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Along with his confirmation of the $e$-positivity of all cycle-chord graphs $\theta_{ab1}$, the third author conjectured the $e$-positivity of all theta graphs $\theta_{abc}$. In this paper, we establish the $e$-positivity of all clock graphs $\theta_{ab2}$ by using the composition method. The key idea is to investigate the fibers of certain partial reversal transformation on compositions with all parts at least $2$., Comment: 15 pages, 4 figures

We find a positive $e_I$-expansion for the chromatic symmetric function of KPKP graphs, which are graphs obtained by connecting a vertex in a complete graph with a vertex in the maximal clique of a lollipop graph by a path. This generalizes the positive $e_I$-expansion for the chromatic symmetric function of lollipops obtained by Tom, for that of KPK graphs obtained by Wang and Zhou, and as well for those of KKP graphs and PKP graphs obtained by Qi, Tang and Wang. As an application, we confirm the $e$-positivity of twinned lollipops, which belong to the graph family that is concerned by Stanley and Stembridge's $e$-positivity conjecture. We also discover the first positive $e_I$-expansion for the chromatic symmetric function of kayak paddle graphs which are formed by connecting a vertex on a cycle and a vertex on another cycle with a path. This refines the $e$-positivity of kayak paddle graphs which was obtained by Aliniaeifard, Wang, and van Willigenburg using an expanded version of Gebhard and Sagan's appendable $(e)$-positivity technique., Comment: 22 pages, 8 figures

Entity matching (EM), the task of identifying whether two descriptions refer to the same entity, is essential in data management. Traditional methods have evolved from rule-based to AI-driven approaches, yet current techniques using large language models (LLMs) often fall short due to their reliance on static knowledge and rigid, predefined prompts. In this paper, we introduce Libem, a compound AI system designed to address these limitations by incorporating a flexible, tool-oriented approach. Libem supports entity matching through dynamic tool use, self-refinement, and optimization, allowing it to adapt and refine its process based on the dataset and performance metrics. Unlike traditional solo-AI EM systems, which often suffer from a lack of modularity that hinders iterative design improvements and system optimization, Libem offers a composable and reusable toolchain. This approach aims to contribute to ongoing discussions and developments in AI-driven data management., Comment: 2 pages, compound ai systems 2024

We establish the $e$-positivity of spider graphs of the form $S(4m+2,\, 2m,\, 1)$, which was conjectured by Aliniaeifard, Wang and van Willigenburg. A key to our proof is the $e_I$-expansion formula of the chromatic symmetric function of paths due to Shareshian and Wachs, where the symbol~$I$ indicates integer compositions rather than partitions. Following the strategy of the divide-and-conquer technique, we pick out one or two positive $e_J$-terms for each negative $e_I$-term in an $e$-expression for the spiders, where $J$ are selected to be distinct compositions obtained by rearranging the parts of $I$.

We establish the $e$-positivity of all cycle-chord graphs by using the composition method that is developed by Zhou and the author very recently. Our method is much simpler than the $(e)$-positivity approach that is used for handling cycle-chords with girth at most $4$. We also provide a combinatorial interpretation of the $e_I$-coefficients of the chromatic symmetric functions of cycle-chords., Comment: 7 pages, 3 figures

For any integers $1\leq k\leq n$, we introduce a new family of parking functions called $k$-vacillating parking functions of length $n$. The parking rule for $k$-vacillating parking functions allows a car with preference $p$ to park in the first available spot in encounters among the parking spots numbered $p$, $p-k$, and $p+k$ (in that order and if those spots exists). In this way, $k$-vacillating parking functions are a modification of Naples parking functions, which allow for backwards movement of a car, and of $\ell$-interval parking functions, which allow a car to park in its preference or up to $\ell$ spots in front of its preference. Among our results, we establish a combinatorial interpretation for the numerator of the $n$th convergent of the continued fraction of $\sqrt{2}$, as the number of non-decreasing $1$-vacillating parking functions of length~$n$. Our main result gives a product formula for the enumeration of $k$-vacillating parking functions of length $n$ based on the number of $1$-vacillating parking functions of smaller length. We conclude with some directions for further research., Comment: 12 pages, 1 figure, to appear in Journal of Combinatorics

This work is inspired by Shareshian and Wachs's exquisite formula for the chromatic symmetric function of paths. We develop a composition method to unearth neat noncommutative analogs of chromatic symmetric functions. A symmetric function is $e$-positive if and only if it has a $\Lambda$-positive noncommutative analog. We bring to light short and sweet $\Lambda$-positive noncommutative analogs for the chromatic symmetric functions of tadpoles and barbells. Using these elegant formulas and the composition method, we discover a new family of $e$-positive graphs and call it hat graphs, which are the unicyclic graphs obtained by adding an edge to a path. We also obtain a compact ribbon Schur analog for cycles.

