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Séminaire Lotharingien de Combinatoire, 82B.47 (2019), 12 pp.

# Duncan Levear

# A bijection for Shi arrangement faces

**Abstract.**
The Shi arrangement of hyperplanes in **R**^{n} is known to have (*n*+1)^{n-1} regions. This remarkable formula was first derived algebraically in 1986 by Shi, and has since been explained bijectively through either parking functions or Cayley trees. Although the lower-dimensional faces have been counted by a finite field method, no bijective correspondence has been established. In this paper, we extend a bijection for regions defined by Bernardi to obtain a correspondence for all Shi faces graded by their dimension. The image of the bijection is a set of decorated binary trees, which can further be converted to a simple set of functions *f*: [*n*-1] -> [*n*+1] known as Prüfer sequences. In the process, we also obtain a correspondence for the faces of the Catalan arrangement, and the results generalize to both extended arrangements.

Received: November 15, 2018.
Accepted: February 17, 2019.
Final version: April 1, 2019.

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