Bingmann-Lovejoy-Osburn’s generating function in the overpartitions

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 Jennings-Shaffer 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.