# Mathematics

## ECM3721 - Combinatorics (2018)

MODULE TITLE CREDIT VALUE Combinatorics 15 ECM3721 Dr Robin Chapman (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 0 11 0
 Number of Students Taking Module (anticipated) 107
DESCRIPTION - summary of the module content

Combinatorics, a branch of pure mathematics, is the study of finite mathematical structures and is used frequently in computer science to obtain estimates on the number of elements of certain sets. This module will be particularly concerned with enumerative questions, that is, counting the number of structures of a particular kind. We will consider structures, including partitions, lattice paths and block designs. Furthermore, we will look at relations with other branches of mathematics, notably probability theory, algebra, matrix theory and number theory. Your goal is to solve combinatorial problems of standard type in the topics covered and to tackle easier problems of non-standard form.

Prerequisite module: ECM1702 and ECM2712 or equivalent

AIMS - intentions of the module

To inspire a genuine engagement with combinatorial methods and their applications in other branches of mathematics

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:
1 solve combinatorial problems of standard type in the topics covered and to tackle easier problems of non-standard form.
Discipline Specific Skills and Knowledge:
2 understand the close links between combinatorial problems and other areas of mathematics including the central topics of algebra and probability theory;
3 appreciate the role of conjecture and investigation within mathematics.
Personal and Key Transferable/ Employment Skills and  Knowledge:
4 develop strategies to solve challenging mathematical problems, which will enhance your time management and problem solving skills.

SYLLABUS PLAN - summary of the structure and academic content of the module

- binomial and multinomial coefficients;

- application to basic enumerative problems and to lattice path counting;

- the pigeonhole principle;

- parity arguments;

- the inclusion-exclusion principle;

- recurrences;

- generating functions;

- the Catalan numbers and Catalan families;

- rook polynomials  - product, recurrence and reciprocity principles for rook polynomials  - designs;

- affine and projective designs;

- Fisher's inequality;

- symmetric designs;

- Steiner triple systems;

- partitions of sets;

- Stirling and Bell numbers;

- partitions of numbers;

- partitions with restricted parts;

- Euler's pentagonal number theorem;

- Euler's recurrence for partition numbers

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 33 117 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled learning and teaching activities 33 Lectures/example classes Guided independent study 117 Guided independent study

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Not applicable

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams 20 80
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method

Written exam – closed book 80 2 hours - Summer Exam Period All Specific comments by markers and general comments on website
Coursework – example sheets 20 60 hours All Specific comments by markers and general comments on website

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-reassessment
All above Written exam (100%) All August Ref/Def period

RE-ASSESSMENT NOTES

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE – http://vle.exeter.ac.uk