Discrete Mathematics

Recursive problems: Hanoi Tower, plane partition, Flavious Josephus problem. Fundamental combinatorial analysis: basic principles, combinatorics formations. Calculus of Finite sums: properties, multiple sums. Discrete calculus: association of calculus and discrete calculus, negative factorial power, differential tables – sums. Binomial coefficients – special numbers: binomial coefficients, sums of multiplications, Stirling numbers, harmonic numbers, Fibonacci, Catalan numbers. Basic principles of number theory: Euclidean division, divisibility, greatest common divisor, linear Diophantine equation, least common multiple, prime numbers, sum of divisors. Integer functions – generating functions: integer part of real numbers, Euler function, Legendre function. Generating functions: exponential generating function, Catalan Numbers generating function, Fibonacci numbers generating function, Stirling Numbers generating function, calculus with generating functions.

The course introduces students to the fundamental concepts of number theory, including prime numbers, Euclidean division, divisibility, greatest common divisor, least common multiple, and the solution of linear Diophantine equations, with the aim of deepening understanding of concepts applicable to Computer Science.

It covers the basic notions of discrete calculus, such as the correspondence between discrete and infinitesimal calculus, negative factorial powers, difference and summation tables, and finite sum calculus.

A significant part of the course focuses on the principles of combinatorial analysis and techniques for solving recursive problems, such as the Tower of Hanoi.

The course also addresses the study and understanding of special numbers and binomial coefficients, including fundamental identities, sums of products, harmonic numbers, Stirling numbers, Fibonacci numbers, and Catalan numbers.

Students are introduced to integer functions, such as arithmetic and multiplicative functions, Euler’s totient function, and Legendre’s function, as well as generating functions, including exponential generating functions, Catalan, Fibonacci, and Stirling generating functions, calculus with generating functions, tables of sequences and their generating functions, and generating functions of special numbers.

Upon successful completion of the course, students will be able to:

• Comprehend basic principles of number theory, such as prime numbers, divisibility, greatest common divisor, and least common multiple.
• Comprehend basic principles of combinatorial analysis.
• Apply techniques to solve linear Diophantine equations. 
• Understand concepts of discrete calculus, including the correspondence between discrete and infinitesimal calculus, negative factorial powers, difference and summation tables, and finite sum calculus.

Recursive problems: Hanoi Tower, plane partition, Flavious Josephus problem. Fundamental combinatorial analysis: basic principles, combinatorics formations. Calculus of Finite sums: properties, multiple sums. Discrete calculus: association of calculus and discrete calculus, negative factorial power, differential tables – sums. Binomial coefficients – special numbers: binomial coefficients, sums of multiplications, Stirling numbers, harmonic numbers, Fibonacci, Catalan numbers. Basic principles of number theory: Euclidean division, divisibility, greatest common divisor, linear Diophantine equation, least common multiple, prime numbers, sum of divisors. Integer functions – generating functions: integer part of real numbers, Euler function, Legendre function. Generating functions: exponential generating function, Catalan Numbers generating function, Fibonacci numbers generating function, Stirling Numbers generating function, calculus with generating functions.

1. Discrete Mathematics: Mathematics of Computer Science, L. Kyrousis, Ch. Bouras and P. Spyrakis, Gutenberg, 1992
2. Discrete Mathematics: Problems and Solutions, C. Voutsadakis, L. Kyrousis, Ch. Bouras and P. Spyrakis, Gutenberg, 1994.
3. Introduction to Combinatorial Mathematics, CL Liu, Mc Graw Hill Ch. Charalambides, Combinatorics (1st issue) Symmetry
4. Elements of Discrete Mathematics, CL Liu, McGraw-Hill, Second Edition.
5. Discrete Mathematics, Seymour Lipschutz Marglipson, McGraw-Hill, Second Edition.
6. Discrete Mathematics A Unified Approach, Stephen A. Wiitala, McGraw-Hill.
7. Discrete Mathematics and Its Applications, Kenneth H. Rosen, McGraw-Hill, Fourth Edition

• Lectures
• Tutorial exercises
• Student participation in tutorials

• Support of the educational process via the e-learning platform Opencourses
• Use of ICT in teaching and education (specialized software, electronic lecture notes)
• Use of ICT for communication with students

Written final examination with problem-solving questions