arXiv subjects

Matthias Beck

Publications and source records attributed to Matthias Beck.

At least 19 recordsLinked to original sources

arXiv2026

Lonely Runner Relations

We study the Lonely Runner Conjecture (LRC), conceived by J\"org M. Wills in the 1960's: Given positive integers $n_1, n_2, \dots, n_k$, there exists a positive real number $t$ such that for all $1 \le j \le k$ the distance of $t \,n_j$ to the nearest integer is at least $\frac{ 1 }{ k+1 }$. We prove that for any counterexample or tight instance $\mathbf{n}$ of LRC, $\mathbf{m} \cdot \mathbf{n} = 0$ for some $\mathbf{m} \in \mathbb{Z}^k$ with $0 < \| \mathbf{m} \|_1 \le \min(2k+3, \ \frac{ k+1 }{ k-1 } \mathrm{flt}(k))$ where $\mathrm{flt}(k)$ denotes Khinchin's (1948) flatness constant limiting the lattice width of a $k$-dimensional convex body without interior integer points. In other words, potential counterexamples to LRC lie on a finite set of hyperplanes in the parameter space. Our proofs use Fourier analysis and a geometric reformulation of LRC, and our results generalize to the situation of shifted lonely runners of varying measures of loneliness. Our results imply and generalize a theorem of Czerwi\'nski (2012) that when we choose $\mathbf{n}$ at random then, with probability tending to 1, the measure of loneliness $\frac{1}{ k+1 }$ can be replaced by $\frac 1 2 - \epsilon$.

math.CO↗

arXiv2026

Bergeron's conjecture & a tale of two binomial coefficients

Bergeron's conjecture states that, if $1\leq a<b<c<d$ are integers with $ad=bc$, then one has the coefficient-wise inequality ${\binom{b+c}b}_q \ge {\binom{a+d}a}_q$ among two Gaussian polynomials. It originated in algebraic combinatorics and is wide open. The corresponding inequality for binomial coefficients (i.e., the case $q=1$) must be known to experts, but we could not find it in the literature. We give two proofs, each generalizing the statement in a separate direction. Binomial coefficients (and Gaussian polynomials) are fundamental combinatorial objects, and so one naturally hopes to see a combinatorial proof of this inequality. However, this seems hard to come by. We nevertheless give a combinatorial proof of a special case.

math.CO↗

arXiv2026

A matroidal twist on a formula of Brion

Brion's Formula realizes the Laurent polynomial of lattice points in a lattice polytope P as the sum of rational functions associated to the vertices of P. In this paper, we consider the special case where P is a generalized permutohedron. We study a modification of the rational functions associated to the vertices of P depending on a given matroid M. Upon summing these rational functions, we describe how the resulting Laurent polynomial Q_M(P) behaves in certain ways like the lattice points of P, exhibiting natural recursive and reciprocity behaviors. Furthermore, upon evaluating Q_M(P) at 1, we recover the matroid Euler characteristic of Larson, Li, Payne, and Proudfoot, so the combinatorial approach in this paper gives new insight into studying these quantities.

math.CO↗

arXiv2025

Generating functions of $q$-chromatic polynomials

Given a graph $G=(V,E)$ and a linear form $\lambda \in \mathbb{Z}_{ > 0 }^V$, Bajo et al. (2025) introduced the $q$-chromatic polynomial $\chi_G^\lambda(q,n) := \sum q^{\sum_{v \in V} \lambda_v c(v)}$ where the sum is over all proper colorings $c: V \to [n] := \{ 1, 2, \dots, n \}$; they showed that $\chi_G^\lambda(q,n)$ is a polynomial in $[n]_q := 1 + q + \dots + q^{ n-1 } $ with coefficients in $\mathbb{Z}(q)$. For $d \in \mathbb{Z}_{>0}$ and the linear form given by $(d,d^2,\ldots,d^d)$, we show that the $q$-chromatic polynomial distinguishes labeled graphs with vertex set $[d]$. Using permutation statistics introduced by Chung--Graham (1995), called $G$-statistics, and polyhedral geometry, we give the multivariate integer point transform for the region of proper colorings of a given graph $G$. This integer point transform allows us to find the generating function for the $q$-chromatic polynomial with respect to any linear form. We further specialize these results to the linear form $(1, 1, \dots, 1)$, which allows us to write the $q$-chromatic polynomial in the $q$-binomial basis, clarifying expressions found by Bajo et al. Moreover, we show that $G$-statistics are compatible with the theory of order polytopes used by Bajo et al. and Chow (1999). This yields further properties for the generating function of $q$-chromatic polynomial with linear form $(1, 1, \dots, 1)$, where certain coefficients of the numerator polynomial are palindromic polynomials in $q$.

