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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., Comment: 27 pages, 12 figures, comments welcome!

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., Comment: 23 pages, 11 figures, Comments welcomed!

Polar activity is shown here to induce a progressive local deformation of linear polymer chains, making a clear distinction between head and tail, while the overall chain conformation gets more compact. This breakdown of self-similarity, provoked by the accumulated tension on the polymer backbone induced by the activity, is shown to occur both in the absence and presence of inertia, although it is stronger in the latter case. Other properties like the relaxation time and diffusion are also largely influenced by the presence of a polar activity., Comment: 6 pages, 4 figures

We introduce and 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 $\lambda \in \mathbb{Z}^V$ 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., Comment: 16 pages, 1 figure

For $\mathbf{b}=(b_1,\dots,b_n)\in \mathbb{Z}_{>0}^n$, a $\mathbf{b}$-parking function is defined to be a sequence $(\beta_1,\dots,\beta_n)$ of positive integers whose nondecreasing rearrangement $\beta'_1\leq \beta'_2\leq \cdots \leq \beta'_n$ satisfies $\beta'_i\leq b_1+\cdots + b_i$. 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., Comment: 27 pages, 4 figures, Comments welcomed!

We analyze correlation structures in financial markets by coarse graining the Pearson correlation matrices according to market sectors to obtain Guhr matrices using Guhr's correlation method according to Ref. [P. Rinn {\it et. al.}, Europhysics Letters 110, 68003 (2015)]. We compare the results for the evolution of market states and the corresponding transition matrices with those obtained using Pearson correlation matrices. The behavior of market states is found to be similar for both the coarse grained and Pearson matrices. However, the number of relevant variables is reduced by orders of magnitude., Comment: 15 pages, 16 figures, version as accepted for publication in Physica Scripta (2024)

We introduce a new PAC-Bayes oracle bound for unbounded losses. This result can be understood as a PAC-Bayesian version of the Cram\'er-Chernoff bound. The proof technique relies on controlling the tails of certain random variables involving the Cram\'er transform of the loss. We highlight several applications of the main theorem. First, we show that our result naturally allows exact optimization of the free parameter on many PAC-Bayes bounds. Second, we recover and generalize previous results. Finally, we show that our approach allows working with richer assumptions that result in more informative and potentially tighter bounds. In this direction, we provide a general bound under a new ``model-dependent bounded CGF" assumption from which we obtain bounds based on parameter norms and log-Sobolev inequalities. All these bounds can be minimized to obtain novel posteriors., Comment: Updated Section 5

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 M\"obius function is a restriction of the M\"obius function of the symmetric group under the weak Bruhat order, and show its ground set is the Jordan-H\"older 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., Comment: 15 pages, 4 figures, 1 table

Panhandle matroids are a specific lattice-path matroid 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-Mel\'endez, 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. Our proofs rely solely on combinatorial techniques which involve determining intricate interpretations of certain set partitions., Comment: 26 pages

The cold posterior effect (CPE) (Wenzel et al., 2020) in Bayesian deep learning shows that, for posteriors with a temperature $T<1$, the resulting posterior predictive could have better performances than the Bayesian posterior ($T=1$). As the Bayesian posterior is known to be optimal under perfect model specification, many recent works have studied the presence of CPE as a model misspecification problem, arising from the prior and/or from the likelihood function. In this work, we provide a more nuanced understanding of the CPE as we show that misspecification leads to CPE only when the resulting Bayesian posterior underfits. In fact, we theoretically show that if there is no underfitting, there is no CPE., Comment: 9 pages, 3 figures, ICLR 2024

The local $h^*$-polynomial of a lattice polytope is an important invariant arising in Ehrhart theory. Our focus in this work 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 relatively small normalized volume. We also provide a thorough analysis of two specific families of such simplices, to illustrate and motivate our main result., Comment: minor edits

We study the polytopes that arise from the convex hulls of stack-sorting on particular permutations. We show that they are simplices and proceed to study their geometry and lattice-point enumeration. First, we prove some enumerative results on $Ln1$ permutations, i.e., permutations of length $n$ whose penultimate and last entries are $n$ and $1$, respectively. Additionally, we then focus on a specific permutation, which we call $L'n1$, and show that the convex hull of all its iterations through the stack-sorting algorithm share 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 $L'n1$ permutations. This then allows us to show that $L'n1$ simplices are Gorenstein of index $2$., Comment: 25 pages, 5 figures, 1 table, comments welcomed!

