Math 121b
 
Combinatorial Analysis
Winter 2014-15
 
TR 1:00 PM // 257 Sloan
Course Description | Policies | Textbooks | Lecture Notes | Topics Covered | Homework | Sections

Instructor:  Eric Rains, 276 Sloan, 626-395-4322, rains@caltech.edu
Office Hours: 
TBA
TA:  TBA
Office Hours:  TBA



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Course Description
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In this second term more quantative and algebraic aspects of combinatorics will be studied.

Most of the time we will be discussing various aspects of counting. A significant part of the course will be devoted to studying generating functions; the most powerful tool to understanding combinatorical sequences that I know. We will also discuss Mobius inversion on general lattices, which is best seen as a generalization of the Principle of Inclusion/Exclusion. Another topic is Polya's Enumeration theory, which counts objects in the presence of symmetry, and is a generalization of Burnside's Lemma. Finally we will discuss some results on partitions and Young Tableaux, which are in essence three dimensional partitions.


Policies
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Grades: Grades will be assessed on the basis of homeworks, which will be given roughly biweekly. There will be no exams (or midterms). Students may discuss problems with each other but have to write down the solutions independently.

The following restrictions hold for all problems of the set: Students may ask questions to the TA (if one exists) and the professor. Resources which you may use while working on the homework include any books and non-interactive websites (i.e. no posting of the questions on internet fora). For calculations which you can do by hand, you may use a computer
algebra program (Mathematica, Maple, etc.); you may also fell free to use a computer to gain intuition (say by solving small instances of the problems), so long as your eventual proof does not depend on such a computation.  If you use significant ideas from any source, other than the books mentioned on this website, you should mention where you got them from.

You can get one extension for one homework set of 1 day, no questions asked, if requested before the deadline. Extensions for a longer period or after the first time you have to provide a letter from the Dean or a doctor (unless you did this for the first extension).


 


Textbooks
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The books for the course are the two volumes of Richard Stanley's excellent book on enumerative (and, despite the title, algebraic) combinatorics:

Enumerative Combinatorics (Vol. 1, 2nd ed.), R. Stanley, ISBN: 1107602629
Enumerative Combinatorics (Vol. 2), R. Stanley, ISBN: 0521789877

A preliminary version of the second edition of volume 1 is online here [http://www-math.mit.edu/~rstan/ec/ec1.pdf].  Exercise numbers may have changed between the two editions and the online version, so I will try to retype the exercises as necessary.  (If I don't, please complain!)

Some additional references for alternate approaches:  For generating functions,

Generatingfunctionology by Herbert Wilf ISBN: 1568812795

(freely downloadable from the link, http://www.math.upenn.edu/~wilf/gfology2.pdf). 

For P\'olya enumeration, there is a brief discussion in Chapter 37 of the book used in 121a:

A Course in Combinatorics, by J.H. van Lint and R.M. Wilson, 2nd
edition. ISBN: 0521006015

You may also find the following lecture notes from 2011 to be helpful:

[lecture notes from the old site]


Lecture Notes
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Topics Covered
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Homework
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Due Date Homework  Solutions
January 22, 2015
Homework 1

February 5, 2015
Homework 2

February 24, 2015
Homework 3

March 10, 2015
Homework 4






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