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### Course Overview

Combinatorics is the branch of mathematics concerned with counting. This includes counting structures of a given kind (enumerative combinatorics), deciding when certain criteria can be met, finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization), and applying algebraic techniques to combinatorial problems (algebraic combinatorics). The course is recommended to students of mathematics and computing science.

### Course Details

Study Type Level Undergraduate 1 Second Sub Session 15 credits (7.5 ECTS credits) None. No Professor Ben MartinDr Alexey Sevastyanov

### Qualification Prerequisites

• Either Programme Level 1 or Programme Level 2

None.

None.

No

### Course Description

* Induction * Growth rates * Subsets and binomial coefficients (subsets of fixed size, properties of binomial coefficients, the binomial therem) * Partitions and Bell numbers * Counting with symmetry * Recurrences * Generating functions * Derangements * Principle of inclusion-exclusion * Extremal problems * The Pigeonhole Principle * Ramsey theory * Graph theory

Syllabus

• Counting and probability, induction, growth rates.
• Permutations, subsets, selections, the Binomial Theorem, partitions, Bell numbers.
• Generating functions, derangements, exponential generating functions.
• Inclusion and exclusion, Stirling numbers of the second kind.
• Extremal set theory, intersecting families, Sperner families.
• The Pigeonhole Principle, Ramsey numbers, Ramsey’s Theorem.
• Graphs, paths, trees, topological graph theory, Euler’s Theorem, the Platonic solids.

Course Aims

### Degree Programmes for which this Course is Prescribed

• Master of Engineering in Computing Science

### Contact Teaching Time

43 hours

This is the total time spent in lectures, tutorials and other class teaching.

### Assessment

1st Attempt: 1 two-hour written examination (80%); in-course assessment (20%). Resit: 1 two-hour written examination paper (maximum of (100%) resit and (80%) resit with (20%) in-course assessment).

### Formative Assessment

Informal assessment of weekly homework through discussions in tutorials.

### Feedback

In-course assignments will normally be marked within one week and feedback provided to students in tutorials.

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