Any permutation statistic $f:{mathfrak{S}}o{mathbb C}$ may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: $f= Sigma_aulambda_f(au)au$. 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 $lambda_p(au) = (-1)^{|au|-|pi|}{p}^{star}(au)$ 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, André 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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机译：任何置换统计量$ f：{ mathfrak {S}} to { mathbb C} $都可以唯一地表示为（经典）置换模式的无限可能的线性组合：$ f = Sigma_ tau lambda_f（ tau） tau $。为了提供对某些统计数据的显式扩展，我们引入了一种新型的排列模式，称为网格模式。直观地，网格模式$ p =（ pi，R）$的出现是排列模式$ pi $的出现，其中$ R $对出现的条目的相对位置指定了其他限制。我们表明，对于任何网格图案$ p =（ pi，R）$，我们都有$ lambda_p（ tau）=（-1）^ {| tau |-| pi |} {p} ^ { star}（ tau）$，其中$ {p} ^ { star} =（ pi，R ^ c）$是具有与$ p $相同的基础排列但具有互补限制的网格模式。我们使用该结果来扩展一些众所周知的置换统计量，例如从左到右的最大值，下降，兴奋，固定点，强固定点和主要指数。我们还显示出交替排列，第一类André排列和simsun排列自然会作为避免某些网格图案的排列而发生。最后，我们提供了新的自然Mahonian统计数据。
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