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This primer covers a remarkably wide range of topics in discrete calculus, from combinatorics to recurrence relations. With examples and exercises throughout, the text´s clarity and rigor are informed by extensive teaching experience and academic discussion.

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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.

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1 Let´s Learn to Count.- 2 Counting Sequences and Collections.- 3 Occupancy Constraints.- 4 Inclusion/Exclusion.- 5 Stirling Numbers and Eulerian Numbers.- 6 Manipulation of Sums.- 7 Formal Power Series.- 8 Generating Formal Series and Applications.- 9 Recurrence Relations.- 10 Linear Recurrence Relations.- 11 Symbolic Calculus.- 12 The Euler-Maclaurin Formulas of Order 1 and 2.- 13 The Euler-Maclaurin Formula of Arbitrary Order.- 14 Cauchy and Riemann Sums, Factorials, Ramanujan Numbers and their Approximations.- 15 Tables and Formulas.- 16 Appendix A.

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Carlo Mariconda was born in 1964. In 1991 he received his PhD in Functional Analysis from the International School for Advanced Studies (SISSA-ISAS), Trieste, Italy. Since 1992 he has been a member of the Department of Mathematics at the University of Padova, where he is now Professor in Mathematical Analysis. His scientific interests mainly focus on the Calculus of Variations and Functional Analysis.



Alberto Tonolo was born in 1962. In 1993 he received his PhD in Algebra from the University of Florence, Italy. Since 1992 he has been a member of the Department of Mathematics at the University of Padova, where he is now Professor in Algebra. His scientific interests mainly focus on Module and Ring Theory and Homological Algebra.

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