Journal of Integer Sequences, Vol. 18 (2015), Article 15.11.5

Partitions with Fixed Number of Sizes

David Christopher
Department of Mathematics
The American College
Tamil Nadu


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.

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(Concerned with sequences A000005 A000203 A002133 A004526 A010052 A117955 A117956.)

Received March 27 2015; revised versions received August 29 2015; October 5 2015; December 7 2015. Published in Journal of Integer Sequences, December 9 2015.

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