The Automorphism Conjecture for Ordered Sets of Dimension 2 and Interval Orders

Let $\lambda \in \left (0,\frac {1}{2} \right ) $ . We prove that, for ordered sets P of order dimension 2 and for interval orders, the ratio of the number of automorphisms to the number of endomorphisms is asymptotically bounded by $2^{-|P|^{\lambda } } $ . The key to the proof is to establish this bound for certain types of lexicographic sums.