Discrete Calculus
Methods for Counting

Authors: Carlo Mariconda, Alberto Tonolo



A new and efficient way to learn combinatorics

Includes effective problem-solving methods

Efficient didactical approach, taking care of the reader

An original collection of important aspects of discrete mathematics, rarely presented in the same book
Includes numerous examples and exercises

Based on the extensive teaching experience of and fruitful discussions between the different authors

This book provides an introduction to combinatorics, finite calculus, formal series, recurrences, and approximations of sums. Readers will find not only coverage of the basic elements of the subjects but also deep insights into a range of less common topics rarely considered within a single book, such as counting with occupancy constraints, a clear distinction between algebraic and analytical properties of formal power series, an introduction to discrete dynamical systems with a thorough description of Sarkovskii’s theorem, symbolic calculus, and a complete description of the Euler-Maclaurin formulas and their applications. Although several books touch on one or more of these aspects, precious few cover all of them. The authors, both pure mathematicians, have attempted to develop methods that will allow the student to formulate a given problem in a precise mathematical framework. The aim is to equip readers with a sound strategy for classifying and solving problems by pursuing a mathematically rigorous yet user-friendly approach. This is particularly useful in combinatorics, a field where, all too often, exercises are solved by means of ad hoc tricks. The book contains more than 400 examples and about 300 problems, and the reader will be able to find the proof of every result. To further assist students and teachers, important matters and comments are highlighted, and parts that can be omitted, at least during a first and perhaps second reading, are identified.

Table of contents (14 chapters)

Front Matter
Pages i-xxi

Let’s Learn to Count
Pages 1-26

Counting Sequences and Collections
Pages 27-64

Occupancy Constraints
Pages 65-82

Inclusion/Exclusion
Pages 83-104

Stirling Numbers and Eulerian Numbers
Pages 105-151

Manipulation of Sums
Pages 153-191

Formal Power Series
Pages 193-273

Generating Formal Series and Applications
Pages 275-318

Recurrence Relations
Pages 319-354

Linear Recurrence Relations
Pages 355-417

Symbolic Calculus
Pages 419-463

The Euler–Maclaurin Formulas of Order 1 and 2
Pages 465-534

The Euler–Maclaurin Formula of Arbitrary Order
Pages 535-578

Cauchy and Riemann Sums, Factorials, Ramanujan Numbers and Their Approximations
Pages 579-618

Back Matter
Pages 619-659

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