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arXiv 1202.4765math.CO
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置换的深度

The depth of a permutation

T. Kyle Petersen, Bridget Eileen Tenner

备注：
to appear in Journal of Combinatorics
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摘要：

对于Coxeter群的元素，我们提出了一个称为深度的统计量，定义为将元素分解为反射的乘积。深度的上界为长度，下界为长度和反射长度的平均值。在这篇文章中，我们集中在对称群的情况下，我们证明了深度等于sum_i max{w（i）-i，0}。我们描述了那些深度等于长度的排列：这些是321-避免排列（因此由加泰罗尼亚数枚举）。我们还描述了那些深度等于反射长度的排列：这些排列避免了321和3412（也称为布尔排列，因此我们也可以枚举）。在这种情况下，它也发生，长度等于反射长度，导致一个新的角度对结果的埃德尔曼。

英文摘要：

For the elements of a Coxeter group, we present a statistic called depth, defined in terms of factorizations of the elements into products of reflections. Depth is bounded above by length and below by the average of length and reflection length. In this article, we focus on the case of the symmetric group, where we show that depth is equal to sum_i max{w(i)-i, 0}. We characterize those permutations for which depth equals length: these are the 321-avoiding permutations (and hence are enumerated by the Catalan numbers). We also characterize those permutations for which depth equals reflection length: these are permutations avoiding both 321 and 3412 (also known as boolean permutations, which we can hence also enumerate). In this case, it also happens that length equals reflection length, leading to a new perspective on a result of Edelman.

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