arXiv subjects

Andrés R. Vindas-Meléndez

Publications and source records attributed to Andrés R. Vindas-Meléndez.

At least 19 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.

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arXiv2026

Lattice slices, Ehrhart polynomials, and magic positivity of generalized parking-function polytopes

For $\mathbf{b}=(b_1,\dots,b_n)\in\mathbb{Z}_{>0}^n$, a $\mathbf{b}$-parking function is a sequence $(β_1,\dots,β_n)$ of positive integers whose nondecreasing rearrangement $β_1'\leβ_2'\le\cdots\leβ_n'$ satisfies $β_i'\le b_1+\cdots+b_i$. The $\mathbf{b}$-parking-function polytope $\mathfrak{X}_n(\mathbf{b})$ is the convex hull of all $\mathbf{b}$-parking functions of length $n$ in $\mathbb{R}^n$. We prove that every lattice slice of $\mathfrak{X}_n(\mathbf{b})$, obtained by fixing one coordinate at an integer value, is itself a $\mathbf{b}'$-parking-function polytope of one dimension less, with an explicit parameter vector $\mathbf{b}'$; this yields a recursion for the number of lattice points of $\mathfrak{X}_n(\mathbf{b})$. We further show that every dilate of a $\mathbf{b}$-parking-function polytope is a translate of another such polytope, that the number of lattice points is a polynomial function of $\mathbf{b}$, and we deduce an explicit formula for the Ehrhart polynomial of $\mathfrak{X}_n(\mathbf{b})$ for arbitrary $\mathbf{b}$ as a finite sum indexed by draconian sequences, resolving a problem of Hanada, Lentfer, and Vindas-Meléndez; an equivalent formula was recently obtained, independently, by Liu and Thawinrak in a closely related setting. In the special case $\mathbf{b}=(a,b,\dots,b)$, we obtain an explicit closed form and a generating function for the Ehrhart polynomial. As an application, we classify magic positivity in the two-parameter family $\mathfrak{X}_n(a,b)=\mathfrak{X}_n(a,b,\dots,b)$: the polytope $\mathfrak{X}_n(a,b)$ is magic positive if and only if $(n,a,b)\ne(2,1,1)$. Thus, we answer a problem posed by Ferroni and Higashitani for $\mathfrak{X}_n(a,b)$. Our result extends recent work of Liu and Zhang on partial permutahedra and leads us to conjecture that magic positivity holds for every $\mathfrak{X}_n(\mathbf{b})$ with $n\ge3$.

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arXiv2026

$q$-Chromatic polynomials

We study a $q$-version of the chromatic polynomial of a given graph $G=(V,E)$, namely, \[ χ_G^λ(q,n) \ := \sum_{\substack{\text{proper colorings}\\ c\,:\,V\to[n]}} q^{ \sum_{ v \in V } λ_v c(v) }, \] where $λ\in \mathbb{Z}_{>0}^V$ is a fixed linear form. Via work of Chapoton (2016) on $q$-Ehrhart polynomials, $χ_G^λ(q,n)$ 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 $χ_G^λ(q,n)$ 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↗

arXiv2026

Generalized snake posets, order polytopes, and lattice-point enumeration

Building from the work of von Bell et al.~(2022), we study the Ehrhart theory of order polytopes arising from a special class of distributive lattices, known as generalized snake posets. We present arithmetic properties satisfied by the Ehrhart polynomials of order polytopes of generalized snake posets along with a computation of their Gorenstein index. Then we give a combinatorial description of the chain polynomial of generalized snake posets as a direction to obtain the $h^*$-polynomial of their associated order polytopes. Additionally, we present explicit formulae for the $h^*$-polynomial of the order polytopes of the two extremal examples of generalized snake posets, namely the ladder and regular snake poset. We then provide a recursive formula for the $h^*$-polynomial of any generalized snake posets and show that the $h^*$-vectors are entry-wise bounded by the $h^*$-vectors of the two extremal cases.

math.CO↗

arXiv2026

Building with Blocks: Enumerating Polyforms on Tilings

In areas as diverse as contemporary art, play structures, climbing equipment, and modular construction toys, we see the presence of building block-like polyhedral complexes, which are generalizations of the pieces in the game Tetris. We give an algorithm for counting the number of $n$-celled structures on polygonal and polyhedral cells of certain periodic two- and three-dimensional tilings; moreover, we count these structures up to translations, rotations, and reflections of the tiling. We describe this algorithm with respect to structures in the snub square tiling, provide numerical data related to existing three-dimensional art and structures, and suggest puzzles based on these constructions.

