Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bijections for faces of the Shi and Catalan arrangements   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.

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The categories of this paper are: atom-ph
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The categories of this paper are: solv-int
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The categories of this paper are: chao-dyn
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-lat
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The categories of this paper are: nlin
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The categories of this paper are: eess
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The categories of this paper are: dg-ga
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The categories of this paper are: nucl-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: cmp-lg
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The categories of this paper are: funct-an
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The categories of this paper are: gr-qc
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The categories of this paper are: quant-ph
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The categories of this paper are: q-bio