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A Bijection from Shi Arrangement Regions to Parking Functions via Mixed Graphs

Consider all the different regions in three dimensions bounded by the planes x1 = x2, x1 = x3 and x2 = x3. This is a three-dimensional braid arrangement. More generally, in dimension n, Bn is the braid arrangement of hyperplanes of the form xi = xj for 1 ≤ i < j ≤ n. A Shi arrangement, Shi(n) is an expansion of Bn that includes the hyperplanes xi = xj + 1. Pak and Stanley have shown that there exists a bijection between the regions of Shi(n) and a combinatorial object called parking functions of length n where both objects have the cardinality of (n+1)n−1. Our goal is to expand on the work of Pak and Stanley and provide a different approach to this bijection. We will establish our bijection by linking the regions of Shi(n) and parking functions to mixed graphs, which are well known in the area of Graph Theory. This work provides an exciting link among three areas of mathematics.