Combinatorics

Code:

M4034

Acronym:

M4034

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2025/2026 - 1S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Master in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
M:M	4	Official study plan since 2024/2025	1	-	6	42	162
			2

Teaching language

Suitable for English-speaking students

Objectives

To convey some of the fundamental methods, principles and results of combinatorics, encompassing the main aspects of the problems in this area: enumeration, existence, construction and optimization. There will be special emphasis on algebraic combinatorics, enhancing the interactions between combinatorics and algebra.

Learning outcomes and competences

Upon completing this course, the student should know and be able to apply the concepts and results covered in the course. It is intended that this unit contribute to the furthering of skills in the field of discrete mathematics and its interaction with algebra. In summary, it is intended that upon completion of this class the student can:

• Understand and apply fundamental combinatorial techniques as well as understand when these can or cannot be applied.
• Use appropriate techniques and problem solving skills on new problems.
• Recognize algebraic structures in combinatorial problems, can formulate them algebraically and solve them using algebraic techniques.
• Be mathematically creative and inquisitive, being capable of formulating interesting new questions in combinatorics.
• Have had contact with some current topics of research in combinatorics.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Familiarity with basic concepts and techiques from linear algebra (vector spaces, matrices, determinants, characteristic polynomial, eigenvalues) and with the concept of a group.

Program

1. Partitions, identities, generating functions, Gaussian integers, pentagonal number theorem, and applications.
2. Sign counting and the use of determinants in counting, including PIE and the Matrix-Tree Theorem.
3. Incidence algebra and Moebius inversion.
4. Symmetric functions.

Mandatory literature

Richard P. Stanley; Algebraic combinatorics. ISBN: 978-1-4614-6997-1
Richard P. Stanley; Enumerative combinatorics. ISBN: 0 521 66351 2
Sebastian M. Cioaba and M. Ram Murty; A first course in graph theory and combinatorics, Texts and Readings in Mathematics 55, Hindustan Book Agency, 2009
Samuel Lopes; Notas fornecidas pelo docente

Complementary Bibliography

Richard A. Brualdi; Introductory combinatorics. ISBN: 0-7204-8610-6
David R. Mazur; Combinatorics. ISBN: 978-0-88385-762-5
Martin Aigner; A course in enumeration
François Bergeron; Combinatorial species and tree-like structures. ISBN: 0-521-57323-8

Teaching methods and learning activities

Exposition by the teacher, discussion of exercises.

Software

sage

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Teste	50,00
Apresentação/discussão de um trabalho científico	25,00
Trabalho escrito	25,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	120,00
Frequência das aulas	42,00
Total:	162,00

Eligibility for exams

No required conditions.

Calculation formula of final grade

The final grade (CF) is obtained using the formula:

CF=.5*E+.25*TPC+.25*A,

where:

E=grade on the regular or appeal exam
TPC=grade for solving exercises and problems asynchronously
A=grade for writing a report on a topic related to Combinatorics and/or its applications, to be agreed with the course instructor, and oral presentation of the report

Of the items listed above, only the exam (E) can be retaken for the purpose of passing the appeal or improving the grade.

Any student may be required to take an oral exam to clarify any questions that may have arisen regarding the exams or assessment assignments.

Special assessment (TE, DA, ...)

Special exams will consist of a written test, which might be preceded by an eliminatory oral test to assess whether the student satisfies minimum requirements to tentatively pass the written test.

Observations

Any student may be required to take an oral examination should there be any doubts concerning his/her performance on certain assessment pieces.

Jury:
Samuel António de Sousa Dias Lopes
Pedro Ventura Alves da Silva