  We introduce, and partially resolve, a conjecture that brings a
three-centuries-old derangements phenomenon and its much younger
two-decades-old analogue under the same umbrella. Through a graph-theoretic
lens, a derangement is a perfect matching in the complete bipartite graph
$K_{n,n}$ with a disjoint perfect matching $M$ removed. Likewise, a deranged
matching is a perfect matching in the complete graph $K_{2n}$ minus a perfect
matching $M'$. With $\mathrm{pm}(\cdot)$ counting perfect matchings, the elder
phenomenon takes the form $\mathrm{pm}(K_{n,n}-M)/\mathrm{pm}(K_{n,n})\to 1/e$
as $n\to\infty$ while its youthful analogue is
$\mathrm{pm}(K_{2n}-M')/\mathrm{pm}(K_{2n})\to 1/\sqrt{e}$. These starting
graphs are both $2n$-vertex `balanced complete $r$-partite' graphs $K_{r \times
{2n}/{r}}$, respectively with $r=2$ and $r=2n$. We conjecture that
$\mathrm{pm}(K_{r\times{2n}/r}-M)/\mathrm{pm}(K_{r\times{2n}/r})\sim
e^{-r/(2r-2)}$ as $n\to\infty$ and establish several substantive special cases
thereof. For just two examples, $r=3$ yields the limit $e^{-3/4}$ while $r=n$
results again in $e^{-1/2}$. Our tools blend combinatorics and analysis in a
medley incorporating Inclusion-Exclusion and Tannery's Theorem.
