Consider a set $S=\{1,2 \ldots n\}$. I am constructing two random multi-sets $X$ and $Y$. $X$ is a uniformly random permutation of $S$ whereas $Y$ is constructed by drawing n samples uniformly randomly from $S$ — hence these are just samples drawn without and with replacement.
My question is what are events which are much more (or less) likely for $X$ to belong to than $Y$; specifically I am looking at separations where the ratio of probabilities is poly(n). An easy example includes the event that there are no repetitions in $Y$. But I am looking for a richer description of such events. Any help or link to relevant material is much appreciated.
I'm understanding your question to mean you have two sets of samples $X = (X_1, \ldots, X_n)$, and $Y = (Y_1, \ldots, Y_n)$ where the entries $X_i$ are sampled without replacement and the entries $Y_i$ are sampled with replacement from $\{1, \ldots, n\}$. I believe you are asking about events on $X$ that are much more/less likely than events of $Y$?
Like you said, the existence of a repetition has probability $0$ for $X$ but non-zero for $Y$, so the ratio is quite high.
Another example would be the probability that $X$ or $Y$ equal a fixed permutation $\pi$. You have
$$ \begin{align*} P(X = \pi) &= \frac{1}{n!} \\ P(Y = \pi) &= \frac{1}{n^n}. \end{align*} $$
As $n$ gets large, the ratio of these two is something like $e^n/\sqrt{2\pi n}$ by Stirling's approximation. Are you trying to come up with a class of events where the ratio is $O(\text{poly}(n))$ at worst?