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I am looking for the number of partitions of some number $n$ into $k$ parts. Following the Wikipedia article on partitions, I ended up with Andrew's book [1]. Judging by Google's preview Chapter 3 seems to contain information about what I need; unfortunately, the preview contains only one page of that chapter. Our library does not have a copy available either.

Even though there may not be an answer for me there I'd like to check available results. Since I can not access the reference list of said chapter: which papers is Andrew's chapter based upon?

Are there other (newer?) books or articles that contain relevant information?

  1. The Theory of Partitions by George E. Andrews (1976)
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Raphael
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2 Answers2

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$\newcommand{\partition}[2]{\genfrac{\lvert}{\rvert}{0pt}{}{#1}{#2}}$ I was not able to track down Andrew's references yet, but I found the following.

Knuth [1, p399] gives as ordinary generating function

$\qquad\displaystyle \sum_{n \geq 0}\ \partition{n}{k} = \frac{z^k}{(1-z)(1-z^2) \cdot \dots \cdot (1-z^k)}$

where $\partition{n}{k}$ is the number of partitions of $n$ into (exactly) $k$ parts.

Flajolet/Sedgewick [2, p47] give the asymptotic

$\qquad\displaystyle [z^n]\frac{1}{(1-z)(1-z^2) \cdot \dots \cdot (1-z^k)} \sim \frac{n^{k-1}}{k!(k-1)!}$,

for the number of partitions of $n$ with at most $k$ parts.

Put together, this gives

$\qquad\displaystyle \partition{n}{k} \sim \frac{(n-k)^{k-1}}{k!(k-1)!}$;

this may be the best known result.

  1. The Art of Computer Programming, Volume 4A by D. E. Knuth (2011)
  2. Analytic Combinatorics by P. Flajolet and R. Sedgewick (2009)
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In case you're a constructivist, a useful and widely known recursion exists:
p(n,k)=p(n,k-1)+p(n-k,k) with p(n,0)=0 and p(0,k)=1.

Wouter M.
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  • Thanks, but it does -- unfortunately -- not help me. (The recurrence is, of course, given in TAoCP, by the way) – Raphael Apr 01 '14 at 22:17