We give a new generalization of the spt-function of G. E. Andrews, namely Sptj(n), and give its combinatorial interpretation in terms of successive lower-Durfee squares. We then generalize the higher order spt-function sptk(n), due to F. G. Garvan, to jsptk(n), thus providing a two-fold generalization of sptn, and give its combinatorial interpretation. Lastly, we show how the positivity of jsptk(n) can be used to generalize Garvan's inequality between rank and crank moments to the moments of j-rank and (j+1)-rank.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory