MATH 7230 - Spring 2017

Partitions, Hypergeometric q-series, and Modular Forms

This is a first course in the theory of integer partitions, and their connections to many other areas of mathematics, including combinatorics, special functions, and number theory.  See the Syllabus and detailed Lecture Schedule for more details.

Course Information

Scheduled Time
Lectures TTh 10:30
Lockett 111
Office Hours T 12:00
Lockett 320

George Andrews, The Theory of Partitions, Cambridge University Press, 1998

Problem Sets

Other references

Papers: (links provided to original sources; subscription or LSU campus-access may be required)

Links to other resources:

Lecture Schedule

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 Odd-Distinct 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 Odd-Distinct 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; Two-parameter 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 q-factorials (Andrews 2.1)
Thurs., Feb. 2
Properties of q-factorials; Partitions with at most m parts; Cauchy's Theorem (Andrews 2.1), proof by series expansion and q-difference 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 q-series; Motivating example: inversions in permutations; q-binomial coefficients
Thurs., Feb. 23
Gaussian polynomials and restricted partitions (Andrews 3.2 and 3.3); q-Binomial Theorem, analytic and combinatorial proofs
Thurs., Mar. 2
q-Chu-Vandermonde identity, analytic proof (Andrews 3.3); q-analog of Hockey Stick Theorem (Wikipedia); Permutations of multisets, the inversion statistic, and q-multinomial 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 q-series 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 Erdos-Lehner 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; Sum-Product Identities and partitions with gap conditions, including Euler, Rogers-Ramanujan, and Schur (Andrews Chapter 7; I. Pak's survey)
Tues., Mar. 28
Proof of Schur's second partition Theorem using recurrences and q-difference 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 gap-product identities, d=3 case (Lehmer's paper); Discussion of Alder-Andrews Conjecture
Tues., Apr. 4
k-modular partition diagrams (Andrews Examples 1.6 - 1.7); Conjugation of 2-modular diagrams; Bressoud's bijective proof of Schur's Theorem (D. Bressoud's paper, I. Pak's survey); Double sum representation for Schur partitions (Andrews-Bringmann-Mahlburg's paper)
Thurs., Apr. 6
Ehrenpreis problem (K. Kadell's paper); Andrews and Baxter's "Motivated proof" of Rogers-Ramanujan (Andrews-Baxter's paper)
Tues., Apr. 18
Proof of Rogers-Ramanujan identities using q-difference equations (Rogers and Ramanujan's paper in archived journal volume); Ramanujan congruences and Dyson's rank statistic (A.O.L. Atkin and P. Swinnerton-Dyer'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 Andrews-Garvan crank statistic and congruences (K. Mahlburg's paper)
Tues., Apr. 22
Circle Method and Hardy-Ramanujan's asymptotic formula for the partition function (G. Hardy and S. Ramanujan's paper); Hardy-Ramanujan 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

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