2DFA that requires many states in equivalent DFA?
The tight bound is $ n(n^n -(n-1)^n) $, which was given in Removing Bidirectionality from Nondeterministic Finite Automata by Christos Kapoutsis (2005).

I am also putting a figure from this paper giving a clear picture:


To get more, I think this paper is a good starting point. To check the related recent developments, I can also recommend to check the papers listed here: dblp: Christos A. Kapoutsis.