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

Let Π s ( n ) denote the set of partitions of the integer n into exactly s parts, and Π s ( 2 ) ( n ) the subset of Π s ( n ) containing all partitions whose two largest parts coincide. We present a bijection between Π s ( 2 ) ( n ) and Π s − 1 ( m ) for a suitable m < n in the cases s = 3 , 4 . Such bijections yield recurrence formulas for the numbers P 3 ( n ) and P 4 ( 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 .