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 $j_{\mathrm{spt}_k(n)}$, 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).
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 $j_{\mathrm{spt}_k(n)}$, 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).