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I just found out that the name "Secretary problem" is given to two different problems. The first one talks about a secretary who mixes letters and envelopes, and ask for the probability that no letter will be put into the right envelope: this is an application of derangements, and the limit value for the probability is $1/e$.

However, Wikipedia defines the Secretary problem as the task to choose the best candidate, if you see them one at a time and cannot keep anybody on hold. Curiously (at least for me), the best algorithm has the identical chance to find the best candidate, that is $1/e$. Is it really a casual correlation - after all, $e$ pops out everywhere - or derangements are somewhat involved?

In the comments at this answer in SO the same question was made, but nobody answered.

mau
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(To differentiate between the two problems, I will call the hiring problem the Sultan's Dowry, as that's the name that is more familiar to me. My apologies if any sultanas are offended by the term.)

It's a coincidence, since $e$ is just that ubiquitous. The order in which the offers are made in the Sultan's Dowry is completely arbitrary, so there is no sense in which derangements come up even subtly in the optimal solution.