Bingmann-Lovejoy-Osburn’s Generating Function in the Overpartitions
[1]
Fazlee Hossain, Department of Mathematics, University of Chittagong, Bangladesh.
[2]
Sabuj Das, Department of Mathematics, Raozan University College, Bangladesh.
[3]
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
[1]
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[7]
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[8]
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