Welcome to Open Science
Contact Us
Home Books Journals Submission Open Science Join Us News
Bingmann-Lovejoy-Osburn’s Generating Function in the Overpartitions
Current Issue
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
Fazlee Hossain, Department of Mathematics, University of Chittagong, Bangladesh.
Sabuj Das, Department of Mathematics, Raozan University College, Bangladesh.
Haradhan Kumar Mohajan, Premier University, Chittagong, Bangladesh.
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.
Components, Congruent, Crank, Non-Negative, Overpartitions, Overlined, Weight
Andrews, G.; Dyson, F. and Rhoades R., On the Distribution of the spt-crank, Mathematics, 1(3): 76–88, 2013.
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.
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.
Bringann, K.; Lovejoy, J. and Osburn, R., Rank and Crank Moments for Overpartitions, J. Number Theory, 129(7):1758–1772, 2009.
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.
Chen, W.Y.C.; Ji, K.Q. and Zang, W.J.T., The spt-crank for Ordinary Partitions, arXiv e-prints, Aug. 2013.
Garvan, F.G. and Shaffer, C.J., The spt-crank for Overpartitions, arXiv:1311.3680v2 [Math. NT], 23 Mar 2014.
Lovejoy, J. and Osburn, R., M2-rank Differences for Partitions without Repeated Odd Parts. J. Theor. Nombres Bordeaux, 21(2): 313–334, 2009.
Open Science Scholarly Journals
Open Science is a peer-reviewed platform, the journals of which cover a wide range of academic disciplines and serve the world's research and scholarly communities. Upon acceptance, Open Science Journals will be immediately and permanently free for everyone to read and download.
Office Address:
228 Park Ave., S#45956, New York, NY 10003
Phone: +(001)(347)535 0661
Copyright © 2013-, Open Science Publishers - All Rights Reserved