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., Comment: 22 pages, five figures, four tables

The twisted hexagonal boron nitride (hBN) bilayer has demonstrated exceptional properties, particularly the existence of electronic flat bands without needing a magic angle, suggesting strong excitonic effects. Therefore, a systematic approach is presented to study the excitonic properties of twisted AA' hBN using the Bethe-Salpeter equation based on single-particle tight-binding wave functions. These are provided by a one-particle Hamiltonian that is parameterized to describe the main features of {\it ab initio} calculations. The Bethe-Salpeter equation is then solved in the so-called excitonic transition representation, which significantly reduces the problem dimensionality by exploiting the system's symmetries. Consequently, the excitonic energies and the excitonic wave functions are obtained from the direct diagonalization of the effective two-particle Hamiltonian of the Bethe-Salpeter equation. We have studied rotation angles as low as $7.34^{\circ}$. The model allows the study of commensurate and incommensurate moir\'e patterns at much lower computational cost than the {\it ab initio} version of the Bethe-Salpeter equation. Here, using the model and effective screening of the Keldysh type, we could obtain the absorption spectra and characterize the excitonic properties of twisted hBN bilayers for different rotation angles, demonstrating how this property affects the excitonic energies and localizations of their wavefunctions., Comment: 32 pages, 16 figures

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., Comment: 18 pages, comments welcomed

This paper introduces a distribution-dependent PAC-Chernoff bound that exhibits perfect tightness for interpolators, even within over-parameterized model classes. This bound, which relies on basic principles of Large Deviation Theory, defines a natural measure of the smoothness of a model, characterized by simple real-valued functions. Building upon this bound and the new concept of smoothness, we present an unified theoretical framework revealing why certain interpolators show an exceptional generalization, while others falter. We theoretically show how a wide spectrum of modern learning methodologies, encompassing techniques such as $\ell_2$-norm, distance-from-initialization and input-gradient regularization, in combination with data augmentation, invariant architectures, and over-parameterization, collectively guide the optimizer toward smoother interpolators, which, according to our theoretical framework, are the ones exhibiting superior generalization performance. This study shows that distribution-dependent bounds serve as a powerful tool to understand the complex dynamics behind the generalization capabilities of over-parameterized interpolators., Comment: 56 pages, 11 figures, Pre-print

We introduce a novel semantics for a multi-agent epistemic operator of knowing how, based on an indistinguishability relation between plans. Our proposal is, arguably, closer to the standard presentation of knowing that modalities in classical epistemic logic. We study the relationship between this new semantics and previous approaches, showing that our setting is general enough to capture them. We also study the logical properties of the new semantics. First, we define a sound and complete axiomatization. Second, we define a suitable notion of bisimulation and prove correspondence theorems. Finally, we investigate the computational complexity of the model checking and satisfiability problems for the new logic., Comment: arXiv admin note: text overlap with arXiv:2106.11492

Data science for social justice (DS4SJ) is data-scientific work that supports the liberation of oppressed and marginalized people. By nature, this work lies at the intersection of technical scholarship and activist practice. We discuss this growing efforts in DS4SJ within the broad mathematics community. We begin by defining terms and offering a series of guiding principles for engaging in critical data science work, providing examples of how these principles play out in practice. We then highlight the roles that DS4SJ can play in the scholarship and pedagogy of practicing mathematicians. We focus in particular on the engagement of early-career mathematicians in DS4SJ, which we illustrate through a series of four personal vignettes. While the primary aim of DS4SJ is to achieve impact for marginalized communities, we also argue that engagement with DS4SJ can benefit the entire mathematical ecosystem, including researchers, instructors, students, departments, institutes, and professional societies. We close with reflections on how these various actors can support ongoing efforts in data science for social justice.