Language Models (LMs) have shown impressive performance in various natural language tasks. However, when it comes to natural language reasoning, LMs still face challenges such as hallucination, generating incorrect intermediate reasoning steps, and making mathematical errors. Recent research has focused on enhancing LMs through self-improvement using feedback. Nevertheless, existing approaches relying on a single generic feedback source fail to address the diverse error types found in LM-generated reasoning chains. In this work, we propose Multi-Aspect Feedback, an iterative refinement framework that integrates multiple feedback modules, including frozen LMs and external tools, each focusing on a specific error category. Our experimental results demonstrate the efficacy of our approach to addressing several errors in the LM-generated reasoning chain and thus improving the overall performance of an LM in several reasoning tasks. We see a relative improvement of up to 20% in Mathematical Reasoning and up to 18% in Logical Entailment., Comment: Accepted at EMNLP 2023 Main Conference, Camera Ready

Multi-criteria recommender systems can improve the quality of recommendations by considering user preferences on multiple criteria. One promising approach proposed recently is multi-criteria ranking, which uses Pareto ranking to assign a ranking score based on the dominance relationship between predicted ratings across criteria. However, applying Pareto ranking to all criteria may result in non-differentiable ranking scores. To alleviate this issue, we proposed a hybrid multi-criteria ranking method by using subsorting. More specifically, we utilize one ranking method as the major sorting approach, while we apply another preference ordering method as subsorting. Our experimental results on the OpenTable and Yahoo!Movies data present the advantages of this hybrid ranking approach. In addition, the experiments also reveal more insights about the sustainability of the multi-criteria ranking for top-N item recommendations.

We obtain the Schur positivity of spider graphs of the forms $S(a,2,1)$ and $S(a,4,1)$, which are considered to have the simpliest structures for which the Schur positivity was unknown. The proof outline has four steps. First, we find noncommutative analogs for the chromatic symmetric functions of the spider graphs $S(a,b,1)$. Secondly, we expand the analogs under the $\Lambda$- and $R$-bases, whose commutative images are the elementary and skew Schur symmetric functions, respectively. Thirdly, we recognize the Schur coefficients via the Littlewood--Richardson rule in terms of norms of multisets of Yamanouchi words. At last we establish the Schur positivity combinatorially together with the aid of computer assistance., Comment: 26 pages, with an appendix of 6 pages

In healthy sleepers, cortical alpha oscillations are present during the transition from wakefulness to sleep, and dissipate at sleep onset. For individuals with insomnia, alpha power is elevated during the wake-sleep transition and can persist throughout the night. Neuromodulation techniques using phase-locked stimulation have been put forth as alternatives to drugs for improving slow-wave sleep quality. Due to technical limitations, this approach has not been tested on faster frequency alpha oscillations. Here we examine the feasibility of using an endpoint-corrected version of the Hilbert Transform (ecHT) algorithm implemented on-device to measure alpha phase and deliver phase-locked auditory stimulation to modulate alpha and promote sleep initiation. First, the ecHT algorithm is implemented on a tabletop electroencephalogram (EEG) device and used to measure the timing of the auditory evoked response and its delivery at precise phases of the alpha oscillation. Secondly, a pilot at-home study tests feasibility to use a wearable version of the neuromodulation device for real-time phase-locked stimulation in the alpha (8-12 Hz) frequency range. Auditory stimulation was delivered at the intended phases of alpha with high precision, and alpha oscillations were affected differently by stimuli delivered at opposing phases. Our wearable system was capable of measuring sleep micro- and macro-events present in the EEG that were appropriate for clinical sleep scoring. Sleep onset latencies were reduced for a subset of subjects displaying sleep onset insomnia symptoms in the stimulation condition. This study demonstrates the feasibility of closed-loop tracking and neuromodulation of alpha oscillations using a wearable EEG device. Preliminary results suggest that this approach could be used to accelerate sleep initiation in individuals with objective insomnia symptoms., Comment: 31 pages, 10 main figures, 5 supplementary figures

