conjugate of an integer partition

Question

Is the conjugate of an integer partition, selected at random from the set of all partitions for n, also a uniform random sample? My results suggest yes, which is encouraging for the sake of quickly generating random partitions of n that are of length s, but I can't explain why that should or shouldn't be.

By the way, my results are based on 1.) generating all partitions for a small n (<70) of a specific length (s) 2.) calculating the variance of each partition as a macrostate descriptor and 3.) comparing the kernel density curve for the variance across the entire feasible set (all partitions for n of length s) against small random samples (i.e. <500 randomly generated partitions of n whose lengths either match s or whose conjugate lengths match s). Kernel density curves for random samples closely match the curve for the entire feasible set (i.e. all partitions of n matching s). This visually illustrates that random samples, the majority of which are conjugate partitions, capture the distribution of variance among partitions of the n and s based feasible set. I just can't explain why it should work as it appears to do; downfall of making a creative leap.

Note: Many other procedures for producing random samples yield a clearly biased sample (i.e. a differently shaped and highly non-overlapping kernel density curve).

Answer 1

Yes. Conjugation is a bijective operation, so each partition maps to a unique conjugate, which in turn maps back to the original partition. Therefore, there can't be any bias introduced by taking the conjugate of a partition selected uniformly at random.

I don't think this helps you generate fixed length partitions at random though - you should probably adapt Nijenhuis & Wilf's algorithm to do this correctly. This shouldn't be very hard to do, since the numbers of partitions of n into k parts can be computed easily, and the random generation algorithm really only depends on this.

Knuth includes an exercise (47) on generating random partitions in section 7.2.4.1 of TAOCP volume 4A. This would be an excellent starting point for an efficient algorithm to generate fixed length partitions uniformly at random.