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## ECM3721 - Combinatorics (2018)

MODULE TITLE | Combinatorics | CREDIT VALUE | 15 |
---|---|---|---|

MODULE CODE | ECM3721 | MODULE CONVENER | Dr Robin Chapman (Coordinator) |

DURATION: TERM | 1 | 2 | 3 |
---|---|---|---|

DURATION: WEEKS | 0 | 11 | 0 |

Number of Students Taking Module (anticipated) | 107 |
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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

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

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.

- 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

Scheduled Learning & Teaching Activities | 33.00 | Guided Independent Study | 117.00 | Placement / Study Abroad | 0.00 |
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Category | Hours of study time | Description |

Scheduled learning and teaching activities | 33 | Lectures/example classes |

Guided independent study | 117 | Guided independent study |

Form of Assessment | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|

Not applicable | |||

Coursework | 20 | Written Exams | 80 | Practical Exams |
---|

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 |

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 |

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.

information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE – http://vle.exeter.ac.uk

Reading list for this module:

Type | Author | Title | Edition | Publisher | Year | ISBN | Search |
---|---|---|---|---|---|---|---|

Set | Bryant, Victor | Aspects of combinatorics : a wide-ranging introduction | Cambridge University Press | 1993 | 000-0-521-41974-3 | [Library] | |

Set | Robert Wilson | Combinatorics: A Very Short Introduction | OUP | 2016 | 9780198723493 | [Library] | |

Set | Cameron, Peter | Combinatorics: Topics, Techniques, Algorithms | Cambridge University Press | 1994 | 000-0-521-45761-0 | [Library] |

CREDIT VALUE | 15 | ECTS VALUE | 7.5 |
---|---|---|---|

PRE-REQUISITE MODULES | ECM2712, ECM1702 |
---|---|

CO-REQUISITE MODULES |

NQF LEVEL (FHEQ) | 6 | AVAILABLE AS DISTANCE LEARNING | No |
---|---|---|---|

ORIGIN DATE | Thursday 06 July 2017 | LAST REVISION DATE | Wednesday 27 February 2019 |

KEY WORDS SEARCH | Combinatorics; generating functions; design theory; partitions. |
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