Congruence Properties of Andrews’SPT- Function

Abstract

Let spt (n) denote the total number of appearances of the smallest part in each partition of n. In 1988, Garvan gave new combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7 and 11 in terms of a crank for weighted vector partitions. This paper shows how to generate the generating functions for spt(n), elaborately and also shows how to prove the relation among the terms spt (n) and. In 2008, Andrews stated Ramanujan- type congruences for the spt- function mod 5, 7 and 13. The new combinatorial interpretations of the spt- congruences mod 5 and 7 are given in this article. These are in terms of the spt- crank but for a restricted set of vector partitions. The proofs depend on relating the spt- crank with the crank of vector partitions.