Combinatorics and Number Theory of Counting Sequences

István Mező 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.

• I Counting sequences related to set partitions and permutationsSet partitions and permutation cycles.Generating functionsThe Bell polynomialsUnimodality, log concavity and log convexityThe Bernoulli and Cauchy numbersOrdered partitionsAsymptotics and inequalitiesII Generalizations of our counting sequencesProhibiting elements from being togetherAvoidance of big substructuresProhibiting elements from being togetherAvoidance of big substructuresAvoidance of small substructuresIII Number theoretical propertiesCongurencesCongruences vial finite field methodsDiophantic resultsAppendix

This book provides an interesting introduction to combinatorics by employing number-theoretic techniques of counting sequences. The level of the presentation often seems elementary, as the author frequently throws out lagniappes suitable for high school students. The text unfolds in three parts. Part 1 covers set partitions, generating functions, Bell polynomials, log-concavity, log-convexity, Bernoulli and Cauchy numbers, ordered partitions, asymptotes, and related inequalities. Part 2 discusses generalizations of counting sequences in three chapters. The final part considers number theoretical properties, including congruences, by way of finite field methods and Diophantine results. Each chapter concludes with an "Outlook" section that gives suggestions about exploring additional topics not covered in the text. Mathematical proof is used throughout the exposition and tends to be "enumerative," again contributing to a sense that the author hopes to engage mathematical novices through this text. However, the more than 250 exercises included in the book are frequently challenging and always interesting. The bibliography comprises more than 600 entries. Anyone who can follow the text is likely to enjoy working through the book.-D. P. Turner, Faulkner University