Date

Lecture Topics / Reading / Handouts

Thurs., Jan. 12

Definition
of partitions (Andrews 1.1); Motivating examples: finitely
generated abelian groups, and conjugacy classes in S_n

Tues., Jan. 17

Partition
function p(n); Euler's OddDistinct Theorem (Andrews 1.2;
page 10 of H. Wilf's lecture
notes); Ferrers Diagrams (Andrews 1.3)

Thurs., Jan. 19

Partition
statistics, including largest part and length; Conjugate of
a partition (Andrews 1.3); Sylvester's proof of Euler's
OddDistinct Theorem (p. 12 of I. Pak's lecture)

Tues., Jan. 24

Sylvester's
proof of Euler's Theorem (cont.); Introduction to generating
functions; Van der Monde's Identity (Wikipedia);
Partition generating functions; Generating function proof of
Euler's Theorem (Andrews 1.2)

Thurs., Jan. 26

Congruences
modulo 2 for partition generating functions; Twoparameter
generating functions; Pentagonal Number Theorem and
Franklin's proof via "almost"involution (Andrews 1.3)

Tues., Jan. 31

Consequences
of Pentagonal Number Theorem, including computing p(n)
recursively; Introduction to qfactorials (Andrews 2.1)

Thurs., Feb. 2

Properties
of qfactorials; Partitions with at most m parts; Cauchy's
Theorem (Andrews 2.1), proof by series expansion and
qdifference equations; Binomial theorem as corollary

Tues., Feb. 7

Euler's
identities, as corollaries of Cauchy's Theorem (Andrews
2.1), and as combinatorial partition identities; Jacobi
Triple Product (Andrews 2.2), and Pentagonal Number Theorem
as corollary

Thurs., Feb. 9

Proof of
Jacobi Triple Product (Andrews 2.2); Heine's transformation
and summation formulas; Basic hypergeometric series (Digital
Library of Mathematical Functions at NIST); Examples of
theta functions

Tues, Feb. 14

Partition
identities from basic hypergeometric series (Andrews 2.3);
Combinatorial proofs, Durfee squares, counting largest part
and number of parts simultaneously

Thurs., Feb. 16

Classical
hypergeometric functions (DLMF
at NIST); Examples and Identities, Euler's transformation;
Basic hypergeometric series

Tues., Feb. 21

Finite/polynomial
qseries; Motivating example: inversions in permutations;
qbinomial coefficients

Thurs., Feb. 23

Gaussian
polynomials and restricted partitions (Andrews 3.2 and 3.3);
qBinomial Theorem, analytic and combinatorial proofs

Thurs., Mar. 2

qChuVandermonde
identity, analytic proof (Andrews 3.3); qanalog of Hockey
Stick Theorem (Wikipedia);
Permutations of multisets, the inversion statistic, and
qmultinomial coefficients (Andrews 3.4)

Tues., Mar. 7

Proof of
generating function for inversion statistic (Andrews 3.4);
Greater index statistic; Reciprocal and unimodal polynomials
(Andrews 3.5, R. Stanley's survey)

Thurs., Mar. 9

Basic
results on reciprocal and unimodal polynomials (D.
Zeilberger's paper);
Combinatorics of inversions, including combinatorial proof
of Andrews' Theorem 3.11

Tues., Mar. 14

Discussion
of O'Hara's qseries proof of unimodality for Gaussian
polynomials (K. O'Hara's paper);
Szekeres' proof of unimodality of restricted partitions (G.
Szekeres' paper);
Relations between restricted partitions and compositions
(Andrews 4.2)

Thurs., Mar. 16

Proof of
ErdosLehner Theorem (Andrews 4.2); Introduction to
Ramanujan congruences (Andrews Chapter 10, S. Ramanujan's paper)

Tues., Mar. 21

Proof of
Ramanujan congruence modulo 5 (Andrews Examples 10.7 
10.13); Discussion of partition congruences, including work
of Atkin, Newman, Ono (K. Ono's survey);
Ramanujan's congruences modulo powers of 5 (Andrews 10.3);
Introduction to product expansions of formal power series

Thurs., Mar. 23

Product
expansions of formal power series; SumProduct Identities
and partitions with gap conditions, including Euler,
RogersRamanujan, and Schur (Andrews Chapter 7; I. Pak's survey)

Tues., Mar. 28

Proof of
Schur's second partition Theorem using recurrences and
qdifference equations (Andrews Examples 7.1  7.4; Andrews'
paper)

Thurs., Mar. 30

Appell's
Comparison Theorem; Heine's transformation and Schur's
theorem; Proof of Lehmer/Alder's result on nonexistence of
gapproduct identities, d=3 case (Lehmer's paper);
Discussion of AlderAndrews Conjecture

Tues., Apr. 4

kmodular
partition diagrams (Andrews Examples 1.6  1.7); Conjugation
of 2modular diagrams; Bressoud's bijective proof of Schur's
Theorem (D. Bressoud's paper, I. Pak's survey);
Double sum representation for Schur partitions
(AndrewsBringmannMahlburg's paper)

Thurs., Apr. 6

Ehrenpreis
problem (K. Kadell's paper);
Andrews and Baxter's "Motivated proof" of RogersRamanujan
(AndrewsBaxter's paper)

Tues., Apr. 18

Proof of
RogersRamanujan identities using qdifference equations
(Rogers and Ramanujan's paper in archived journal
volume); Ramanujan congruences and Dyson's rank
statistic (A.O.L. Atkin and P. SwinnertonDyer's paper)

Thurs., Apr. 20

Guest
Speaker: Ken Ono, Partitions with restricted smallest parts
and arithmetic densities (K. Ono, R. Schneider, and I.
Wagner's preprint);
Garvan and AndrewsGarvan crank statistic and congruences
(K. Mahlburg's paper)

Tues., Apr. 22

Circle
Method and HardyRamanujan's asymptotic formula for the
partition function (G. Hardy and S. Ramanujan's paper);
HardyRamanujan Tauberian Theorem; Asymptotic behavior of
log(P(q))

Thurs., Apr. 27

Wright's
Circle Method (K. Bringmann and K. Mahlburg's paper), definition of Major
and Minor Arc; Modular inversion formula for P(q); Proof of
asymptotic main term for p(n), and error term
