Abstract

In 1916, Ramanujan’s showed the spt-crank for marked overpartitions. The corresponding special functions S z, x , S1z, x and S2 z, x are found in Ramanujan’s notebooks, part 111. In 2009, Bingmann, Lovejoy and Osburn defined the generating functions for sptn , spt n 1 and spt n 2 . 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, not overlined and odd, 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 functions sptn , spt n 1 , spt n 2 , M (m,n) S , ( , ) 1 M m n S ( , ) 2 M m n S are shown with the corresponding results in terms of modulo 3, where the generating functions M (m,n) S , ( , ) 1 M m n S ( , ) 2 M m n S are collected from Ramanujan’s notebooks, part 111. This paper shows how to prove the Theorem 1 in terms of M (m,n) S ,Theorem 2 in terms of ( , ) 1 M m n S and Theorem 3 in terms of ( , ) 2 M m n S respectively with the numerical examples, and shows how to prove the Theorems 4,5 and 6 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 results with the help of 6 SP -partition pairs of 3, help of 20 SP1 -partition pairs of 5 and help of 15 SP2 -partition pairs of 8 respectively

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