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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Andrews, G.E.: Problems and prospects for basic hypergeometric functions. In: Askey, R.A. (ed.) Theory and Application of Special Functions, pp. 191–224. Academic Press, New York (1975)
Andrews, G.E.: q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics and Computer Algebra. C.B.M.S. Regional Conference Series in Math, vol. 66. Am. Math. Soc., Providence (1986)
Andrews, G.E.: Partitions, Durfee symbols, and the Atkin–Garvan moments of ranks. Invent. Math. 169, 37–73 (2007)
Andrews, G.E.: The number of smallest parts in the partition of n. J. Reine Angew. Math. 624, 133–142 (2008)
Andrews, G.E., Garvan, F.G.: Dyson’s crank of a partition. Bull., New Ser., Am. Math. Soc. 18, 167–171 (1988)
Atkin, A.O.L., Garvan, F.: Relations between the ranks and cranks of partitions. Ramanujan J. 7, 343–366 (2003)
Dyson, F.J.: Some guesses in the theory of partitions. Eureka (Cambridge) 8, 10–15 (1944)
Dyson, F.J.: Mappings and symmetries of partitions. J. Comb. Theory, Ser. A 51, 169–180 (1989)
Garvan, F.G.: New combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7 and 11. Trans. Am. Math. Soc. 305, 47–77 (1988)
Garvan, F.G.: Generalizations of Dyson’s rank and non-Rogers–Ramanujan partitions. Manuscr. Math. 84, 343–359 (1994)
Garvan, F.G.: Higher order spt-functions. Adv. Math. 228, 241–265 (2011)
Warnaar, S.O.: 50 years of Bailey’s lemma. In: Betten, A., et al. (eds.) Algebraic Combinatorics and Applications, pp. 333–347. Springer, Berlin (2001)
Acknowledgements
The authors sincerely thank Bruce C. Berndt for several suggestions which improved the quality of this paper. This work was done while the second author was visiting University of Queensland. She thanks Ole Warnaar for his warm hospitality.
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The second author was partially supported by National Security Agency Grant H98230-10-1-0205 and by the Australian Research Council.
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Dixit, A., Yee, A.J. Generalized higher order spt-functions. Ramanujan J 31, 191–212 (2013). https://doi.org/10.1007/s11139-012-9434-2
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DOI: https://doi.org/10.1007/s11139-012-9434-2