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{\bf Petter Br\"and\'en and Anders Claesson}
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{\bf Mesh Patterns and the Expansion of Permutation Statistics as Sums of Permutation Patterns}
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Any permutation statistic $f:{\mathfrak{S}}\to{\mathbb C}$ 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.



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