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Partitions with Fixed Number of Sizes
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David Christopher

Department of Mathematics

The American College

Madurai

Tamil Nadu

India

**Abstract:**

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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