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

Let Πs(n)Πs(n) denote the set of partitions of the integer nn into exactly ss parts, and Πs(2)(n) the subset of Πs(n)Πs(n) containing all partitions whose two largest parts coincide. We present a bijection between Πs(2)(n) and Πs−1(m)Πs−1(m) for a suitable m