arXiv subjects

Andrés R. Vindas-Meléndez

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

32 records · Page 2Linked to original sources

arXiv2024

Combinatorics of generalized parking-function polytopes

For $\mathbf{b}=(b_1,\dots,b_n)\in \mathbb{Z}_{>0}^n$, a $\mathbf{b}$-parking function is defined to be a sequence $(β_1,\dots,β_n)$ of positive integers whose nondecreasing rearrangement $β'_1\leq β'_2\leq \cdots \leq β'_n$ satisfies $β'_i\leq b_1+\cdots + b_i$. 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 $\mathbb{R}^n$. Geometric properties of $\mathfrak{X}_n(\mathbf{b})$ were previously explored in the specific case where $\mathbf{b}=(a,b,b,\dots,b)$ and were shown to generalize those of the classical parking-function polytope. In this work, we study $\mathfrak{X}_n(\mathbf{b})$ in full generality. We present a minimal inequality and vertex description for $\mathfrak{X}_n(\mathbf{b})$, prove it is a generalized permutahedron, and study its $h$-polynomial. Furthermore, we investigate $\mathfrak{X}_n(\mathbf{b})$ through the perspectives of building sets and polymatroids, allowing us to identify its combinatorial types and obtain bounds on its combinatorial and circuit diameters.

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arXiv2023

Combinatorial Results on Barcode Lattices

A barcode is a finite multiset of intervals on the real line. Jaramillo-Rodriguez (2023) previously defined a map from the space of barcodes with a fixed number of bars to a set of multipermutations, which presented new combinatorial invariants on the space of barcodes. A partial order can be defined on these multipermutations, resulting in a class of posets known as combinatorial barcode lattices. In this paper, we provide a number of equivalent definitions for the combinatorial barcode lattice, show that its Möbius function is a restriction of the Möbius function of the symmetric group under the weak Bruhat order, and show its ground set is the Jordan-Hölder set of a labeled poset. Furthermore, we obtain formulas for the number of join-irreducible elements, the rank-generating function, and the number of maximal chains of combinatorial barcode lattices. Lastly, we make connections between intervals in the combinatorial barcode lattice and certain classes of matchings.

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arXiv2023

On the critical group of hinge graphs

Let $G$ be a finite, connected, simple graph. The critical group $K(G)$, also known as the sandpile group, is the torsion subgroup of the cokernel of the graph Laplacian $\operatorname{cok}(L)$. We investigate a family of graphs with relatively simple non-cyclic critical group with an end goal of understanding whether multiple divisors, i.e., formal linear combinations of vertices of $G$, generate $K(G)$. These graphs, referred to as hinge graphs, can be intuitively understood by taking multiple base shapes and ``gluing" them together by a single shared edge and two corresponding shared vertices. In the case where all base shapes are identical, we compute the explicit structure of the critical group. Additionally, we compute the order of three special divisors. We prove the structure of the critical group of hinge graphs when variance in the number of vertices of each base shape is allowed, generalizing many of the aforementioned results.

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arXiv2023

Generalized parking function polytopes

A classical parking function of length $n$ is a list of positive integers $(a_1, a_2, \ldots, a_n)$ whose nondecreasing rearrangement $b_1 \leq b_2 \leq \cdots \leq b_n$ satisfies $b_i \leq i$. The convex hull of all parking functions of length $n$ is an $n$-dimensional polytope in $\mathbb{R}^n$, which we refer to as the classical parking function polytope. Its geometric properties have been explored in (Amanbayeva and Wang 2022) in response to a question posed in (Stanley 2020). We generalize this family of polytopes by studying the geometric properties of the convex hull of $\mathbf{x}$-parking functions for $\mathbf{x}=(a,b,\dots,b)$, which we refer to as $\mathbf{x}$-parking function polytopes. We explore connections between these $\mathbf{x}$-parking function polytopes, the Pitman-Stanley polytope, and the partial permutahedra of (Heuer and Striker 2022). In particular, we establish a closed-form expression for the volume of $\mathbf{x}$-parking function polytopes. This allows us to answer a conjecture of (Behrend et al. 2022) and also obtain a new closed-form expression for the volume of the convex hull of classical parking functions as a corollary.

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arXiv2023

Ehrhart Theory of Paving and Panhandle Matroids

We show that the base polytope $P_M$ of any paving matroid $M$ can be systematically obtained from a hypersimplex by slicing off certain subpolytopes, namely base polytopes of lattice path matroids corresponding to panhandle-shaped Ferrers diagrams. We calculate the Ehrhart polynomials of these matroids and consequently write down the Ehrhart polynomial of $P_M$, starting with Katzman's formula for the Ehrhart polynomial of a hypersimplex. The method builds on and generalizes Ferroni's work on sparse paving matroids. Combinatorially, our construction corresponds to constructing a uniform matroid from a paving matroid by iterating the operation of stressed-hyperplane relaxation introduced by Ferroni, Nasr, and Vecchi, which generalizes the standard matroid-theoretic notion of circuit-hyperplane relaxation. We present evidence that panhandle matroids are Ehrhart positive and describe a conjectured combinatorial formula involving chain forests and Eulerian numbers from which Ehrhart positivity of panhandle matroids will follow. As an application of the main result, we calculate the Ehrhart polynomials of matroids associated with Steiner systems and finite projective planes, and show that they depend only on their design-theoretic parameters: for example, while projective planes of the same order need not have isomorphic matroids, their base polytopes must be Ehrhart equivalent.

