I cannot find a satisfactory way to mathematically model the following situation.
When entering a restaurant, $n$ people leave their jackets at the entrance. After eating, they leave, and go to collect them. The member of the staff whose in charge to give back the jackets to the respective owners does that sloppily, according to the following procedure: he starts randomly (uniformly) permuting the $n$ jackets, and giving them back to the guests following the obtained order. This way, $L_1$ people get their own jacket, while the remaining $n-L_1$ don't. He then goes on the same way, randomly (uniformly) permuting the $n-L_1$ remaining jackets, and giving them back to the remaining $n-L_1$ guests following the obtained order. This way, $L_2$ more people get their own jacket, while $n-L_1-L_2$ don't.
We would like to find $\mathbb{E}(\tau)$, where $\tau$ is the random time necessary for all the guests to get their own jacket.
Is there maybe a way to find a martingale around there, and then use the Doob's optional stopping theorem?