How do I generate a uniform random integer partiti

Question

A Google search reveals plenty about generating all possible partitions of an integer n into m parts, but I haven't found anything about sampling a uniformly distributed random partition of n into m parts.

Answer 1

I have implemented the above solution and found that it works very well if one wants to calculate integer partitions for n but not with respect to m. If working with large n, recursion limits and call stacks may need to be increased a lot.

However, you don't need the first function because count_partitions(n, limit) will actually equal the number of partitions of 'n+limit' with 'limit' number of parts. Some mathematical software have very fast functions for finding the number of partition of n into m parts.

I have recently derived a definitely unbiased, very simple, and very fast method (using memoization) to solve your exact question: An algorithm for randomly generating integer partitions of a particular length, in Python?

It's based on knowing something about lexically ordered partitions of n having m parts and uses a similar approach to well-accepted algorithms (e.g. Nijenhuis and Wilf 1978) that find random partitions of n, and is conceptually similar to the above.

In short, if there are x partitions of n with m parts, then we choose a random number between 1 and x. That random number will code for one and only one partition satisfying n and m. I hope this helps.