Generalized higher order spt-functions |
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Authors: | Atul Dixit Ae Ja Yee |
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Institution: | 1. Department of Mathematics, University of Illinois, Urbana, IL, 61801, USA 2. Department of Mathematics, Tulane University, New Orleans, LA, 70118, USA 3. Department of Mathematics, Penn State University, University Park, PA, 16802, USA
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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. |
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