  Partially ordered sets (posets) are fundamental combinatorial objects with
important applications in computer science. Perhaps the most natural
algorithmic task, given a size-$n$ poset, is to compute its number of linear
extensions. In 1991 Brightwell and Winkler showed this problem to be
$\#P$-hard. In spite of extensive research, the fastest known algorithm is
still the straightforward $O(n 2^n)$-time dynamic programming (an adaptation of
the Bellman-Held-Karp algorithm for the TSP). Very recently, Dittmer and Pak
showed that the problem remains $\#P$-hard for two-dimensional posets, and no
algorithm was known to break the $2^n$-barrier even in this special case. The
question of whether the two-dimensional problem is easier than the general case
was raised decades ago by M\"ohring, Felsner and Wernisch, and others. In this
paper we show that the number of linear extensions of a two-dimensional poset
can be computed in time $O(1.8172^n)$.
  The related jump number problem asks for a linear extension of a poset,
minimizing the number of neighboring incomparable pairs. The problem has
applications in scheduling, and has been widely studied. In 1981 Pulleyblank
showed it to be NP-complete. We show that the jump number problem can be solved
(in arbitrary posets) in time $O(1.824^n)$. This improves (slightly) the
previous best bound of Kratsch and Kratsch.
