# Combinatorial Mathematics

By Dr. Resmi T   |   KKTM Govt. College, Pullut, Kodungallur, Thrissur
Learners enrolled: 216

Combinatorial Mathematics is a branch of Mathematics which deals with counting, arrangements, selection and operations of finite systems or discrete systems. It finds its applications in Computer Science, Cryptography, Communications networks, Computational molecular biology, simulation and so on. It has vide variety of applications in Probability and Statistics, Graph theory, logic, topology etc. Combinatorics  is used in many optimization problems in operations research. Combinatorial Matematics play different roles in these fields.  For example,  in computer science, combinatorial problems arise in numerical analysis.  Combinatorial techniques are used in the design of computers. Different divide and conquer algorithms and recurrence relations in Comb. Maths are used to handle big data. Combinatorial tools are used to the storage of information and its retrieval.

This course will help the students to acquire a good knowledge in combinatorial mathematics which will also result a better understanding of the subjects  where the combinatorial tools play lead roles.

Summary
 Course Status : Upcoming Course Type : Elective Duration : 16 weeks Category : Mathematics Credit Points : 5 Level : Undergraduate Start Date : 01 Jul 2024 End Date : 31 Oct 2024 Enrollment Ends : 31 Aug 2024 Exam Date : 14 Dec 2024 IST EXAM SHIFT : SHIFT 1

Note: This exam date is subjected to change based on seat availability. You can check final exam date on your hall ticket.

### Course layout

Week 1 Day 1 Module 1 : Basic Counting Principles
Day 2 Module 2 : Permutations - Part 1
Day 3 Module 3 : Permutations - Part 2
Day 4 Discussion
Day 5 Assignments

Week 2 Day 1 Module 4 : COMBINATIONS WITHOUT REPETITIONS
Day 2 Module 5 : COMBINATIONS WITHOUT REPETITIONS, COUNTING SUBSETS of a Set
Day 3 Module 6 : COMBINATIONS WITH REPETITIONS
Day 4 Discussion
Day 5 Assignments

Week 3 Day 1 Module 7 : BINOMIAL THEOREM PART - 1
Day 2 Module 8 : BINOMIAL THEOREM PART - 2
Day 3 Module 9 : BINOMIAL THEOREM PART - 3
Day 4 Discussion
Day 5 Assignments

Week 4 Day 1 Module 10 : Recurrence relations-Recurrence relation models
Day 2 Module 11 : Solution of recurrence relations - Method of iteration
Day 3 Module 12 : Solution of First and Second order Linear Difference Equations
Day 4 Discussion
Day 5 Assignments

Week 5 Day 1 Module 13 : Solution of Second-Order Homogeneous and Non
Homogeneous Linear Difference Equations
Day 2 Module 14 : Divide and conquer relations
Day 3 Module 15 : Principle of inclusion and exclusion
Day 4 Discussion
Day 5 Assignments

Week 6   Day 1 Module 16 : DERANGEMENTS AND INVERSION FORMULA
Day 2 Module 17 : Algibra of Formal Power Series
Day 3 Module 18 : Generating functions -Part I
Day 4 Discussion
Day 5 Assignments

Week 7   Day 1 Module 19 : Generating functions -Part 2
Day 2 Module 20 : Calculating generating functions
Day 3 Module 21 : Exponential generating functions
Day 4 Discussion
Day 5 Assignments

Week 8   Day 1 Module 22 : Solutions of Recurrence Relations by Generating Functions – Part 1
Day 2 Module 23: Solutions of Recurrence Relations by Generating Functions – Part 2
Day 3 Module 24 : Set partitions part 1
Day 4 Discussion
Day 5 Assignments

Week 9   Day 1 Module 25 : Set partitions part 2
Day 2 Module 26 : Stirling numbers Part 1
Day 3 Module 27: Stirling numbers Part 2
Day 4 Discussion
Day 5 Assignments

Week 10   Day 1 Module 28 : Integer partitions 1
Day 2 Module 29 : Integer partitions 2
Day 3 Module 30 : Integer partitions 3
Day 4 Discussion
Day 5 Assignments

Week 11   Day 1 Module 31 : Multinomial Theorem
Day 2 Module 32 : Systems of Distinct Representatives - Part 1
Day 3 Module 33: Systems of Distinct Representatives - Part 2
Day 4 Discussion
Day 5 Assignments

Week 12   Day 1 Module 34 : Latin Squares Part 1
Day 2 Module 35 : Latin Squares Part 2
Day 3 Module 36 : Hadamard Matrices
Day 4 Discussion
Day 5 Assignments

Week 13   Day 1 Module 37 : CYCLE INDEX - PART 1
Day 2 Module 38 : CYCLE INDEX - PART 2
Day 3 Module 39 : CYCLE INDEX - PART 3
Day 4 Discussion
Day 5 Assignments

Week 14  Day 1 Module 40 : Burnside's Lemma Part 1
Day 2 Module 41 : Burnside's Lemma Part 2
Day 3 Module 42 : Burnside's Lemma Part 3
Day 4 Discussion
Day 5 Assignments

Week 15  Day 1 Module 43 : POLYA'S THEOREM
Day 2 Module 44 : APPLICATIONS OF POLYA'S THEOREM
Day 3 Module 45 : Necklace Problem
Day 4 Discussion
Day 5 Assignments

Week 16  Day 1 Module 46 : Combinatorial Designs - t design Part 1
Day 2 Module 47 : Combinatorial Designs - t design Part 2
Day 3 Module 48 : Combinatorial Designs - BIBDs
Day 4 Discussion
Day 5 Assignments

Week 17  Day 1 Module 49 : Combinatorial Designs - SBIBDs
Day 2 Module 50 : Some Special Combinatorial Design
Day 3
Day 4 Discussion
Day 5 Assignments

### Books and references

1. J.H. van Lint and R.M. Wilson, A Course in Combinatorics, 2nd Ed., Cambridge University Press, 2001.

2. V. Krishnamurthy, Combinatorics, Theory and Application, Affiliated East-West Press 1985.

3. P.J. Cameron, Combinatorics, Topics, Techniques, Algorithms, Cambridge University Press, 1995.

4. M. Jr. Hall, Combinatorial Theory, 2nd Ed., John Wiley & Sons, 1986.

5. S.S. Sane, Combinatorial Techniques, Hindustan Book Agency, 2013.

6. R.A. Brualdi, Introductory Combinatorics, 5th Ed., Pearson Education Inc., 2009.

NPETL Course videos.

### Dr. Resmi T

KKTM Govt. College, Pullut, Kodungallur, Thrissur
Dr. Resmi T has been working as an Associate Professor in KKTM Govt. College, Pullut. She has 18 years of teaching experience in UG and PG levels. Her research area is Stochastic Processes and Queueing Theory. Dr. Resmi had published 4 research papers and presented papers in 7 international conferences. She is interested in the topics Calculus, Linear Algebra, Topology, Analysis, Differential Geometry, etc.

### Course certificate

Course Completion  will carry 70% weightage of end term exam and 30% weightage of interal assessments. A minimum 40% in each is required to qualify for the Course Completion Certifcate.