We consider a variant of the classic coupon collectors problem.
In our setting, one player collects coupons. There are $n$ different coupons, but he does not stop when he has collected each of the $n$ coupon once. Each time he gets a new coupon at a time $t$ he notes which coupons he has seen the least often so far. These coupons then form the set $\min(t)$.
He keeps collecting until he observes the following event A:
A: $\exists m \in \min(t)$ with $m \in min(t-i)$ for $i \in [n^2]$ . That is, a coupon was in the set of the rarest coupons for the duration of $n^2$ draws.
What is the expected number of draws until the game ends?