**Atul Dixit**

Tulane University, USA

January 3, 2013

**Two problems in the Theory of Partitions**:
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 u