Asymptotics of Redistricting the $n\times n$ grid

How many ways are there to redistrict the $n\times n$ grid into $n$ districts of equal size? How many of these redistricting plans are compact? In this article we give asymptotic bounds on the number of plans, showing there are exponentially many in $n^2$. We then use the lower bound to show that most plans are as least compact as possible.
Comment: 11 pages, 11 figures