Instructor: Dr. Dezső Miklós
Text: handouts and Miklós Bóna: A walk through combinatorics
Basic counting rules (product rule, sum rule, permutations, combinations, Pascal's triangle, occupancy problems).
Generating functions (definition, operations on generating functions,
applications to counting, binomial theorem,
exponential generating functions).
Recurrences (Fibonacci numbers, derangements, recurrences involving more than one sequence, the method of generating functions).
Principle of inclusion and exclusion (the principle and applications,
occupancy problems with distinguishable balls and
cells, derangements, the number of objects having exactly m properties).
Introductory graph theory ( quick overview of fundamental concepts, connectedness, graph coloring, trees; Cayley's theorem on the number of trees).
Pigeonhole principle and Ramsey theory (Ramsey's theorem, bounds on Ramsey numbers, applications).
Symmetric combinatorial structures, block designs (definition, latin squares, finite projective planes).
Remark. In case of high number of students registering for introductory combinatorics, two versions of it may be offered. This is the syllabus of the faster space one, therefore - depending on the maturity of the audience - the scope of the topics finally covered (in any of the courses) will be "bounded above" by this list.