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., Comment: 21 pages, 2 figures

We investigate a tantalizing symmetry on Catalan objects. 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 two proofs of this symmetry: an algebraic proof using generating functions, and a combinatorial proof which makes heavy use of the cycle lemma and an alternate interpretation of the numbers $w_{n,k,m}$ using plane trees. In particular, our combinatorial proof expresses the numbers $w_{2k+1,k,m}$ in terms of Narayana numbers, and we generalize this to a relationship between the numbers $w_{n,k,m}$ and a family of generalized Narayana numbers due to Callan. Some further generalizations and applications of our combinatorial proofs are explored. Finally, we investigate properties of the polynomials $W_{n,k}(t)= \sum_{m=0}^k w_{n,k,m} t^m$, including real-rootedness, $\gamma$-positivity, and a symmetric decomposition., Comment: 25 pages, 1 table, 6 figures, Comments welcome!

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., Comment: 29 pages, 3 figures, comments welcome!

In this work, we explore the dynamics of active entangled chains using molecular dynamics simulations of a modified Kremer-Grest model. The active chains are diluted in a mesh of very long passive linear chains, to avoid constraint release effects, and an active force is applied to the monomers in a way that it imparts a constant polar drift velocity along the primitive path. The simulation results show that, over a wide range of activity values, the conformational properties of the chains and the tubes are not affected, but the dynamics of the chains are severely modified. Despite not having an explicit tube, the simulations verify all the predictions of the theory about all possible observables very accurately, including a diffusion coefficient that becomes independent of the molecular weight at moderate values of the activity. Overall, this work provides novel information on the study of active entangled polymers, giving a route map for studying this phenomenon and an efficient way of obtaining a polar activity that reproduces the physics of the active reptation theory., Comment: 45 pages, 9 figures

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., Comment: 24 pages, 15 figures, Comments welcomed!

We revisit a model of semiflexible Gaussian chains proposed by Winkler \textit{et al}, solve the dynamics of the discrete description of the model and derive exact algebraic expressions for some of the most relevant dynamical observables, such as the mean-square displacement of individual monomers, the dynamic structure factor, the end-to-end vector relaxation and the shear stress relaxation modulus. The mathematical expressions are verified by comparing them with results from Brownian dynamics simulations, reporting an excellent agreement. Then, we generalize the model to linear polymer chains with arbitrary stiffness. In particular, we focus on the case of a linear polymer with stiffness that changes linearly from one end of the chain to the other, and study the same dynamical functions previously presented. We discuss different approaches to check whether a polymer has constant or heterogeneous stiffness along its contour. Overall, this work presents a new insight for a well known model for semiflexible chains and provides tools that can be exploited to compare the predictions of the model with simulations of coarse grained semiflexible polymers.

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., Comment: 30 pages, 1 Figure, to appear in Advances in Geometry

There is considerable anecdotal evidence suggesting that software engineers enjoy engaging in solving puzzles and other cognitive efforts. A tendency to engage in and enjoy effortful thinking is referred to as a person's 'need for cognition.' In this article we study the relationship between software engineers' personality traits and their need for cognition. Through a large-scale sample study of 483 respondents we collected data to capture the six 'bright' personality traits of the HEXACO model of personality, and three `dark' personality traits. Data were analyzed using several methods including a multiple Bayesian linear regression analysis. The results indicate that ca. 33% of variation in developers' need for cognition can be explained by personality traits. The Bayesian analysis suggests four traits to be of particular interest in predicting need for cognition: openness to experience, conscientiousness, honesty-humility, and emotionality. Further, we also find that need for cognition of software engineers is, on average, higher than in the general population, based on a comparison with prior studies. Given the importance of human factors for software engineers' performance in general, and problem solving skills in particular, our findings suggest several implications for recruitment, working behavior, and teaming.

Ensembles are widely used in machine learning and, usually, provide state-of-the-art performance in many prediction tasks. From the very beginning, the diversity of an ensemble has been identified as a key factor for the superior performance of these models. But the exact role that diversity plays in ensemble models is poorly understood, specially in the context of neural networks. In this work, we combine and expand previously published results in a theoretically sound framework that describes the relationship between diversity and ensemble performance for a wide range of ensemble methods. More precisely, we provide sound answers to the following questions: how to measure diversity, how diversity relates to the generalization error of an ensemble, and how diversity is promoted by neural network ensemble algorithms. This analysis covers three widely used loss functions, namely, the squared loss, the cross-entropy loss, and the 0-1 loss; and two widely used model combination strategies, namely, model averaging and weighted majority vote. We empirically validate this theoretical analysis with neural network ensembles.