Geodesic
Partitions with fixed number of sizes
Journal of integer sequences, Tome 18 (2015) no. 11
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: (i) $s = k - 1$ 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 of its partial sum is derived. Finally, a parity result for $t(n, 2)$ is obtained.
Classification : 05A17, 11P81
Keywords: partition, size of a partition, parity, asymptotic estimate 
@article{JIS_2015__18_11_a4,
     author = {Christopher,  David},
     title = {Partitions with fixed number of sizes},
     journal = {Journal of integer sequences},
     year = {2015},
     volume = {18},
     number = {11},
     zbl = {1328.05011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2015__18_11_a4/}
}
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Christopher,  David. Partitions with fixed number of sizes. Journal of integer sequences, Tome 18 (2015) no. 11. http://geodesic.mathdoc.fr/item/JIS_2015__18_11_a4/