University of Connecticut College of Liberal Arts and Sciences
Department of Mathematics
Tom Roby's Math 5020 Home Page (Fall 2014)
Algebraic Combinatorics

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Class Information

COORDINATES: Classes meet Tuesdays and Thursdays 2:00–3:15 in MSB 315. The registrar calls this Section 001.

PREREQUISITES: Math 5210 or equivalent experience.

TEXTS: Richard P. Stanley: Enumerative Combinatorics 1 (2nd edition) and 2. A near-final pdf version of EC1ed2 is available and useful for searching.

OTHER REFERENCES: The following texts and treatises are at the graduate or advanced undergraduate level. They may provide useful alternative points of view or more leisurely presentations of some of the material we will cover. (I'm not including here the much longer list of books concerned with more specialized topics related to representation theory, polytopes, etc.)

  • M. Aigner: Combinatorial Theory (Classics in Mathematics), Springer, 1979 (reprint 2013).
  • M. Aigner: A Course in Enumeration (GTM), Springer, 2007.
  • M. Bona: Combinatorics of Permutations, Chapman and Hall/CRC, 2004.
  • M. Bona: Introduction to Enumerative Combinatorics, McGraw-Hill, 2005.
  • M. Bona: A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory, World Scientific, 2002, 2006, 2011.
  • L. Comtet: Advanced Combinatorics, Reidel/Kluwer, 1974.
  • I. Goulden & D. Jackson Combinatorial Enumeration, Wiley, 1983; Dover reprint, 2004.
  • J. van Lint & R. Wilson: A Course in Combinatorics, Cambridge, 2001.
  • R. Merris: Combinatorics, Wiley, 2003.
  • R. Stanley: Algebraic Combinatorics: Walks, Trees, Tableaux, and More, Springer UTM, 2013.
  • H. Wilf: Generatingfunctionology, Academic Press 1990, 1994: A. K. Peters, 2006. (2nd edition is freely downloadable).

WEBPAGE: The homepage for this course is http://www.math.uconn.edu/~troby/Math5020F14/. Some others web resources are listed below.

SOFTWARE: While doing some computations by hand is generally essential for learning, having software that can do bigger computations or check one's work is very useful. The free open-source computer algebra system Sage has an extensively-developed collection of objects and algorithms implemented for algebraic combinatorics. See for example, their thematic tutorials. Also, Christian Krattenthaler has kindly compiled a comprehensive list of Combinatorial software and databases

GRADING: Your grade will be based on several HW assignments, and possibly a presentation given in class.

The text is loaded with exercises, most of them with solutions, though the latter often are brief outlines, sometimes with pointers to the research literature. Stanley's knowledge of the subject is encylopedic, and he was able include a large amount of interesting material this way. If I assign an exercise whose solution is already included, I'll expect much more detail in what you hand in. I recommend you browse through many of the unassigned exercises to get a sense for what's there.

You may find some homework problems to be challenging, and collaborating to find solutions is certainly fine. It's OK to get significant help from other resources, but in the end, please write your own solution in your own words. Copying someone else's work without credit is plagiarism and will be dealt with according to university policy. It also is a poor learning strategy.

ACADEMIC INTEGRITY: Please make sure you are familiar with and abide by The Student Code governing Academic Integrity in Undergraduate Education and Research.

ACKNOWLEDGEMENTS: Many people have influenced my approach to this material, particularly Richard Stanley, but also Ira Gessel, Jim Propp, and Gian-Carlo Rota.

CONTENT: Although interesting counting problems go back to Archimedes and beyond, as its own field of study combinatorics really came into its own in the second half of the twentieth century. The seminal work of Paul Erdős on the graph theory and extremal side connected it with the young field of probability and a number of new problems arising in the fledgling field of computer science. On the algebraic side, Gian-Carlo Rota, Richard Stanley, and their students connected problems in enumeration and the structure of posets to many more established fields of mathematics: representation theory, algebraic topology, commutative algebra, and algebraic geometry. All sides of combinatorics have been aided by the rapid advances in computing power, which allow the exploration of many more examples than one could do by hand. Conversely, many problems arising in computer science have been solved by combinatorial means, in some cases guiding the discover of new techniques.

EC1 and EC2 contain easily enough material for two semesters, and choosing among topics will be difficult. If there's a topic you would particularly enjoy seeing, please let me know early on. At the moment, I plan to spend a good deal of time in chapters 1 and 3, some in chapters 2, 4, and 5. Chapter 7 on symmetric functions could easily fill an entire semester on its own, but some of you may have already seen this elsewhere.

ACCESSIBILITY & DISABILITY ISSUES: Please contact me and UConn's Center for Students with Disabilities as soon as possible if you have any accessibility issues, have a (documented) disability and wish to discuss academic accommodations, or if you would need assistance in the event of an emergency.

HOMEWORK ASSIGNMENTS:

  • [Due 9/18] HW1: EC1ed2, Ch. 1: #4, 5, 16, 17, 18, 20, 21, 26, 28, 29.
  • [Due 10/24] HW2: EC1ed2, Ch. 1: #38, 44, 47, 55, 112, 121, 126, 132, 192.
  • [Due 12/8] HW3: EC1 Ed2 Ch. 3: Ex#14, 34, 42, 45a, 62a-e, 70ab, 85, 144, 187a, 212

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