arXiv 1102.42262011-02math.CO

网孔模式与置换统计量作为置换模式和的展开

Mesh patterns and the expansion of permutation statistics as sums of permutation patterns

Petter Brändén, Anders Claesson

摘要：

任何置换统计$f:\sym\to\CC$可以唯一地表示为（经典）置换模式的可能无限的线性组合：$f= Σ_τλ_f(τ)τ$。为了提供明确的扩展某些统计，我们引入了一种新的类型的排列模式，我们称之为网格模式。直观地，网格模式$p=(π,R)$的出现是置换模式$π$的出现，其具有由$R$对出现的条目的相对位置指定的附加限制。我们表明，对于任何网格模式$p=(π,R)$，我们有$λ_p(τ) = (-1)^{|τ|-|π|}p^{\star}(τ)$，其中$p^{\star}=(π,R^c)$是网格模式与$p$相同的基本置换，但互补的限制。利用这个结果，我们推广了一些著名的置换统计量，如从左到右的最大值、下降点、超出点、不动点、强不动点和主指数的个数.我们还表明，交替置换，安德烈置换的第一类和simsun置换自然发生的置换，避免某些网格模式。最后，我们提供了新的自然Mahonian统计。

英文摘要：

Any permutation statistic $f:\sym\to\CC$ may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: $f= Σ_τλ_f(τ)τ$. 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 $p=(π,R)$ is an occurrence of the permutation pattern $π$ with additional restrictions specified by $R$ on the relative position of the entries of the occurrence. We show that, for any mesh pattern $p=(π,R)$, we have $λ_p(τ) = (-1)^{|τ|-|π|}p^{\star}(τ)$ where $p^{\star}=(π,R^c)$ 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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