Math 121b  
Combinatorial
Analysis 

Winter 201415  

Instructor: Eric
Rains,
276
Sloan,
6263954322, rains@caltech.edu

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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. 

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. 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).


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://wwwmath.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] 






