DEVELOPMENT OF ANDREWS-GARVAN-LIANG’S SELF-CONJUGATE S-PARTITIONS

In 2013, Andrews, Garvan and Liang defined Self-conjugate S-partitions. In 2011, Andrews stated the definition of spt(n). This paper shows how to find the Self-conjugate S-partitions of 4 and 5 respectively, and proves the Corollary-1 that is ‘The number of Self-conjugate S-partitions counted according to the weight w is congruent to the total number of smallest parts in all the partitions of n modulo 2’. This paper shows how to generate the generating functions for Msc(n) in different ways. This paper shows how to find the number of partitions of n with an odd number of smallest parts and a total number of even (respectively odd) parts, and shows how to find the number of partitions of n in odd parts without gaps. This paper shows how to prove the further three Corollaries with the help of examples, and shows how to prove the three Theorems by easy algebraic method.