论文标题
什叶派和加泰罗尼亚人的面孔的射击
Bijections for faces of the Shi and Catalan arrangements
论文作者
论文摘要
1986年，Shi得出了著名的公式$（n+1）^{n-1} $，用于SHI布置的数量，这是$ \ Mathbb {r}^n $的超平面布置。这个公式至少有两种不同的射击解释，一个由帕克和斯坦利，另一个由Athanasiadis和Linusson组成的。 1996年，Athanasiadis使用有限的野外方法来得出任何$ k $的$ k $维面的公式，用于任何$ k $。到目前为止，Athanasiadis的公式还没有任何徒解释。在本文中，我们扩展了Bernardi定义的区域的培养，以在任何$ K $和一套装饰的二进制树上的SHI布置的$ K $维面之间进行两次射击。此外，我们展示了如何将这些树转换为$ f：[n-1] \ to [n+1] $的简单函数，以及$ \ text {im}（im}（f）$的标记子集。这种对应关系给出了第一个族裔雅典迪斯公式的徒证明。在此过程中，我们还获得了加泰罗尼亚排列面孔的两次培养和计数公式。我们所有的结果都概括为两个扩展安排。
In 1986, Shi derived the famous formula $(n+1)^{n-1}$ for the number of regions of the Shi arrangement, a hyperplane arrangement in $\mathbb{R}^n$. There are at least two different bijective explanations of this formula, one by Pak and Stanley, another by Athanasiadis and Linusson. In 1996, Athanasiadis used the finite field method to derive a formula for the number of $k$-dimensional faces of the Shi arrangement for any $k$. Until now, the formula of Athanasiadis did not have a bijective explanation. In this paper, we extend a bijection for regions defined by Bernardi to obtain a bijection between the $k$-dimensional faces of the Shi arrangement for any $k$ and a set of decorated binary trees. Furthermore, we show how these trees can be converted to a simple set of functions of the form $f: [n-1] \to [n+1]$ together with a marked subset of $\text{Im}(f)$. This correspondence gives the first bijective proof of the formula of Athanasiadis. In the process, we also obtain a bijection and counting formula for the faces of the Catalan arrangement. All of our results generalize to both extended arrangements.
