Hossain, Fazlee and Das, Sabuj and Mohajan, Haradhan (2014): BingmannLovejoyOsburn’s generating function in the overpartitions. Published in: Open Science Journal of Mathematics and Application , Vol. 2, No. 4 (31 December 2014): pp. 3743.

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Abstract
In 2009, Bingmann, Lovejoy and Osburn defined the generating function for spt(n). In 2012, Andrews, Garvan and Liang defined the sptcrank in terms of partition pairs. In this article the number of smallest parts in the overpartitions of n with smallest part not overlined is discussed, and the vector partitions and S ̅ partitions with 4 components, each a partition with certain restrictions are also discussed. The generating function for spt(n), and the generating function for Ms(m, n) are shown with a result in terms of modulo 3. This paper shows how to prove the Theorem 1 in terms of Ms(m, n) with a numerical example, and shows how to prove the Theorem 2 with the help of sptcrank in terms of partition pairs. In 2014, Garvan and JenningsShaffer are able to define the sptcrank for marked overpartitions. This paper also shows another result with the help of 6 SP partition pairs of 3 and shows how to prove the Corollary with the help of 42 marked overpartitions of 6.
Item Type:  MPRA Paper 

Original Title:  BingmannLovejoyOsburn’s generating function in the overpartitions 
English Title:  BingmannLovejoyOsburn’s generating function in the overpartitions 
Language:  English 
Keywords:  Components, Congruent, Crank, NonNegative, Overpartitions, Overlined, Weight 
Subjects:  C  Mathematical and Quantitative Methods > C3  Multiple or Simultaneous Equation Models ; Multiple Variables C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling 
Item ID:  83044 
Depositing User:  Haradhan Kumar Mohajan 
Date Deposited:  26 Dec 2017 08:52 
Last Modified:  26 Dec 2017 08:53 
References:  1. Andrews, G.; Dyson, F. and Rhoades R., On the Distribution of the sptcrank, Mathematics, 1(3): 76–88, 2013. 2. Andrews, G.E.; Garvan, F.G. and Liang, J., Combinatorial Interpretations of Congruences for the sptfunction, Ramanujan J. 29(1–3): 321–338, 2012. 3. Berkovich, A. and Garvan, F.G., K. Saito’s Conjecture for Nonnegative eta Products and Analogous Results for other Infinite Products. J. Number Theory, 128(6): 1731–1748, 2008. 4. Bringann, K.; Lovejoy, J. and Osburn, R., Rank and Crank Moments for Overpartitions, J. Number Theory, 129(7):1758–1772, 2009. 5. Bringann, K.; Lovejoy, J. and Osburn, R., Automorphic Properties of Generating Functions for Generalized Rank Moments and Durfee Symbols, Int. Math. Res. Not. IMRN, (2): 238–260, 2010. 6. Chen, W.Y.C.; Ji, K.Q. and Zang, W.J.T., The sptcrank for Ordinary Partitions, arXiv eprints, Aug. 2013. 7. Garvan, F.G. and Shaffer, C.J., The sptcrank for Overpartitions, arXiv:1311.3680v2 [Math. NT], 23 Mar 2014. 8. Lovejoy, J. and Osburn, R., M2rank Differences for Partitions without Repeated Odd Parts. J. Theor. Nombres Bordeaux, 21(2): 313–334, 2009. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/83044 