I hadn't been able to figure out why the shuffling helped you, either. Before Eily's post, I was going to lean down the road of suggesting comparing your monotonically increasing ramp of weights with a monotonically decreasing ramp of weights, and see whether collisions were affected. And then recommend trying a list of weights which was half ramp + half random (actually, three sets of half-ramp: #1:first half increasing, #2:middle half increasing (quarter random on each side), #3: final half increasing. (And compare all those to one of your fully-shuffled weight setes.) If you saw that the down-ramp and the the half-ramps all had worse collisions than your shuffle, then I would suggest trying a mutator that tried to get rid of rampy-segments: if you found a sequence of weights that were strictly increasing (or generally increasing, with occasional excursions), I would suggest having the mutator pick new random primes or just re-shuffle the rampy section.

Also, I have found that histograms can hide some critical information. With your ordered set of 736000 combinations from the first example, you might want to comapre the time-series of the generated signatures -- probably with the multiple weight lists: ramp up, ramp down, 3 half ramps, and one or two completely random weights, to see if any patterns jump out and give you a hint for what your mutator could do.

Finally, as a last thought: as Eily said, sums of primes don't approach uniqueness. However, products of primes do. I know that purely multiplying primes is unique. I spent some time while I couldn't sleep last night trying to come up with a reasonable way to use that, since you'd obviously quickly overrun your 64bit integer with a product of K=1000 primes. :-) I think I came up with something (after looking up a few primey facts today) that would be a product of two primes.

• To fit in 64bits, each prime would have to be less than 2**32. The biggest 32bit prime number ⁽¹⁾ is 4_294_967_291, which is the 203_280_221^{st (2)} prime.
• That number of multiplyable-primes is bigger than your expected N≥200_000, so in theory, we could assign two primes (or 1000) to each of your N things, and multiple two primes together.
• If you didn't want to just start doling out primes randomly, you could do something like
  • @prime[1 .. 200_000] = import_from_list_of_first_million_primes()
  • generate two 1000-weight lists @wa and @wb, using your randomly shuffled primes
  • for each $comb, generate two sigs $sa and $sb in your sig function
    • even if one of your sigs had a collsion, the chances that both collide at the same time are significanlty smaller
    • return $product = $prime[$sa] * $prime[$sb]

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