  Solving the first nonmonotonic, longer-than-three instance of a classic
enumeration problem, we obtain the generating function $H(x)$ of all
1342-avoiding permutations of length $n$ as well as an {\em exact} formula for
their number $S_n(1342)$. While achieving this, we bijectively prove that the
number of indecomposable 1342-avoiding permutations of length $n$ equals that
of labeled plane trees of a certain type on $n$ vertices recently enumerated by
Cori, Jacquard and Schaeffer, which is in turn known to be equal to the number
of rooted bicubic maps enumerated by Tutte in 1963. Moreover, $H(x)$ turns out
to be algebraic, proving the first nonmonotonic, longer-than-three instance of
a conjecture of Zeilberger and Noonan. We also prove that $\sqrt[n]{S_n(1342)}$
converges to 8, so in particular, $lim_{n\rightarrow
\infty}(S_n(1342)/S_n(1234))=0$.
