Combinatorics and Number Theory of Counting Sequences  book cover
1st Edition

Combinatorics and Number Theory of Counting Sequences





ISBN 9781032475356
Published January 21, 2023 by CRC Press
498 Pages 10 B/W Illustrations

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Book Description

Combinatorics and Number Theory of Counting Sequences is an introduction to the theory of finite set partitions and to the enumeration of cycle decompositions of permutations.



The presentation prioritizes elementary enumerative proofs. Therefore, parts of the book are designed so that even those high school students and teachers who are interested in combinatorics can have the benefit of them. Still, the book collects vast, up-to-date information for many counting sequences (especially, related to set partitions and permutations), so it is a must-have piece for those mathematicians who do research on enumerative combinatorics.



In addition, the book contains number theoretical results on counting sequences of set partitions and permutations, so number theorists who would like to see nice applications of their area of interest in combinatorics will enjoy the book, too.



Features







  • The Outlook sections at the end of each chapter guide the reader towards topics not covered in the book, and many of the Outlook items point towards new research problems.






  • An extensive bibliography and tables at the end make the book usable as a standard reference.






  • Citations to results which were scattered in the literature now become easy, because huge parts of the book (especially in parts II and III) appear in book form for the first time.




Table of Contents

I Counting sequences related to set partitions and permutations



Set partitions and permutation cycles.



Generating functions



The Bell polynomials



Unimodality, log concavity and log convexity



The Bernoulli and Cauchy numbers



Ordered partitions



Asymptotics and inequalities



II Generalizations of our counting sequences



Prohibiting elements from being together



Avoidance of big substructures



Prohibiting elements from being together



Avoidance of big substructures



Avoidance of small substructures



III Number theoretical properties



Congurences



Congruences vial finite field methods



Diophantic results



Appendix



 

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Author(s)

Biography

István Mezo is a Hungarian mathematician. He obtained his PhD in 2010 at the University of Debrecen. He was working in this institute until 2014. After two years of Prometeo Professorship at the Escuela Politécnica Nacional (Quito, Ecuador) between 2012 and 2014 he moved to Nanjing, China, where he is now a full-time research professor.