Instructor | Graham Denham |
---|---|

Office hours | WF, 2:30-3:30pm |

Class times | MWF 11:30-12:30pm |

Class location | MC 107 |

Textbook | Introductory Combinatorics, 5th edition, Richard Brualdi, available at the bookstore |

Prerequisites | 0.5 course from: Mathematics 2120A/B, 2155A/B, 2211A/B, Applied Mathematics 2811B, or permission of the Department. |

Midterm exam date |
October 25 and 27 (two parts; in class) |

Final exam | TBA |

Evaluation | 40% Final exam; 30% midterm; 30% assignments |

- 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.

- 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

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)

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.

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

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