Friday, October 12, 2012: 7:40 PM
Hall 4E/F (WSCC)
Consider the regions of the xy-plane that are on different sides of line x = y. Now
consider all the different regions in three dimensions bounded by the planes x = y, x = z and
y = z. These are examples of braid arrangements. 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 Sn is
an expansion of Bn, which has tangible applications such as ordering the printing preference
of color to non-color in a collection of printers. We establish an equivalence between a
mixed graph, which is well known in the area of Graph Theory, and parking functions, which
are staples in the area of Combinatorics. This connection is established by linking each of
them to the Shi arrangement, which is entirely geometric. The work provides an exciting link
among three areas of mathematics.
consider all the different regions in three dimensions bounded by the planes x = y, x = z and
y = z. These are examples of braid arrangements. 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 Sn is
an expansion of Bn, which has tangible applications such as ordering the printing preference
of color to non-color in a collection of printers. We establish an equivalence between a
mixed graph, which is well known in the area of Graph Theory, and parking functions, which
are staples in the area of Combinatorics. This connection is established by linking each of
them to the Shi arrangement, which is entirely geometric. The work provides an exciting link
among three areas of mathematics.