Random integer partitions with restricted numbers of parts

Currently, no mathematical software includes a function to do this and from combing the literature and peer help cites (stackexchange), it doesn't appear that this problem has ever been solved. It, at least, has not been implemented into a straightforward computing algorithm.

I derive the algorithm, provide a generic code representation for it, provide a specific Python based implementation, and compare the performance to the random partitioning function of Sage (a powerful and generally very fast computer algebra environment). Finally, and the reason I tagged macroecology, maxent, and economics, I reveal how random integer partitioning algorithms can be used to address current and meaningful problems in the natural and social sciences.