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arXiv2026

A Discrete-Time Model of the Academic Pipeline in Mathematical Sciences with Constrained Hiring in the United States

The field of the mathematical sciences relies on a continuous academic pipeline in which individuals progress from undergraduate study through graduate training and postdoctoral program to long term faculty employment. National statistics report trends in bachelor's, master's, and doctoral degree awards, but these data alone do not explain how individuals move through the academic system or how structural constraints shape downstream career outcomes. Persistent growth in postdoctoral appointments alongside relatively stable faculty employment indicates that degree production alone is insufficient to characterize workforce dynamics. In this study, we develop a discrete time compartmental model of the academic pipeline in the field of the mathematical sciences that links observed degree flows to latent population stocks. Undergraduate and graduate populations are reconstructed directly from nationally reported degree data, allowing postdoctoral and faculty dynamics to be examined under completion, exit, and hiring processes. Advancement to faculty positions is modeled as vacancy limited, with competition for permanent positions depending on downstream population size. Numerical simulations show that increases in degree inflow do not translate into proportional faculty growth when hiring is constrained by limited turnover. Instead, excess supply accumulates primarily at the postdoctoral stage, leading to sustained congestion and elevated competition. Sensitivity analyses indicate that long run workforce outcomes are governed mainly by faculty exit rates and hiring capacity rather than by degree production alone. These results demonstrate the central role of vacancy limited hiring in shaping academic career trajectories within the field of the mathematical sciences.

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arXiv2026

A dynamical model of the U.S. mathematics graduate degree pipeline

We present a latent-stock compartmental framework for modeling degree production systems when only completion flows, rather than enrollments, are observed. Applied to U.S.\ mathematics degrees from 1969 to 2017, the model treats master's and PhD populations as latent compartments -- unobserved state variables that are inferred indirectly because they generate the observed completion flows -- with time-varying routing fractions and completion hazards. Using information-criterion model comparison across a grid of specifications, we find strong support for smooth nonlinear time variation in routing fractions and hazards, while models with explicit international forcing are disfavored. The preferred model achieves a log-scale root mean squared error of approximately 0.036, corresponding to a typical multiplicative error of about 4\% in fitted degree counts, and highlights key structural shifts in the graduate pipeline: the master's pathway became increasingly central to PhD production through the late twentieth century before weakening, while direct bachelor's-to-PhD entry remained small but persistent. Estimated completion hazards for both degrees rise over time, indicating faster effective turnover in the graduate compartments. Methodologically, our main contribution is a latent stock dynamical approach that recasts linked degreecompletion time series as a coherent stock-flow system when intermediate enrollments are unobserved, making explicit both what features of pipeline dynamics are identifiable from completion data alone and what limitations such data impose.

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arXiv2025

The Ehrhart polynomial of a matroid specializes to the beta invariant

We show that the linear coefficient of the Ehrhart polynomial of a matroid base polytope evaluated at $t-1$ is equal to, up to normalization, the $β$-invariant of the matroid. This yields a lattice-point counting formula for the $β$-invariant and establishes a new and unexpected positivity property of Ehrhart polynomials of matroid polytopes.

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arXiv2025

Unitary actions and equivariant volumes of symmetric edge polytopes

The symmetric edge polytope ($\mathrm{SEP}$) of a finite simple graph $G$ is a centrally symmetric lattice polytope whose vertices are defined by the edges of the graph. Among the information encoded by these polytopes are the symmetries of the graph, which appear as symmetries of the polytope. We describe the rigid symmetries of these polytopes, and show that $\mathrm{SEP}$s are unitarily equivalent exactly when their associated graphs are isomorphic. We then find an explicit relationship between the relative volumes of the subsets of the symmetric edge polytope $\mathrm{SEP}$ fixed by the natural action of symmetric group elements and the symmetric edge polytopes of smaller graphs to which the subsets are linearly equivalent. We also provide a vertex description of the fixed polytopes and find a description of the symmetric edge polytopes to which they are equivalent, in terms of contractions of the graph $G$ induced by the cycle decompositions of the permutations under which the subsets are fixed. Specializations of our results provide equivalence and volume relationships for fixed polytopes of symmetric edge polytopes of complete graphs (equivalently, for fixed polytopes of root polytopes of type $A_n$), and describe the symmetry group of this family of polytopes.

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arXiv2025

On the geometry of stack-sorting simplices

We show that all stack-sorting polytopes are simplices. Furthermore, we show that the stack-sorting polytopes generated from $Ln1$ permutations have relative volume 1. We establish an upper bound for the number of lattice points in a stack-sorting polytope. In particular, stack-sorting polytopes generated from $2Ln1$ permutations have no interior points.

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arXiv2025

Ehrhart Bounds for Panhandle and Paving Matroids Through Enumeration of Chain Forests

Panhandle matroids are a specific family of lattice-path matroids corresponding to panhandle-shaped Ferrers diagrams. Their matroid polytopes are the subpolytopes carved from a hypersimplex to form matroid polytopes of paving matroids. It has been an active area of research to determine which families of matroid polytopes are Ehrhart positive. We prove Ehrhart positivity for panhandle matroid polytopes, thus confirming a conjecture of Hanely, Martin, McGinnis, Miyata, Nasr, Vindas-Meléndez, and Yin (2023). Another standing conjecture posed by Ferroni (2022) asserts that the coefficients of the Ehrhart polynomial of a connected matroid are bounded above by those of the corresponding uniform matroid. We prove Ferroni's conjecture for paving matroids -- a class conjectured to asymptotically contain all matroids. These results follow from purely enumerative statements, the main one being conjectured by Hanely et. al concerning the enumeration of a certain class of ordered chain forests.

