show that the resulting graded quotient algebra has a basis where each element is indexed by colored permutations $(π,ε)\in\mathbb{Z}_r\wr S_n$ and each element encodes the negative descent and negative major index statistics on $(π,ε)$. This gives an algebraic interpretation of these statistics which was previously unknown. This basis of the $\mathbb{Z}_r\wr S_n$-quotients allows us to recover certain combinatorial identities involving Euler-Mahonian distributions of statistics.

math.CO↗