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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