Haradhan Kumar Mohajan
In 2009, Bingmann, Lovejoy and Osburn have shown the generating function for . In 2012, Andrews, Garvan, and Liang have defined the in terms of partition pairs. In this article the number of smaller parts in the overpartitions of n with smallest part not overlined and even are discussed, and the vector partitions and -partitions with 4 components, each a partition with certain restrictions are also discussed. The generating function for , and the generating function for are shown with a result in terms of modulo 3. This paper shows how to prove the Theorem 1, in terms of with a numerical example, and shows how to prove the Theorem 2, with the help of in terms of partition pairs. In 2014, Garvan and Jennings-Shaffer are capable to define the for marked overpartitions. This paper also shows another result with the help of 15 -partition pairs of 8 and shows how to prove the Corollary with the help of 15 marked overpartitions of 8.