Partitions with Fixed Number of Sizes

Let t(n, s) and t(n, k, s), respectively, be the number of partitions of n with s different sizes, and the number of partitions of n with exactly k parts and s different sizes. In this article, an asymptotic estimate for t(n, k, s) is presented for the following two cases: (1) s = k - 1 and (ii) when k is a prime number with s = 2. Further, the enumeration of uniform partitions with exactly 2 sizes is considered and the estimate for its partial sum is derived. Finally, a parity result for t(n, 2) is obtained.