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arXiv2025

A combinatorial proof of a symmetry for a refinement of the Narayana numbers

We establish a tantalizing symmetry of certain numbers refining the Narayana numbers. In terms of Dyck paths, this symmetry is interpreted in the following way: if $w_{n,k,m}$ is the number of Dyck paths of semilength $n$ with $k$ occurrences of $UD$ and $m$ occurrences of $UUD$, then $w_{2k+1,k,m}=w_{2k+1,k,k+1-m}$. We give a combinatorial proof of this fact, relying on the cycle lemma, and showing that the numbers $w_{2k+1,k,m}$ are multiples of the Narayana numbers. We prove a more general fact establishing a relationship between the numbers $w_{n,k,m}$ and a family of generalized Narayana numbers due to Callan. A closed-form expression for the even more general numbers $w_{n,k_{1},k_{2},\ldots , k_{r}}$ counting the semilength-$n$ Dyck paths with $k_{1}$ $UD$-factors, $k_{2}$ $UUD$-factors, $\ldots$ , and $k_{r}$ $U^{r}D$-factors is also obtained, as well as a more general form of the discussed symmetry for these numbers in the case when all rise runs are of certain minimal length. Finally, we investigate properties of the polynomials $W_{n,k}(t)= \sum_{m=0}^k w_{n,k,m} t^m$, including real-rootedness, $γ$-positivity, and a symmetric decomposition.

math.CO↗

arXiv2025

Stack-sorting simplices: geometry and lattice-point enumeration

We initiate the study of subpolytopes of the permutahedron that arise as the convex hulls of stack-sorting on permutations. We primarily focus on $Ln1$ permutations, i.e., permutations of length $n$ whose penultimate and last entries are $n$ and $1$, respectively. First, we present some enumerative results on $Ln1$ permutations. Then we show that the polytopes that arise from stack-sorting on $Ln1$ permutations are simplices and proceed to study their geometry and lattice-point enumeration. In addition, we pose questions and problems for further investigation. Particular focus is then taken on the $Ln1$ permutation $23\cdots n1$. We show that the convex hull of all its iterations through the stack-sorting algorithm shares the same lattice-point enumerator as that of the $(n-1)$-dimensional unit cube and lecture-hall simplex. Lastly, we detail some results on the real lattice-point enumerator for variations of the simplices arising from stack-sorting on the permutation $23\cdots n1$. This then allows us to show that those simplices are Gorenstein of index $2$.

math.CO↗

arXiv2025

Local $h^*$-polynomials for one-row Hermite normal form simplices

The local $h^*$-polynomial of a lattice polytope is an important invariant arising in Ehrhart theory. Our focus is on lattice simplices presented in Hermite normal form with a single non-trivial row. We prove that when the off-diagonal entries are fixed, the distribution of coefficients for the local $h^*$-polynomial of these simplices has a limit as the normalized volume goes to infinity. Further, this limiting distribution is determined by the coefficients for a particular choice of normalized volume. We also provide an analysis of two specific families of such simplices to illustrate and motivate our main result.

math.CO↗

arXiv2024

Generating Trees and Fibonacci Polyominoes

We study a new class of polyominoes, called $p$-Fibonacci polyominoes, defined using $p$-Fibonacci words. We enumerate these polyominoes by applying generating functions to capture geometric parameters such as area, semi-perimeter, and the number of inner points. Additionally, we establish bijections between Fibonacci polyominoes, binary Fibonacci words, and integer compositions with certain restrictions.

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

Matching polytopes, Gorensteinness, and the integer decomposition property

The matching polytope of a graph $G$ is the convex hull of the indicator vectors of the matchings on $G$. We characterize the graphs whose associated matching polytopes are Gorenstein, and then prove that all Gorenstein matching polytopes possess the integer decomposition property. As a special case study, we examine the matching polytopes of wheel graphs and show that they are not Gorenstein, but do possess the integer decomposition property.

math.CO↗

arXiv2024

Quantifying and Documenting Inequity in PhD-granting Mathematical Sciences Departments in the United States

We provide an example of the application of quantitative techniques, tools, and topics from mathematics and data science to analyze the mathematics community itself in order to quantify and document inequity in our discipline. This work is a contribution to the new and growing interdisciplinary field recently termed "mathematics of Mathematics," or "MetaMath." Using data about PhD-granting institutions in the United States and publicly available funding data from the National Science Foundation, we highlight inequalities in departments at U.S. institutions of higher education that produce PhDs in the mathematical sciences. Specifically, we determine that a small fraction of mathematical sciences departments receive a large majority of federal funding awarded to support mathematics in the United States. Additionally, we identify the extent to which women faculty members are underrepresented in mathematical sciences PhD-granting institutions in the United States. We also show that this underrepresentation of women faculty is even more pronounced in departments that received more federal grant funding.

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