  In the classical secretary problem, $n$ ranked items arrive one by one, and
each item's rank relative to its predecessors is noted. The observer must
select or reject each item as it arrives, with the object of selecting the item
of highest rank. For $M_n\in\{0,1,\cdots, n-1\}$, let $\mathcal{S}(n,M_n)$
denote the strategy whereby the observer rejects the first $M_n$ items, and
then selects the first later-arriving item whose rank is higher than that of
any of the first $M_n$ items (if such an item exists). If the ranked items
arrive in a uniformly random order, it is well-known that the limiting optimal
probability of success is $\frac1e$, which occurs if $M_n\sim\frac ne$. It has
been shown that when the ranked items arrive according to certain non-uniform
distributions on the set of permutations, $\frac1e$ serves as a lower bound for
the optimal probability. There is a fundamental reason for this phenomenon. We
consider certain distributions for which that reason does not apply. We begin
by noting a cooked-up class of distributions for which $\mathcal{S}(n,M)$
yields the lowest possible probability of success -- namely $\frac1n$, for all
$M$. We then consider the uniform distribution over all permutations avoiding a
particular pattern of length three. In the case of the pattern 231 or 132, for
any choice of $M_n$, the strategy $\mathcal{S}(n,M_n)$ yields the very same
probability of success; namely $\frac{n+1}{2(2n-1)}$, which gives a limiting
probability of $\frac14$. For the pattern 213, the optimal strategy is obtained
for $M\in\{0,1\}$, also yielding a limiting probability of $\frac14$. For the
pattern 123, the optimal strategy is obtained for $M=1$, yielding a limiting
probability of $\frac34$. For the other two patterns, 312 and 321, an optimal
strategy will yield a limiting probability of at least $\frac7{16}$.
