International audience Eulerian numbers (and ''Alternate Eulerian numbers'') are often interpreted as distributions of statistics defined over the Symmetric group. The main purpose of this paper is to define a way to represent permutations that provides some other combinatorial interpretations of these numbers. This representation uses a one-to-one correspondence between permutations and the so-called \emphsubexceedant functions.

A topological index is a molecular descriptor derived from the molecular structure of a chemical substance. These indices can be used to analyze mathematical values and predict various physical properties of drugs. This article discusses tofacitinib, leflunomide, upadacinib, baricitinib, filgotinib, methotrexate, and other drugs used to treat rheumatoid arthritis, and the goal of the QSPR study is to calculate the relationship between the properties under investigation (e.g., boiling point, polarity, and molar volume) and molecular descriptors. Topological indices (TIs) were imposed on said drugs to calculate the correlation with physicochemical properties in this course.

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A topological index is a real number derived from the structure of a chemical graph. It is helpful to determine the physicochemical and biological properties of a wide range of drugs, and it better reflects the theoretical properties of organic compounds. This is accomplished using degree-based topological indices. Vitiligo is a common, acquired skin pigmentation disorder that significantly impacts the quality of life. It frequently embodies a therapeutic challenge, resulting in interest in alternative treatments based on vitamin and herbal supplements. In this article, azathioprine, clobetasol, desonide, hydrocortisone valerate, and other drugs utilized to cure vitiligo have discoursed, and the goal of QSPR revision is to determine the mathematical relationship between properties under investigation (e.g., polarity and enthalpy) and diverse descriptors associated with the drugs’ molecule. The QSPR model will help to predict physical properties. In this study, topological indices (TIs) imposed on said drugs were found to have a good correlation with physicochemical properties in this course. Finally, this work can be helpful to design and synthesize new vitiligo treatments and other disease drugs.

Counting polynomials are closely related to certain features of chemical graphs and provide an elegant means of expressing graph topological invariants. The current paper aims to calculate four polynomials for double benzenoid chains, Sadhana, omega, theta, and Padmakar–Ivan (PI). The edge-cut method is used to derive analytical closed expressions for these polynomials.