## Math 3152a/Math 9043a -- Combinatorial mathematics (Fall 2017)

Instructor Graham Denham WF, 2:30-3:30pm MWF 11:30-12:30pm MC 107 Introductory Combinatorics, 5th edition, Richard Brualdi, available at the bookstore 0.5 course from: Mathematics 2120A/B, 2155A/B, 2211A/B, Applied Mathematics 2811B, or permission of the Department. October 25 and 27 (two parts; in class) TBA 40% Final exam; 30% midterm; 30% assignments

Wilf's book on generating functions is a good secondary reference, available for free here.  The course synopsis below will contain other specific suggestions as the course progresses.

## Synopsis

A week-by-week record of what's going on:
• 9-8: Introduction and overview. Our focus will be mostly on counting. What, though, constitutes a solution to a counting problem?
• 9-11, 9-13, 9-15: permutations and combinations.  We can organize some classical counting problems in terms of additive and multiplicative principles.  $$k$$-element sequences of entries from $$[n]=\{1,2,...,n\}$$ can also be thought of as functions $$f\colon[k]\to [n]$$.  From this point of view permutations are the injective functions.  The discussion extends to permutations of multisets, which we see can be counted by multinomial coefficients.  Subsets of $$[n]$$ are in one-to-one correspondence with permutations of a multiset with just two kinds of element.  This gets us back to classical binomial coefficients.
• 9-18: the pigeonhole principle 9-20: fancy variations and applications like the Erdos-Szekeres Theorem
• 9-22: generating permutations: the Johnson-Trotter algorithm.

## Syllabus

This is an intermediate course in enumerative combinatorics, the study of counting.  There are not many formal prerequisites, but you will enjoy the course best if you have some enthusiasm for problem-solving and hands-on math. We will review the basics -- how to count permutations and combinations of labelled and unlabelled objects.  We will see how to use formal power series (also known as generating functions) to solve counting problems easily and systematically.  The usual topics include:
• the pigeonhole principle
• permutations and combinations of sets and multisets
• introducing formal power series
• ordinary and exponential generating functions
• solving recurrence relations
• counting graphs and trees
• Joyal's theory of species
• counting in the presence of symmetry (Polya theory) or Lagrange inversion

## Assignments

Learning the art of counting requires, above all, practice.  Accordingly, there will be regular homework assignments.  This is the most important part of the course.  Please note that no late assignments will be accepted.  See the homework page for an up-to-date list.

Some of the assignment problems will be routine, and some will take some thought.  Collaborating with other people can add a lot to the experience of doing math, and I encourage you to do so.  (Research-level mathematics can be done alone, but is probably more often done in groups of two or three.)  Just make sure to write your own solutions, your own way, and to acknowledge any debts you may have.  Ask me if in doubt, since presenting the work of others as your own constitutes a serious academic offence.

Sometimes it can be useful to use some symbolic computation software, for example to evaluate a few terms of a power series.  Try Maple or Mathematica, if you have access or familarity.  You can also use Sage, an open-source symbolic computation tool, online and for free.  For example, create a Sage notebook, and enter the following:
var('t')
f = e^(e^t-1)
f.taylor(t,0,10)

This will give you the first ten terms of the exponential generating function for the Bell numbers, which we will learn about in early November.

## Exams

There will be one midterm which we will schedule at the start of week 2.

## Math 9043a

The MSc version of this course includes slightly different homework problems, and an additional self-directed written project, to be chosen at the start of term. In this case, the evaluation is weighted as 30% final exam; 25% midterm; 25% assignments; 20% project.  The project is due on the first Monday after the last lecture.

## Further information

Academic dishonesty: Scholastic offences are taken seriously and students are directed to read the official policy

Accessibility Statement: Please contact the course instructor if you require material in an alternate format or if you require any other arrangements to make this course more accessible to you. You may also wish to contact Services for Students with Disabilities (SSD) at 661-2111 ext. 82147 for any specific question regarding an accommodation.

Support Services: Learning-skills counsellors at the Student Development Centre are ready to help you improve your learning skills. Students who are in emotional/mental distress should refer to Mental Health@Western for a complete list of options about how to obtain help. Additional student-run support services are offered by the USC. The website for Registrarial Services is http://www.registrar.uwo.ca.

Eligibility: You are responsible for ensuring that you have successfully completed all course prerequisites and that you have not taken an antirequisite course. Unless you have either the requisites for this course or written special permission from your Dean to enroll in it, you may be removed from this course and it will be deleted from your record. This decision may not be appealed. You will receive no adjustment to your fees in the event that you are dropped from a course for failing to have the necessary prerequisites.