Abstract
In 1916, Ramanujan’s showed the spt-crank for marked overpartitions. The corresponding special functions , and are found in Ramanujan’s notebooks, part 111.
 In 2009, Bingmann, Lovejoy and Osburn defined the generating functions for ,
 and . In 2012, Andrews, Garvan, and Liang defined the 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 - partitions with 4 components, each a partition with certain restrictions are also discussed. The generating functions , , , , are shown with the corresponding results in terms of modulo 3, where the generating functions , are collected from Ramanujan’s notebooks, part 111. This paper shows how to prove the Theorem 1 in terms of ,Theorem 2 in terms of and Theorem 3 in terms of respectively with the numerical examples, and shows how to prove the Theorems 4,5 and 6 with the help of in terms of partition pairs. In 2014, Garvan and Jennings-Shaffer are able to defined the for marked overpartitions. This paper also shows another results with the help of 6 -partition pairs of 3, help of 20 -partition pairs of 5 and help of 15 -partition pairs of 8 respectively.
Highlights
In this paper we give some related definitions of spt (n), spt1 n, spt2 n, various product notations, vector partitions and S - partitions, M (m, n), M (m,t, n), M (m, n), M (m,t, n), S1 M (m, n) S2 M (m,t, n), S2z, x, S1 z, x, S 2 z, x, marked partition and sptcrank for marked overpartitions.We discuss the generating functions for spt (n), spt1 n, spt2 n and prove the Corollaries 1, 2 and 3 with the help of generating functions for M (m, n), M (m, n) and M (m, n) respectively
In 2014, Garvan and Jennings-Shaffer are able to defined the sptcrank for marked overpartitions. This paper 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 Keywords: Components, congruent, crank, overpartitions, overlined, weight
We prove the Theorems 1, 2 and 3 with the help of various generating functions and establish the Corollaries 4, 5 and 6 with the help special series S z, x, S1 z, x and S 2 z, x respectively, where the special series S z, x, S1 z, x and S 2 z, x are collected from Ramanujan’s notebooks, part 111, and prove the Theorems 4, 5 and 6 with the help of sptcrank in terms of partition pairs 1, 2 when 0 s( 1) s( 2 )
Summary
In this paper we give some related definitions of spt (n), spt n , spt n , various product notations, vector partitions and S - partitions, M (m, n) , M (m,t, n) , M (m, n) , M (m,t, n) , S1. S. M (m,t, n) : The number of vector partitions of n in S with crank congruent to m modulo t S counted according to the weight is exactly M (m,t, n). M (m, n) : The number of vector partitions S2 of with crank m counted according to the weight is exactly M (m, n). THE GENERATING FUNCTION FOR spt (n): spt n is the number of smallest parts in the overpartitions of n with smallest part not overlined and even like-.
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