Math 68, Fall 2019

Algebraic Combinatorics
Last updated October 14, 2019 14:21:53 EDT

General Information



Scheduled Lectures

The following is a tentative schedule that will be updated frequently throughout the quarter. The class meets on MWF 11:30-12:35. The room for the class is 307, the geometry lab.

Date Content References
September 16 Introduction
September 18 What combinatorics is [EC1], §1.1 and online lecture notes
September 20 What combinatorics is (continued), basic combinatorial statistics [EC1], §1.1 and online lecture notes+ lecture notes on Catalan numbers
September 23 Counting principles: Twelvefold way [EC1], §1.2 and 1.9
September 24 (x-hour) Intro to computer programming for math SageMath page
September 25 Generating functions, the basics [Wil94], §1
September 27 Counting partitions and Sortware demo: Guessing [Sage], online worksheet
September 30 Posets: bacis concepts and examples, lattices [EC1], §3.1, 3.3, [Sag20+], §5.1
October 2 Properties of lattices [EC1], §3.3, 3.4, [Sag20+], §5.3
October 4 Incidence algebra and Möbius function, part 1 [EC1], §3.6, 3.7, [Sag20+], §5.4, 5.5
October 7 Möbius inversion [EC1], §3.7, 3.8, 3.9, [Sag20+], §5.8
October 9 Brief intro: hyperplane arrangements through posets [EC1], 3.11
October 11 Matroids and posets
October 14 Group acting on sets: intro, Burnside's Lemma [Sta18], §7, [Sag20+], §6.2
October 16 Group acting on sets: Pólya theory [Sta18], §7, [Sag20+], §6.4
October 18 Necklaces [Sta18], §7
October 21 Young tableaux, Symmetric group acting on Young tableaux, bumping and sliding [Ful97], §1, 7.1
October 23 Robinson-Schensted bijection [Ful97], §4
October 25 Increasing sequences [Sag01], §3.3, 3.5
October 28 A crash course of representation theory through examples [Sag01], §1
October 30 Specht modules through tabloids [Sag01], §2.1, 2.3, 2.4, [Ful97], §7.2
November 1 Basis of Specht modules, dimension [Sag01], §2.5
November 4 Branching rule for the symmetric group, induced and restricted representations [Sag01], §2.8
November 6 The decomposition of a permutation module [Sag01], §2.9
November 8 The Murnaghan-Nakayama rule [Sag01], §4.10, [EC2], §7.17
November 11 Student presentations
November 13 Student presentations
November 15 Student presentations
November 18 Student presentations
November 18-27 Final exam (take-home)

References

There is no textbook required for this class. Instead, the references listed here are used to prepare the classes. To know what reference was used for a given lecture, refer to the schedule.

[EC1] R. P. Stanley. Enumerative combinatorics. Volume 1, volume 49 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 2012. Freely available online.

[EC2] R. P. Stanley. Enumerative combinatorics. Volume 2, volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999.

[Ful97] W. Fulton. Young tableaux. With applications to representation theory and geometry, volume 35 of London Mathematical Society Student Texts. Cam- bridge University Press, Cambridge, 1997.

[Sag01] B. E. Sagan. The symmetric group. Representations, combinatorial algo- rithms, and symmetric functions. Graduate Texts in Mathematics. Springer- Verlag, New York, second edition edition, 2001.

[Sag20+] B. E. Sagan. Combinatorics: The Art of Counting. In preparation. Freely available preliminary version online.

[Sage] The Sage Developers. SageMath, the Sage Mathematics Software System. http://www.sagemath.org

[Sta18] R. P. Stanley. Algebraic combinatorics. Walks, trees, tableaux, and more, second edition. Undergraduate Texts in Mathematics. Springer, Cham, 2018. Freely available without exercices online.

[Wil94] H. S. Wilf. generatingfunctionology. Academic Press, Inc., Boston, MA, second edition, 1994. Freely available online. Only the first edition is available at the library.

Honor principle

Assignments and examinations serve both the purpose of fixing your ideas on a given subject and, for the evaluator, to assess the comprehension you have of the class and how the class material has been understood. None of these goals can be achieved if you hand out a solution that is taken from someone else or from a publicly available source.

Nevertheless, students are allowed to work together on problems to explore their solutions. The homework you submit must reflect your own understanding, and you thus need to write your own copy of the solution, in your own words.

For the final exam: Students are not allowed to give or receive any assistance of any form, except from their instructor.

More information on the Academic Honor Principle can be found on Dartmouth's website.

More policies

Policies about attendance, religious observance, accessibility and accomodation, mental health and financial needs can be found in the syllabus.