Название: Advances in Applied Combinatorics
Автор: Stefano Spezia
Издательство: Arcler Press
Год: 2020
Страниц: 346
Язык: английский
Формат: pdf (true)
Размер: 11.31 MB

Advances in Applied Combinatorics talks about the subject of binomial coefficients, permutations, the combinational proofs, the graph theory, derangements, partitions, linear recurrences, graph algorithms and permutation groups, to give a far-fetched insight on applied combinatorics. This book also discusses about the fractional sums and the differences, harmonic numbers and the cubed binomial coefficients, the recursive algorithms, linear recurrences and the fibonacci numbers. The generating functions and the sequence of numbers and polynomials.

Combinatorics is the science of combinations. It is a very important topic in the field of Discrete Mathematics. Among many other things, it helps us to formulate methods for enumerating a wide range of objects that fulfil a certain feature of interest in a given field. In particular, combinatorics has links to physical sciences, data processing, probability, statistics, numerical analysis, information and coding theory, and various other domains.

Section 1 of Advances in Applied Combinatorics book begins with the introduction of the binomial coefficients, the permutations and the combinatorial proofs. Among them, it discusses of fractional calculus by making use of the binomial theorem, of spectral norms of circulant matrices whose entries are binomial coefficients combined with either Fibonacci numbers or Lucas numbers, and of optimal algorithms for sorting a signed permutation by short operations. In the end, Section 1 presents a combinatorial enumeration argument for proving the applicability of stability control in epidemics complex networks.

Section 2 focuses on graph theory and partially ordered sets (posets). In particular, it initially provides a study about the total dominator chromatic number of paths, cycles and ladder graph. Then, it treats of the modular Leech trees and of recursive algorithms for phylogenetic tree counting. Lastly, it presents a new proof of Dilworth’s theorem based upon the min-flow/max-cut property in flow networks, and its applications to h-partite graphs with multiple partial orders.

Sections 3 treats initially of some identities of derangement polynomials also by using umbral calculus. In the end, it focuses on the generalized Euler’s totient, its connections to other totients and with counting formulae.
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Finally, the last Section 7 discusses the permutations groups. In particular, it provides a brief survey of primitive groups of prime power degree. In the end, it presents a quantum formalism that does not involve any concepts associated with actual infinities, because formulated in constructive finite terms by using a unitary representation of a finite group.