Consider a discrete distribution $\mathcal{D}$ with probabilities $(p_1,\ldots,p_n)$ over numbers $1, \ldots, n$. In a sequence of i.i.d. draws from $\mathcal{D}$, $X_1, X_2, \ldots$, let $X^{(i)}$ denote the $i$th unique number that is observed in the sequence (e.g., in $2,1,2,3,\ldots$, we have $X^{(1)} = 2$, $X^{(2)} = 1$, $X^{(3)}=3$).
Is there a nice way to characterize explicitly or bound the probability of a particular order of appearance of the support in an i.i.d drawn sequence, such as
$$ \mathrm{Pr}_{ \{ X_i \}_{i=1}^{\infty} \sim \mathcal{D}^{\infty}}(X^{(1)}=1, \ldots, X^{(n)}=n) \quad ? $$