A new kind of signed permutations is proposed in this paper. Based on the standard permutation defined by {1,2,…,n}, we assign a sign (either positive or negative) to each number, including the boundary elements at both ends, the resulting permutation is called a signed permutation. Eight combinatorial statistics about ascents, descents, sign changes with respect to the signs are introduced. With the help of the context-free grammar theory, a series of combinatorial properties of such signed permutations are presented, mainly in the aspects of (homogeneous) γ-positivity, (homogeneous) bi-γ-positivity, determinantal expressions, Pólya-frequency, unimodal, log-concave, real-rooted, asymptotic normality.
The development of the theory of the second-order Eulerian polynomials began with the works of Buckholtz and Carlitz in their studies of an asymptotic expansion. Gessel-Stanley introduced Stirling permutations and provided combinatorial interpretations for the second-order Eulerian polynomials in terms of Stirling permutations. The Stirling permutations have been extensively studied by many researchers. The motivation of this paper is to develop a general method for finding equidistributed statistics on Stirling permutations. Firstly, we show that the up-down-pair statistic is equidistributed with the ascent-plateau statistic, and that the exterior up-down-pair statistic is equidistributed with the left ascent-plateau statistic. Secondly, we introduce the Stirling permutation code (called SP-code). A large number of equidistribution results follow from simple applications of the SP-codes. In particular, we find that six bivariable set-valued statistics are equidistributed on the set of Stirling permutations, and we generalize a classical result on trivariate version of the second-order Eulerian polynomial, which was independently established by Dumont and Bóna. Thirdly, we explore the bijections among Stirling permutation codes, perfect matchings and trapezoidal words. We then show the e-positivity of the enumerators of Stirling permutations by left ascent-plateaux, exterior up-down-pairs and right plateau-descents. In the final part, the e-positivity of the multivariate k-th order Eulerian polynomials is established, which improves a classical result of Janson-Kuba-Panholzer and generalizes a recent result of Chen-Fu. These e-positive expansions are derived from the combinatorial theory of context-free grammars.
A new ketone, volubilisone B (1), along with nine known compounds, including 4-hydroxy-4-methylpentan-2-one (diacetone alcohol) (2), 1-triacontanol (3), 4,7-dimethoxy-5-(2-propenyl)benzo[d][1,3]dioxole (apiole) (4), 4,7-dimethoxy-5-(1-propenyl)benzo[d][1,3]dioxole (isoapiole) (5), methyl caffeate (6), syringaldehyde (7), 3′,4′,5′-trimethoxycinnamyl alcohol (8), apigenin (9), and kaempferol (10), were isolated from the seed shells of Plukenetia volubilis L. (Euphorbiaceae). The structure of the new pentanone derivative (1) was elucidated by chemical and physical evidence.
Retrieval-augmented generation (RAG) enables large language models (LLMs) to dynamically access external information, which is powerful for answering questions over previously unseen documents. Nonetheless, they struggle with high-level conceptual understanding and holistic comprehension due to limited context windows, which constrain their ability to perform deep reasoning over long-form, domain-specific content such as full-length books. To solve this problem, knowledge graphs (KGs) have been leveraged to provide entity-centric structure and hierarchical summaries, offering more structured support for reasoning. However, existing KG-based RAG solutions remain restricted to text-only inputs and fail to leverage the complementary insights provided by other modalities such as vision. On the other hand, reasoning from visual documents requires textual, visual, and spatial cues into structured, hierarchical concepts. To address this issue, we introduce a multimodal knowledge graph-based RAG that enables cross-modal reasoning for better content understanding. Our method incorporates visual cues into the construction of knowledge graphs, the retrieval phase, and the answer generation process. Experimental results across both global and fine-grained question answering tasks show that our approach consistently outperforms existing approaches on both textual and multimodal benchmarks.
It is well known that ascents, descents and plateaux are equidistributed over the set of classical Stirling permutations. Their common enumerative polynomials are the second-order Eulerian polynomials, which have been extensively studied by many researchers. This paper is divided into three parts. The first part gives a convolution formula for the second-order Eulerian polynomials, which simplifies a result of Gessel. As an application, a determinantal expression for the second-order Eulerian polynomial is obtained. We then investigate a convolution formula of the trivariate second-order Eulerian polynomials. Among other things, by introducing three new statistics: proper ascent-plateau, improper ascent-plateau and trace, we discover that a six-variable enumerative polynomial over restricted Stirling permutations equals a six-variable Eulerian-type polynomial over signed permutations. By special parametrizations, we make use of Stirling permutations to give a unified interpretation of the (p, q)-Eulerian polynomials and derangement polynomials of types A and B. The third part presents a box sorting algorithm which leads to a bijection between the terms in the expansion of (cD)ncand ordered weak set partitions, where cis a smooth function in the indeterminate x and D is the derivative with respect to x. Using a map from ordered weak set partitions to standard Young tableaux, we find an expansion of (cD)nc in terms of standard Young tableaux. Combining this with context-free grammars, we provide three new interpretations of the second-order Eulerian polynomials. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
