MATH 581 (Spring 2019)
This course provides an introduction to combinatorics,
probability theory, and related areas from the perspective of
Topics include: algorithms for the manipulation of
generating functions; effective analytic methods for
asymptotics; transcendence proofs; rational, algebraic,
rational diagonal, D-finite, and differential-algebraic
generating functions and their algebraic/analytic/arithmetic
properties; computability questions in enumeration.
Applications from various areas of combinatorics and
mathematics will be discussed, tailored to the interests of
participants. Familiarity with advanced methods in symbolic
computation / computer algebra will not be assumed, however
students taking this course for credit will be expected to
perform basic coding in a computer algebra system (Maple,
Sage, Mathematica, etc.)
Combinatorics and Symbolic Computation
Expected Background: Students should be comfortable with the basics of real analysis (sequences and series), complex analysis (Cauchy residue theorem), and algebra (linear algebra, rings and fields). Exposure to courses in computer science, combinatorics, probability theory, or computer algebra would be beneficial. Interested students unsure of their background should email the instructor for more information.
A good idea of the selection of topics can be found in notes from a recent two week summer school, available here.
Will be posted later.
The main reference for this class is a manuscript draft which
will be distributed, based on the PhD thesis:
Analytic Combinatorics in Several Variables: Effective Asymptotics and Lattice Path Enumeration
S. Melczer. PhD Thesis, University of Waterloo and ENS Lyon, 2017.
Additional details on certain topics can be found in the following sources, all of which are freely available.
P. Flajolet and B. Sedgewick. Cambridge University Press, 2009.
H. Wilf. Academic Press, 1990.
Analytic Combinatorics in Several Variables
R. Pemantle and M. Wilson. Cambridge University Press, 2013.
Algorithmes Efficaces en Calcul Formel [Efficient Algorithms in Computer Algebra]
A. Bostan, F. Chyzak, M. Giusti, R. Lebreton, G. Lecerf, B. Salvy and E. Schost, 2017.