Let's say I have a set of integers permutated to create different series. Let's say $1$ through $6$. Each series of length $N$. Say $N =64$.
$6^{64}$ is ~ $6\times10^{49}$ possibilities.
If I randomly generate a series to evaluate, how can I first check I haven't already evaluated it without storing the complete series?
For example, could I do a few simple arithmetic operations in the new series to compare with past series, assessing uniqueness? (Storing just the metrics from the old series, not the actual old series.)
Like sum(New) and sum(diff(new)) and product(New) NOT = sum(Old) and sum(diff(Old)) and product(Old)
This "checksum" approach does not work perfectly.
I you compress the $64$ digits ($n$ possibilities) in a single integer $\in [1,m]$ ($m$ possibilities), every integer will be the same for $\dfrac nm$ sequences (assuming a uniform distribution).
If you obtain a checksum not seen before, you are sure that the sequence is different. But if it matches another checksum, you are stuck. The loss of information is unrecoverable.
On another hand, you can generate a huge number of random sequences (much more than a computer could store) before the probability of a hit becomes significant. So it is questionable whether detecting duplicates makes any sense.