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Bingmann-Lovejoy-Osburn’s Generating Function in the Overpartitions
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Volume 2, 2014
Issue 4 (August)
Pages: 37-43   |   Vol. 2, No. 4, August 2014   |   Follow on
Paper in PDF Downloads: 20   Since Aug. 28, 2015 Views: 1600   Since Aug. 28, 2015
Authors
[1]
Fazlee Hossain, Department of Mathematics, University of Chittagong, Bangladesh.
[2]
Sabuj Das, Department of Mathematics, Raozan University College, Bangladesh.
[3]
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 M_s ̅ (m,n) are shown with a result in terms of modulo 3. This paper shows how to prove the Theorem 1 in terms of M_s ̅ (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 Jennings-Shaffer are able to defined 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.
Keywords
Components, Congruent, Crank, Non-Negative, Overpartitions, Overlined, Weight
Reference
[1]
Andrews, G.; Dyson, F. and Rhoades R., On the Distribution of the spt-crank, Mathematics, 1(3): 76–88, 2013.
[2]
Andrews, G.E.; Garvan, F.G. and Liang, J., Combinatorial Interpretations of Congruences for the spt-function, 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 spt-crank for Ordinary Partitions, arXiv e-prints, Aug. 2013.
[7]
Garvan, F.G. and Shaffer, C.J., The spt-crank for Overpartitions, arXiv:1311.3680v2 [Math. NT], 23 Mar 2014.
[8]
Lovejoy, J. and Osburn, R., M2-rank Differences for Partitions without Repeated Odd Parts. J. Theor. Nombres Bordeaux, 21(2): 313–334, 2009.
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