, Volume 31, Issue 1-2, pp 191-212

Generalized higher order spt-functions

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Abstract

We give a new generalization of the spt-function of G.E. Andrews, namely $\operatorname {Spt}_{j}(n)$ , and give its combinatorial interpretation in terms of successive lower-Durfee squares. We then generalize the higher order spt-function $\operatorname {spt}_{k}(n)$ , due to F.G. Garvan, to ${}_{j\!}\operatorname {spt}_{k}(n)$ , thus providing a two-fold generalization of $\operatorname {spt}(n)$ , and give its combinatorial interpretation. Lastly, we show how the positivity of j spt k (n) can be used to generalize Garvan’s inequality between rank and crank moments to the moments of j-rank and (j+1)-rank.

The second author was partially supported by National Security Agency Grant H98230-10-1-0205 and by the Australian Research Council.