A Hardy-Ramanujan formula for restricted partitions

Abstract In this paper, we extend the Hardy-Ramanujan-Rademacher formula for p ( n ), the number of partitions of n . In particular we provide such formulas for p ( j , n ), the number of partitions of j into at most n parts and for A ( j , n , r ), the number of partitions of j into at most n parts each ≤ r .