  This paper investigates pattern avoidance in linear extensions of a certain
class of partially ordered set. Since the question of enumerating pattern
avoiding linear extensions of posets in general is a very hard one, we focus
instead on certain partially ordered sets called combs. Combs consist of a
fully ordered spine, and several fully ordered teeth, where each tooth
coincides with a corresponding element of the spine. We consider two natural
assignments of integers to elements of the combs; we refer to the resulting
integer posets as type-alpha combs and type-beta combs. In this paper, we
enumerate the linear extensions of type-alpha and type-beta combs which avoid
some of the length-three pattern. Most notably, the number of linear extensions
of type-beta combs which avoid 312 is shown to be the same as the number 1/(st
+ 1)*(s(t+1) choose s) of (t+1)-ary trees on s nodes, where t is the length of
each tooth, and s is the length of the comb spine, or equivalently, the number
of its teeth. We also investigate the enumeration of linear extensions of
type-alpha and type-beta combs avoiding multiple length-three patterns.
