Higher Order Overpartition Spt Functions

When/Where:

April 8, 2014, at 1:55pm, in LIT 368

Abstract:

In 2003 Atkin and Garvan introduced the \(k\)-th rank and crank moments \(M_k(n) = \displaystyle\sum_{m}m^k M(m,n)\) and \(N_k(n) = \displaystyle\sum_{m}m^k N(m,n)\), where \(N(m,n)\) is the number of partitions of \(n\) with rank \(m\) and \(M(m,n)\) is the number of partitions of \(n\) with crank \(m\). A symmetrized version of the rank moments was studied by Andrews in 2006. In 2011 Garvan used symmetrized rank and crank moments to define a higher order \(\mathrm{spt}\) function, noting Andrews’ \(\mathrm{spt}\) function is half the difference of the second crank and rank moment. This also established an inequality between crank and rank moments.

We show this idea extends to the \(\mathrm{spt}\) functions for overpartitions, overpartitions with smallest part even, and partitions with smallest part even and without repeated odd parts. In this talk we give the necessary definitions, in particular what are the symmetrized moments, and review the steps for the ordinary \(\mathrm{spt}\) function. We then continue on with the three additional \(\mathrm{spt}\) functions. The crucial identities follow from a theorem in Garvan’s work that is an application of Bailey pairs and Bailey’s Lemma.