Two problems in the Theory of Partitions

Abstract:
The Theory of Partitions has blossomed into a wonderful
subject with incredibly many ramifications and applications, for
example, in q-series, Theory of Modular Forms, Mock Theta functions
etc.  We consider two recent topics of interest in this field. The
first one concerns the smallest parts function spt(n), introduced by
George Andrews in 2008, which has attracted a lot of attention. We
give a new generalization of this function, namely Spt_j(n), and give
its combinatorial interpretation in terms of successive lower-Durfee
squares. We then generalize the higher order spt-function spt_k(n),
due to F. G. Garvan, to j_spt_k(n), thus providing a two-fold
generalization of spt(n), and give its combinatorial interpretation.
This also allows us to generalize Garvan's famous inequality between
2k-th moments of rank and crank to an inequality between 2k-th moments
of j-rank and (j+1)-rank. This is joint work with Ae Ja Yee
(Pennsylvania State University).
   The second topic deals with certain useful partial
differential equations associated with partition statistics. In 2003,
A. O. L. Atkin and F.G. Garvan obtained a PDE linking rank and crank
generating functions. The method of deriving this PDE was elementary
and used Theory of Elliptic Functions. Higher order PDEs were recently
found by S. P. Zwegers using ideas motivated from the Theory of Jacobi
Forms. Here, we show that these higher order PDEs may be obtained from
a generalized Lambert series identity, which proves them much in the
spirit of Atkin and Garvan’s proof of the Rank-Crank PDE. This is
joint work with Song Heng Chan (Nanyang Technological University) and
F. G. Garvan (University of Florida).