Bijections and recurrences for integer partitions in a bounded number of parts

Let Πs(n) denote the set of partitions of the integer n into exactly s parts, and image the subset of Πs(n) containing all partitions whose two largest parts coincide. We present a bijection between image and Πs−1(m) for a suitable m<n in the cases s=3,4. Such bijections yield recurrence formulas for the numbers P3(n) and P4(n) of partitions of n into 3 and 4 parts. Furthermore, we show that the present approach can be extended to the case s=5,6.