Number of partitions in decreasing order

Question

I want to find out the number of partitions which are strictly in decreasing order.

Eg. Partition of 4 is :

Here there's only one partition {3, 1} which is in strictly decreasing order, call this function f.I am interested in f(200)

I tried to list all possible partitions and search for the one which are in decreasing order, but it's too slow to work. I believe there is some analytical combinatorics that I'm missing.

I would have thought that the first partition on your list, 4, would also be considered to be in strictly decreasing order. — paw88789, Oct 19 '16 at 15:34
@paw88789 for simplicity, just ignore the first number i.e. the number itself — Mayur Kulkarni, Oct 19 '16 at 15:35

Robert Z · Accepted Answer · 2016-10-19T15:50:33.870

1

Strictly decreasing order means that the parts are all different. Then the generating function of this sequence is $$F(x)=\prod_{k=1}^{\infty}(1+x^k).$$ Now $f(200)+1$ (you ignore the number itself) is the coefficient of $x^{200}$ in $F$, but don't see an easy way to find it by hand.

P.S. Maple says that $[x^{200}]F(x)=487067746$.

edited Oct 19 '16 at 15:50

answered Oct 19 '16 at 15:32

Robert Z

147,345

It is also the number of partitions of $200$ with odd parts. – Dietrich Burde Oct 19 '16 at 15:46
@Robert that works flawlessly, but can you tell me how did you arrive at this generating function – Mayur Kulkarni Oct 19 '16 at 15:58
@Mayur Kulkarni This is a classic. See for example here: http://math.stackexchange.com/questions/54961/the-number-of-partitions-of-n-into-distinct-parts-equals-the-number-of-partiti – Robert Z Oct 19 '16 at 16:12

Number of partitions in decreasing order

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