math.CO↗

arXiv2025

MacMahon's Double Vision: Partition Diamonds Revisited

Plane partition diamonds were introduced by Andrews, Paule, and Riese (2001) as part of their study of MacMahon's $\Omega$-operator in search for integer partition identities. More recently, Dockery, Jameson, Sellers, and Wilson (2024) extended this concept to $d$-fold partition diamonds and found their generating function in a recursive form. We approach $d$-fold partition diamonds via Stanley's (1972) theory of $P$-partitions and give a closed formula for a bivariate generalization of the Dockery--Jameson--Sellers--Wilson generating function; its main ingredient is the Euler--Mahonian polynomial encoding descent statistics of permutations.

math.CO↗

arXiv2025

A Closer Look at Chapoton's q-Ehrhart Polynomials

If $\mathcal{P}$ is a lattice polytope (i.e., $\mathcal{P}$ is the convex hull of finitely many integer points in $\mathbb{R}^d$), Ehrhart's famous theorem (1962) asserts that the integer-point counting function $|t \mathcal{P} \cap \mathbb{Z}^d|$ is a polynomial in the integer variable $t$. Chapoton (2016) proved that, given a fixed integral form $\lambda: \mathbb{Z}^d \to \mathbb{Z}$, there exists a polynomial $\text{cha}_\mathcal{P}^\lambda(q,x) \in \mathbb{Q}(q)[x]$ such that the refined enumeration function $\sum_{ \mathbf{m} \in t \mathcal{P} } q^{ \lambda(\mathbf{m}) }$ equals the evaluation $\text{cha}_\mathcal{P}^\lambda (q, [t]_q)$ where, as usual, $[t]_q := \frac{ q^t - 1 }{ q-1 }$; naturally, for $q=1$ we recover the Ehrhart polynomial. Our motivating goal is to view Chapoton's work through the lens of Brion's Theorem (1988), which expresses the integer-point structure of a given polytope via that of its vertex cones. It turns out that this viewpoint naturally yields various refinements and extensions of Chapoton's results, including explicit formulas for $\text{cha}_\mathcal{P}^\lambda(q,x)$, its leading coefficient, and its behavior as $t \to \infty$. We also prove an analogue of Chapoton's structural and reciprocity theorems for rational polytopes (i.e., with vertices in $\mathbb{Q}^d$).

math.CO↗

arXiv2025

Musical Chords by the Numbers

The mathematics of musical intervals and scales has been extensively studied. Vastly simplified, our ears seem to prefer intervals whose frequency ratios have small numerator and denominator, such as 2:1 (octave), 3:2 (perfect fifth), 4:3 (perfect fourth), and so on. While there also have been numerous studies on the mathematics of musical chords, very few of them consider a model that measures consonance/dissonance of a given chord in analogy with this simple-fractions perspective. Our aim is to develop a measure for the consonance of a chord with crucial symmetry features, including invariance under chord translation, inversion, and interval sets. We apply our model to chords in various musical scales and compare it to existing models and empirical studies.

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arXiv2024

Canon Permutation Posets

A permutation of the multiset $\{1^m,2^m,\dots,n^m\}$ is a {\em canon permutation} if the subsequence formed by the $j$th copy of each element of $[n]:=\{1,2,\dots,n\}$ is identical for all $j\in[m]$. Canon permutations were introduced by Elizalde and are motivated by pattern-avoiding concepts such as (quasi-)Stirling permutations. He proved that the descent polynomial of canon permutations exhibits a surprising product structure; as a further consequence, it is palindromic. Our goal is to understand canon permutations from the viewpoint of Stanley's $(P,\omega)$-partitions, along the way generalizing Elizalde's definition and results. We start with a labeled poset $P$ and extend it in a natural way to canon labelings of the product poset $P \times [n]$. The resulting descent polynomial has a product structure which arises naturally from the theory of $(P,\omega)$-partitions and simplifies existing proofs. When $P$ is graded, this theory also implies palindromicity. We include results on weak descent polynomials, an amphibian construction between canon permutations and multiset permutations, giving rise to \emph{dissonant canon permutations}, as well as $\gamma$-positivity and interpretations of descent polynomials of canon permutations.

