Terry Tao's question https://mathoverflow.net/questions/86118 suggests
the following. What classes $C_n$ of functions $f\colon \lbrace
1,2,\dots,n\rbrace \to \lbrace 1,2,\dots,n\rbrace$
have the property that if $a(n)$ is the number of functions in $C_n$
and $b(n)$ is the number without fixed points, then
$\lim_{n\to\infty} a(n)/b(n) = e$? Examples include all functions,
permutations, and alternating permutations. (I don't know a simple
proof for alternating permutations.) Rather than lots of examples, a
general theorem that includes all (or most) of these examples would be
more interesting.