Module Specification

The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
1. Module Title COMBINATORICS
2. Module Code MATH344
3. Year Session 2023-24
4. Originating Department Mathematical Sciences
5. Faculty Fac of Science & Engineering
6. Semester Second Semester
7. CATS Level Level 6 FHEQ
8. CATS Value 15
9. Member of staff with responsibility for the module
Dr V Guletskii Mathematical Sciences vladimir.guletskii@liverpool.ac.uk
10. Module Moderator
11. Other Contributing Departments  
12. Other Staff Teaching on this Module
Dr SA Fairfax Mathematical Sciences Simon.Fairfax@liverpool.ac.uk
Dr N Koseki Mathematical Sciences Naoki.Koseki@liverpool.ac.uk
13. Board of Studies
14. Mode of Delivery
15. Location Main Liverpool City Campus
    Lectures Seminars Tutorials Lab Practicals Fieldwork Placement Other TOTAL
16. Study Hours 36

  12

      48
17.

Private Study

102
18.

TOTAL HOURS

150
 
    Lectures Seminars Tutorials Lab Practicals Fieldwork Placement Other
19. Timetable (if known)            
 
20. Pre-requisites before taking this module (other modules and/or general educational/academic requirements):

MATH102 CALCULUS II; MATH101 Calculus I; MATH103 Introduction to Linear Algebra
21. Modules for which this module is a pre-requisite:

 
22. Co-requisite modules:

 
23. Linked Modules:

 
24. Programme(s) (including Year of Study) to which this module is available on a mandatory basis:

25. Programme(s) (including Year of Study) to which this module is available on a required basis:

26. Programme(s) (including Year of Study) to which this module is available on an optional basis:

27. Aims
 

To provide an introduction to the problems and methods of Combinatorics, particularly to those areas of the subject with the widest applications such as pairings problems, the inclusion-exclusion principle, recurrence relations, partitions and the elementary theory of symmetric functions.

 
28. Learning Outcomes
 

(LO1) Recognise the types of problem to which the methods of combinatorics apply and model these problems.

 

(LO2) Solve counting and arrangement problems.

 

(LO3) Solve general recurrence relations using the generating function method.

 

(LO4) Apply the elementary theory of partitions to the study of symmetric functions.

 
29. Teaching and Learning Strategies
 

Material is presented during lectures (3 hours per week). Tutorials (1 hour per week) are used for consolidation and practice, and for help with individual questions.

 
30. Syllabus
   

Principles of counting.
Selections, permutations and multinomial coefficients.
The inclusion-exclusion principle.
Pairings and Hall's theorem.
Recurrences, power series and generating functions.
Generating functions and Stirling numbers.
Graphs and Euler's theorem.
Species, a modern view of generating functions.
Symmetric groups and symmetric polynomials.
Signature and alternating groups.
Cycle types and conjugacy classes in symmetric groups.
Young diagrams and irreducible representations of S_n.

 
31. Recommended Texts
  Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module.
 

Assessment

32. EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
  Final Exam 120 50
33. CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
  Homework 1 0 10
  Homework 2 0 10
  Homework 3 0 10
  Homework 4 0 10
  Homework 5 0 10