We show that the number $A(n,m)$ of partitions with $m$ even parts and largest hook length $n$ is strongly unimodal with mode [(n-1)/4] for $n\ge 6$. We establish this result by induction, using a $5$-term recurrence due to Lin, Xiong and Yan, and two $4$-term recurrences obtained by Zeilberger's algorithm. The sequence $A(n,m)$ is not log-concave. Using M\"obius transformation and the method of interlacing zeros, we obtain that every zero of every generating function $\sum_m A(n,m)z^m$ lies on the left half part of the circle |z-1|=2. Moreover, as a direct application of Wang and Zhang's characterization of root geometry of polynomial sequences that satisfy a recurrence of type $(1,1)$, we see that all these zeros are densely distributed on the half circle.

Accreting protoplanets represent a window into planet formation processes. We report H{\alpha} differential imaging results from the deepest and most comprehensive accreting protoplanet survey to date, acquired with the Magellan Adaptive Optics (MagAO) system's VisAO camera. The fourteen transitional disks targeted are ideal candidates for protoplanet discovery due to their wide, heavily depleted central cavities, wealth of non-axisymmetric circumstellar disk features evocative of ongoing planet formation, and ongoing stellar accretion. To address the twin challenges of morphological complexity in the target systems and PSF instability, we develop novel approaches for frame selection and optimization of the Karhounen-Loeve Image Processing algorithm pyKLIP. We detect one new candidate protoplanet, CS Cha "c", at a separation of 75mas and a Delta mag of 5.1 and robustly recover the HD142527 B and HD100453 B low mass stellar companions across multiple epochs. Though we cannot rule out a substantial scattered light contribution to its emission, we also recover LkCa 15 b. Its presence inside of the cleared disk cavity and consistency with a forward-modeled point source suggest that it remains a viable protoplanet candidate. The protoplanet PDS 70 c was marginally recovered under our conservative general methodology. However, through targeted optimization in H-alpha} imagery, we tentatively recover PDS 70 c in three epochs and PDS 70 b in one epoch. Of the many other previously-reported companions and companion candidates around objects in the sample, we do not recover any additional robust candidates. However, lack of recovery at moderate H-alpha contrast does not rule out the presence of protoplanets at these locations, and we report limiting H-alpha contrasts in such cases., Comment: Final version, in press at AJ

Extensive empirical studies show that the long distribution tail of travel time and the corresponding unexpected delay can have much more serious consequences than expected or moderate delay. However, the unexpected delay due to the distribution tail of travel time has received limited attention in recent studies of the valuation of travel time variability. As a complement to current valuation research, this paper proposes the concept of the value of travel time distribution tail, which quantifies the value that travelers place on reducing the unexpected delay for hedging against travel time variability. Methodologically, we define the summation of all unexpected delays as the unreliability area to quantify travel time distribution tail and show that it is a key element of two well-defined measures accounting for unreliable aspects of travel time. We then formally derive the value of distribution tail, show that it is distinct from the more established value of reliability (VOR), and combine it and the VOR in an overall value of travel time variability (VOV). We prove theoretically that the VOV exhibits diminishing marginal benefit in terms of the traveler's punctuality requirements under a validity condition. This implies that it may be economically inefficient for travelers to blindly pursue a higher probability of not being late. We then proceed to develop the concept of the travel time variability ratio, which gives the implicit cost of the punctuality requirement imposed on any given trip. Numerical examples reveal that the cost of travel time distribution tail can account for more than 10% of the trip cost, such that its omission could introduce non-trivial bias into route choice models and transportation appraisal more generally.

We prove Gebhard and Sagan's $(e)$-positivity of the line graphs of tadpoles in noncommuting variables. This implies the $e$-positivity of these line graphs. We then extend this $(e)$-positivity result to that of certain cycle-chord graphs, and derive the bivariate generating function of all cycle-chord graphs., Comment: 15 pages, 4 figures

We investigate the $e$-positivity and Schur positivity of the chromatic symmetric functions of some spider graphs with three legs. We obtain the positivity classification of all broom graphs and that of most double broom graphs. The methods involve extracting particular $e$-coefficients of the chromatic symmetric function of these graphs with the aid of Orellana and Scott's triple-deletion property, and using the combinatorial formula of Schur coefficients by examining certain special rim hook tabloids. We also propose some conjectures on the $e$-positivity and Schur positivity of trees., Comment: 19 pages, 6 figures