Partitions with Fixed Number of Sizes
Department of Mathematics
The American College
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.
Full version: pdf,
(Concerned with sequences
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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