Andrews-Garvan-Liang’s Spt-crank for Marked Overpartitions

In 2009, Bingmann, Lovejoy and Osburn have shown the generating function for spt2(n). In 2012, Andrews, Garvan, and Liang have 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 and even are discussed, and the vector partitions and S-partitions with 4 components, each a partition with certain restrictions are also discussed. The generating function for spt2(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 capable to define the sptcrank for marked overpartitions. This paper also shows another result with the help of 15 SP2-partition pairs of 8 and shows how to prove the Corollary with the help of 15 marked overpartitions of 8