arXiv subjects

Max Hlavacek

Publications and source records attributed to Max Hlavacek.

10 recordsLinked to original sources

arXiv2026

Ehrhart $h^*$-distributions

Every polynomial with real non-negative coefficients yields a finite probability distribution after normalization. The Ehrhart $h^*$-polynomial of a lattice polytope $P$ is a non-negative integer polynomial that encodes the integer-point counts for positive integer dilations of $P$. We study the corresponding finite distributions, which we call $h^*$-distributions. We determine the mean and variance of these distributions, establish a connection between higher moments and Ehrhart polynomial coefficients, and study their cluster points in the $d$-dimensional probability simplex. We consider the special case of real-rooted $h^*$-distributions, applying existing tail bounds to obtain new linear inequalities for the coefficients of real-rooted $h^*$-polynomials arising from reflexive polytopes. We conclude by establishing sufficient conditions under which a sequence of real-rooted $h^*$-distributions is asymptotically normal, and we apply our results to various families of polytopes, including zonotopes and Pitman-Stanley polytopes.

math.CO↗

arXiv2025

The poset of maximal tubings of the cycle graph is a lattice

The poset of maximal tubings of a graph generalizes several well-known and remarkable partial orders. Notable examples include the weak Bruhat order and the Tamari lattice, posets of maximal tubings for the complete graph and the path graph, respectively. It is an open problem to characterize graphs for which the poset of maximal tubings is a lattice. In this paper, we prove that the poset of maximal tubings for the cycle graph is a lattice, and moreover that it is semidistributive and congruence uniform. As main tools, we characterize all order relations in the poset, and introduce a useful map from maximal tubings of the cycle graph to maximal tubings of the path graph.

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↗

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↗

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↗

arXiv2020

Subdivisions of Shellable Complexes

In geometric, algebraic, and topological combinatorics, the unimodality of combinatorial generating polynomials is frequently studied. Unimodality follows when the polynomial is (real) stable, a property often deduced via the theory of interlacing polynomials. Many of the open questions on stability and unimodality of polynomials pertain to the enumeration of faces of cell complexes. In this paper, we relate the theory of interlacing polynomials to the shellability of cell complexes. We first derive a sufficient condition for stability of the $h$-polynomial of a subdivision of a shellable complex. To apply it, we generalize the notion of reciprocal domains for convex embeddings of polytopes to abstract polytopes and use this generalization to define the family of stable shellings of a polytopal complex. We characterize the stable shellings of cubical and simplicial complexes, and apply this theory to answer a question of Brenti and Welker on barycentric subdivisions for the well-known cubical polytopes. We also give a positive solution to a problem of Mohammadi and Welker on edgewise subdivisions of cell complexes. We end by relating the family of stable line shellings to the combinatorics of hyperplane arrangements. We pose related questions, answers to which would resolve some long-standing problems while strengthening ties between the theory of interlacing polynomials and the combinatorics of hyperplane arrangements.

math.CO↗

arXiv2019

Classification of crescent configurations

Let $n$ points be in crescent configurations in $\mathbb{R}^d$ if they lie in general position in $\mathbb{R}^d$ and determine $n-1$ distinct distances, such that for every $1 \leq i \leq n-1$ there is a distance that occurs exactly $i$ times. Since Erdős' conjecture in 1989 on the existence of $N$ sufficiently large such that no crescent configurations exist on $N$ or more points, he, Pomerance, and Palásti have given constructions for $n$ up to $8$ but nothing is yet known for $n \geq 9$. Most recently, Burt et. al. had proven that a crescent configuration on $n$ points exists in $\mathbb{R}^{n-2}$ for $n \geq 3$. In this paper, we study the classification of these configurations on $4$ and $5$ points through graph isomorphism and rigidity. Our techniques, which can be generalized to higher dimensions, offer a new viewpoint on the problem through the lens of distance geometry and provide a systematic way to construct crescent configurations.

math.CO↗

arXiv2018

Summand minimality and asymptotic convergence of generalized Zeckendorf decompositions

Given a recurrence sequence $H$, with $H_n = c_1 H_{n-1} + \dots + c_t H_{n-t}$ where $c_i \in \mathbb{N}_0$ for all $i$ and $c_1, c_t \geq 1$, the generalized Zeckendorf decomposition (gzd) of $m \in \mathbb{N}_0$ is the unique representation of $m$ using $H$ composed of blocks lexicographically less than $σ= (c_1, \dots, c_t)$. We prove that the gzd of $m$ uses the fewest number of summands among all representations of $m$ using $H$, for all $m$, if and only if $σ$ is weakly decreasing. We develop an algorithm for moving from any representation of $m$ to the gzd, the analysis of which proves that $σ$ weakly decreasing implies summand minimality. We prove that the gzds of numbers of the form $v_0 H_n + \dots + v_\ell H_{n-\ell}$ converge in a suitable sense as $n \to \infty$, furthermore we classify three distinct behaviors for this convergence. We use this result, together with the irreducibility of certain families of polynomials, to exhibit a representation with fewer summands than the gzd if $σ$ is not weakly decreasing.

math.NT↗

arXiv2016

Random Matrix Ensembles with Split Limiting Behavior

We introduce a new family of $N\times N$ random real symmetric matrix ensembles, the $k$-checkerboard matrices, whose limiting spectral measure has two components which can be determined explicitly. All but $k$ eigenvalues are in the bulk, and their behavior, appropriately normalized, converges to the semi-circle as $N\to\infty$; the remaining $k$ are tightly constrained near $N/k$ and their distribution converges to the $k \times k$ hollow GOE ensemble (this is the density arising by modifying the GOE ensemble by forcing all entries on the main diagonal to be zero). Similar results hold for complex and quaternionic analogues. We isolate the two regimes by using matrix perturbation results and a nonstandard weight function for the eigenvalues, then derive their limiting distributions using a modification of the method of moments and analysis of the resulting combinatorics.

math-ph↗

arXiv2016

A Decomposition of Parking Functions by Undesired Spaces

There is a well-known bijection between parking functions of a fixed length and maximal chains of the noncrossing partition lattice which we can use to associate to each set of parking functions a poset whose Hasse diagram is the union of the corresponding maximal chains. We introduce a decomposition of parking functions based on the largest number omitted and prove several theorems about the corresponding posets. In particular, they share properties with the noncrossing partition lattice such as local self-duality, a nice characterization of intervals, a readily computable Möbius function, and a symmetric chain decomposition. We also explore connections with order complexes, labeled Dyck paths, and rooted forests.

math.CO↗