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arXiv2023

The Mathematics of Mathematics: Using Mathematics and Data Science to Analyze the Mathematical Sciences Community and Enhance Social Justice

We present and discuss a curated selection of recent literature related to the application of quantitative techniques, tools, and topics from mathematics and data science that have been used to analyze the mathematical sciences community. We engage in this project with a focus on including research that highlights, documents, or quantifies (in)equities that exist in the mathematical sciences, specifically, and STEM (science, technology, engineering, and mathematics) more broadly. We seek to enhance social justice in the mathematics and data science communities by providing numerous examples of the ways in which the mathematical sciences fails to meet standards of equity, equal opportunity and inclusion. We introduce the term ``mathematics of Mathematics" for this project, explicitly building upon the growing, interdisciplinary field known as ``Science of Science" to interrogate, investigate, and identify the nature of the mathematical sciences itself. We aim to promote, provide, and posit sources of productive collaborations and we invite interested researchers to contribute to this developing body of work.

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arXiv2023

On definitions of "mathematician"

The definition of who is or what makes a ``mathematician" is an important and urgent issue to be addressed in the mathematics community. Too often, a narrower definition of who is considered a mathematician (and what is considered mathematics) is used to exclude people from the discipline -- both explicitly and implicitly. However, using a narrow definition of a mathematician allows us to examine and challenge systemic barriers that exist in certain spaces of the community. This paper explores and illuminates tensions between narrow and broad definitions and how they can be used to promote both inclusion and exclusion simultaneously. In this article, we present a framework of definitions based on identity, function, and qualification and exploring several different meanings of ``mathematician". By interrogating various definitions, we highlight their risks and opportunities, with an emphasis on implications for broadening and/or narrowing participation of underrepresented groups.

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arXiv2021

Enumerating $k$-Naples Parking Functions Through Catalan Objects

This paper studies a generalization of parking functions named $k$-Naples parking functions, where backward movement is allowed. One consequence of backward movement is that the number of ascending $k$-Naples is not the same as the number of descending $k$-Naples. This paper focuses on generalizing the bijections of ascending parking functions with combinatorial objects enumerated by the Catalan numbers in the setting of both ascending and descending $k$-Naples parking functions. These combinatorial objects include Dyck paths, binary trees, triangulations of polygons, and non-crossing partitions. Using these bijections, we enumerate both ascending and descending $k$-Naples parking functions.

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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.

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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.

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arXiv2020

Counting $k$-Naples parking functions through permutations and the $k$-Naples area statistic

We recall that the $k$-Naples parking functions of length $n$ (a generalization of parking functions) are defined by requiring that a car which finds its preferred spot occupied must first back up a spot at a time (up to $k$ spots) before proceeding forward down the street. Note that the parking functions are the specialization of $k$ to $0$. For a fixed $0\leq k\leq n-1$, we define a function $φ_k$ which maps a $k$-Naples parking function to the permutation denoting the order in which its cars park. By enumerating the sizes of the fibers of the map $φ_k$ we give a new formula for the number of $k$-Naples parking functions as a sum over the permutations of length $n$. We remark that our formula for enumerating $k$-Naples parking functions is not recursive, in contrast to the previously known formula of Christensen et al [CHJ+20]. It can be expressed as the product of the lengths of particular subsequences of permutations, and its specialization to $k=0$ gives a new way to describe the number of parking functions of length $n$. We give a formula for the sizes of the fibers of the map $φ_0$, and we provide a recurrence relation for its corresponding logarithmic generating function. Furthermore, we relate the $q$-analog of our formula to a new statistic that we denote $\texttt{area}_k$ and call the $k$-Naples area statistic, the specialization of which to $k=0$ gives the $\texttt{area}$ statistic on parking functions.

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arXiv2020

The equivariant Ehrhart theory of the permutahedron

Equivariant Ehrhart theory enumerates the lattice points in a polytope with respect to a group action. Answering a question of Stapledon, we describe the equivariant Ehrhart theory of the permutahedron, and we prove his Effectiveness Conjecture in this special case.

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arXiv2019

The equivariant volumes of the permutahedron

We consider the action of the symmetric group $S_n$ on the permutahedron $Π_n$. We prove that if $σ$ is a permutation of $S_n$ which has $m$ cycles of lengths $l_1, \ldots, l_m$, then the subpolytope of $Π_n$ fixed by $σ$ has normalized volume $n^{m-2} \gcd(l_1, \ldots, l_m)$.

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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).

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