  Any permutation statistic $f:\sym\to\CC$ may be represented uniquely as a,
possibly infinite, linear combination of (classical) permutation patterns: $f=
\Sigma_\tau\lambda_f(\tau)\tau$. To provide explicit expansions for certain
statistics, we introduce a new type of permutation patterns that we call mesh
patterns. Intuitively, an occurrence of the mesh pattern $p=(\pi,R)$ is an
occurrence of the permutation pattern $\pi$ with additional restrictions
specified by $R$ on the relative position of the entries of the occurrence. We
show that, for any mesh pattern $p=(\pi,R)$, we have $\lambda_p(\tau) =
(-1)^{|\tau|-|\pi|}p^{\star}(\tau)$ where $p^{\star}=(\pi,R^c)$ is the mesh
pattern with the same underlying permutation as $p$ but with complementary
restrictions. We use this result to expand some well known permutation
statistics, such as the number of left-to-right maxima, descents, excedances,
fixed points, strong fixed points, and the major index. We also show that
alternating permutations, Andr\'e permutations of the first kind and simsun
permutations occur naturally as permutations avoiding certain mesh patterns.
Finally, we provide new natural Mahonian statistics.