math.CO↗

arXiv2024

Acyclotopes and Tocyclotopes

There is a well-established dictionary between zonotopes, hyperplane arrangements, and their (oriented) matroids. Arguably one of the most famous examples is the class of graphical zonotopes, also called acyclotopes, which encode Minkowski summands of the type-A permutahedron. Stanley (1991) gave a general interpretation of the coefficients of the Ehrhart polynomial (integer-point counting function for a polytope) of a zonotope via linearly independent subsets of its generators. Applying this to the graphical case shows that Ehrhart coefficients count forests of the graph of fixed sizes. Our first goal is to extend and popularize this story to other root systems, which on the combinatorial side is encoded by signed graphs analogously to the work by Greene and Zaslavsky (1983). We compute the Ehrhart polynomial of the acyclotope in the signed case. Our second goal is to translate matroid duality concepts not only to hyperplanes but also to zonotopes in such a way that the Ehrhart polynomial is uniquely defined. In the case of (signed) graphs we construct tocyclotopes and compute their Ehrhart polynomials. Applying the same duality construction to a general integral matrix gives rise to a lattice Gale zonotope, whose face structure was studied by McMullen (1971) and whose duality concepts give a special instance of D'Adderio--Moci's arithmetic matroids (2013). We describe its Ehrhart polynomials in terms of the given matrix.

math.CO↗

arXiv2024

Polyhedral geometry of refined $q,t$-Catalan numbers

We study a refinement of the $q,t$-Catalan numbers introduced by Xin and Zhang (2022, 2023) using tools from polyhedral geometry. These refined $q,t$-Catalan numbers depend on a vector of parameters $\vec{k}$ and the classical $q,t$-Catalan numbers are recovered when $\vec{k} = (1,\ldots,1)$. We interpret Xin and Zhang's generating functions by developing polyhedral cones arising from constraints on $\vec{k}$-Dyck paths and their associated area and bounce statistics. Through this polyhedral approach, we recover Xin and Zhang's theorem on $q,t$-symmetry of the refined $q,t$-Catalan numbers in the cases where $\vec{k} = (k_1,k_2,k_3)$ and $(k,k,k,k)$, give some extensions, including the case $\vec{k} = (k,k+m,k+m,k+m)$, and discuss relationships to other generalizations of the $q,t$-Catalan numbers.

math.CO↗

arXiv2024

$q$-Chromatic polynomials

We study a $q$-version of the chromatic polynomial of a given graph $G=(V,E)$, namely, \[ \chi_G^\lambda(q,n) \ := \sum_{\substack{\text{proper colorings}\\ c\,:\,V\to[n]}} q^{ \sum_{ v \in V } \lambda_v c(v) }, \] where $\lambda \in \mathbb{Z}_{>0}^V$ is a fixed linear form. Via work of Chapoton (2016) on $q$-Ehrhart polynomials, $\chi_G^\lambda(q,n)$ turns out to be a polynomial in the $q$-integer $[n]_q$, with coefficients that are rational functions in $q$. Additionally, we prove structural results for $\chi_G^\lambda(q,n)$ and exhibit connections to neighboring concepts, e.g., chromatic symmetric functions and the arithmetic of order polytopes. We offer a strengthened version of Stanley's conjecture that the chromatic symmetric function distinguishes trees, which leads to an analogue of $P$-partitions for graphs.

math.CO↗

arXiv2023

Signed Poset Polytopes

Stanley introduced in 1986 the order polytope and the chain polytope for a given finite poset. These polytopes contain much information about the poset and have given rise to important examples in polyhedral geometry. In 1993, Reiner introduced signed posets as natural type-B analogues of posets. We define and study signed order and chain polytopes. Our results include convex-hull and halfspace descriptions, unimodular triangulations, Ehrhart $h^*$-polynomials and their relations to signed permutation statistics, and a Gorenstein characterization of signed order and chain polytopes.

math.CO↗

arXiv2023

Deep lattice points in zonotopes, lonely runners, and lonely rabbits

Let $K \subseteq \mathbb{R}^d$ be a convex body and let $\mathbf{w} \in \operatorname{int}(K)$ be an interior point of $K$. The coefficient of asymmetry $\operatorname{ca}(K,\mathbf{w}) := \min\{ \lambda \geq 1 : \mathbf{w} - K \subseteq \lambda (K - \mathbf{w}) \}$ has been studied extensively in the realm of Hensley's conjecture on the maximal volume of a $d$-dimensional lattice polytope that contains a fixed positive number of interior lattice points. We study the coefficient of asymmetry for lattice zonotopes, i.e., Minkowski sums of line segments with integer endpoints. Our main result gives the existence of an interior lattice point whose coefficient of asymmetry is bounded above by an explicit constant in $\Theta(d \log\log d)$, for any lattice zonotope that has an interior lattice point. Our work is both inspired by and feeds on Wills' lonely runner conjecture from Diophantine approximation: we make intensive use of a discrete version of this conjecture, and reciprocally, we reformulate the lonely runner conjecture in terms of the coefficient of asymmetry of a zonotope.

