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By using the calculus of finite differences methods and the umbral calculus, we construct recurrence relations for a new class of special numbers. Using this recurrence relation, we define generating functions for this class of special numbers and also new classes of special polynomials. We investigate some properties of these generating functions. By using these generating functions with their functional equations, we obtain many new and interesting identities and relations related to these classes of special numbers and polynomials, the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers. Finally, some derivative formulas and integral formulas for these classes of special numbers and polynomials are given. In general, this article includes results that have the potential to be used in areas such as discrete mathematics, combinatorics analysis and their applications. [ABSTRACT FROM AUTHOR]

Permutation exists in various domains such as mathematics, combinatorics, and computer science. Enumerating each permutation, as well as the multivariate information among different items, allows us, for example, to observe distribution, similarity, and dissimilarity of all possible permutations and select a satisfactory permutation or solution. However, the number of permutations increases dramatically along with the number of items in the permutation, which makes it challenging for users to evaluate potential solutions and identify interesting insights. In this paper, we propose PermVizor, a novel and scalable visualization system that aims assisting users exploring the arrangement, distribution, and comparison of permutations. Necessary and comprehensive analysis of requirements is presented for visualization of permutations. PermVizor enables users to explore overall distribution of each permutation with a glyph-based MDS view, investigate statistical information of selected permutations with a parallel coordinates view, and examine detailed arrangement of the items as well the multivariate information among them for each permutation with pixel-based and block-based PermView. Case studies are conducted on classical datasets such as the axis reordering issue in parallel coordinate data and permutation of traveling salesman problem, which shows that PermVizor could facilitate users in exploring unexpected and desired permutations and confirm their finding and decisions in expected permutations. [ABSTRACT FROM AUTHOR]