The Comb Poset and the Parsewords Function

In this paper we explore some of the properties of the comb poset, whose notion was first introduced by J. M. Pallo. We show that three binary functions that are not well-behaved in the Tamari lattice are remarkably well-behaved within an interval of the comb poset: rotation distance, meets and joins, and the common parse words function for a pair of trees. We conclude by giving explicit expressions for the number of common parse words for a pair of trees within an interval of the comb poset, a problem whose generalization is known to be equivalent to the Four Color theorem.