Suppose we have a deck of cards, ordered [1,52] (using e.g. the bridge ordering). Shuffle the cards (uniformly at random). Turn over the top card, it has value $v_0$.
We will turn over cards one by one and place them either on the stack or in the discard.
The stack starts off empty. Let $v_t$ be the value of the top card in the stack (or $\infty$ when the stack is empty).
Now we begin turning over additional cards. For each card $i$ with value $v_i$, if $v_0 < v_i < v_t$ then we place card $i$ on the top of the stack, so that now $v_t = v_i$. Otherwise, we place card $i$ in the discard.
After turning over $k$ cards, what is the expected number of cards in the stack?