Spanning trees in complete uniform hypergraphs and a connection to extended r-Shi hyperplane arrangements

TL;DR

This paper derives a formula for counting spanning trees in complete r-uniform hypergraphs, establishes a bijection with parking functions, and links these to the regions of extended Shi arrangements.

Contribution

It introduces a Cayley type formula for hypergraph spanning trees and connects them to parking functions and hyperplane arrangements in a novel way.

Findings

Cayley type formula for spanning trees in hypergraphs

Bijection between hypergraph spanning trees and parking functions

Connection to the number of regions in extended Shi arrangements

Abstract

We give a Cayley type formula to count the number of spanning trees in the complete r-uniform hypergraph for all r >= 3. Similar to the bijection between spanning trees in complete graphs and Parking functions, we derive a bijection from spanning trees of the complete (r+1)-uniform hypergraph which arise from a fixed r-perfect matching and r-Parking functions. We observe a simple consequence of this bijection in terms of the number of regions of the extended Shi arrangement.