math.MG↗

arXiv2022

Inequalities for $f^*$-vectors of Lattice Polytopes

The Ehrhart polynomial $\text{ehr}_P(n)$ of a lattice polytope $P$ counts the number of integer points in the $n$-th integral dilate of $P$. The $f^*$-vector of $P$, introduced by Felix Breuer in 2012, is the vector of coefficients of $\text{ehr}_P(n)$ with respect to the binomial coefficient basis $ \left\{\binom{n-1}{0},\binom{n-1}{1},...,\binom{n-1}{d}\right\}$, where $d = \dim P$. Similarly to $h/h^*$-vectors, the $f^*$-vector of $P$ coincides with the $f$-vector of its unimodular triangulations (if they exist). We present several inequalities that hold among the coefficients of $f^*$-vectors of polytopes. These inequalities resemble striking similarities with existing inequalities for the coefficients of $f$-vectors of simplicial polytopes; e.g., the first half of the $f^*$-coefficients increases and the last quarter decreases. Even though $f^*$-vectors of polytopes are not always unimodal, there are several families of polytopes that carry the unimodality property. We also show that for any polytope with a given Ehrhart $h^*$-vector, there is a polytope with the same $h^*$-vector whose $f^*$-vector is unimodal.

math.CO↗

arXiv2022

Boundary $h^\ast$-polynomials of rational polytopes

If $P$ is a lattice polytope (i.e., $P$ is the convex hull of finitely many integer points in $\mathbb{R}^d$) of dimension $d$, Ehrhart's famous theorem (1962) asserts that the integer-point counting function $|nP \cap \mathbb{Z}^d|$ is a degree-$d$ polynomial in the integer variable $n$. Equivalently, the generating function $1 + \sum_{n\geq 1} |nP \cap \mathbb{Z}^d| \, z^n$ is a rational function of the form $\frac{ h^\ast(z) }{ (1-z)^{ d+1 } }$; we call $h^\ast(z)$ the $h^\ast$-polynomial of $P$. There are several known necessary conditions for $h^\ast$-polynomials, including results by Hibi (1990), Stanley (1991), and Stapledon (2009), who used an interplay of arithmetic (integer-point structure) and topological (local $h$-vectors of triangulations) data of a given polytope. We introduce an alternative ansatz to understand Ehrhart theory through the $h^\ast$-polynomial of the boundary of a polytope, recovering all of the above results and their extensions for rational polytopes in a unifying manner. We include applications for (rational) Gorenstein polytopes and rational Ehrhart dilations.

math.CO↗

arXiv2022

Lattice zonotopes of degree 2

The Ehrhart polynomial $ehr_P (n)$ of a lattice polytope $P$ gives the number of integer lattice points in the $n$-th dilate of $P$ for all integers $n\geq 0$. The degree of $P$ is defined as the degree of its $h^\ast$-polynomial, a particular transformation of the Ehrhart polynomial with many useful properties which serves as an important tool for classification questions in Ehrhart theory. A zonotope is the Minkowski (pointwise) sum of line segments. We classify all Ehrhart polynomials of lattice zonotopes of degree $2$ thereby complementing results of Scott (1976), Treutlein (2010), and Henk-Tagami (2009). Our proof is constructive: by considering solid-angles and the lattice width, we provide a characterization of all $3$-dimensional zonotopes of degree $2$.

math.CO↗

arXiv2021

Bivariate Chromatic Polynomials of Mixed Graphs

The bivariate chromatic polynomial $\chi_G(x,y)$ of a graph $G = (V, E)$, introduced by Dohmen-P\"{o}nitz-Tittmann (2003), counts all $x$-colorings of $G$ such that adjacent vertices get different colors if they are $\le y$. We extend this notion to mixed graphs, which have both directed and undirected edges. Our main result is a decomposition formula which expresses $\chi_G(x,y)$ as a sum of bivariate order polynomials (Beck-Farahmand-Karunaratne-Zuniga Ruiz 2020), and a combinatorial reciprocity theorem for $\chi_G(x,y)$.

math.CO↗

arXiv2021

Rational Ehrhart Theory

The Ehrhart quasipolynomial of a rational polytope $\mathsf{P}$ encodes fundamental arithmetic data of $\mathsf{P}$, namely, the number of integer lattice points in positive integral dilates of $\mathsf{P}$. Ehrhart quasipolynomials were introduced in the 1960s, satisfy several fundamental structural results and have applications in many areas of mathematics and beyond. The enumerative theory of lattice points in rational (equivalently, real) dilates of rational polytopes is much younger, starting with work by Linke (2011), Baldoni-Berline-Koeppe-Vergne (2013), and Stapledon (2017). We introduce a generating-function ansatz for rational Ehrhart quasipolynomials, which unifies several known results in classical and rational Ehrhart theory. In particular, we define $\gamma$-rational Gorenstein polytopes, which extend the classical notion to the rational setting and encompass the generalized reflexive polytopes studied by Fiset-Kasprzyk (2008) and Kasprzyk-Nill (2012).

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