Suppose you have $n$ (blue) wires linearly arrayed at junction box $A$,
connected to a remote junction box $B$, where the wires are now
arrayed
along a line in a randomly permuted order, i.e.,
each of the $n!$ permutations is equally likely at $B$.
Now you tie together every other wire at $A$ and at $B$
with a (green) connector, like this:
What is the probability that you have formed a single cycle (as illustrated)?
More generally, what are the combinatorics of the cycle structures
achievable in this manner?
(It may be best to separate out the $n$-even case from $n$ odd.)
I came upon this thinking of the wires as an arrangement of lines, where each line crosses every other before reaching junction box $B$, in which case, for $n$ even, one necessarily arrives at $n/2$ cycles, each containing two (blue) wires. All $n$ wires in a single cycle is in some sense the obverse situation.
Update. Will Swain's argument shows, as Noam points out, that the probability of a single cycle is asymptotically $\frac{1}{\sqrt{n}}$. I would be interested to learn if there is a way to see this intuitively without Will's explicit calculation. Perhaps an assessment of the probability of repeatedly avoiding premature closing of a loop as one criss-crosses from $A$ to $B$...?
