Standard paths in another composition poset

Bergeron, Bousquet-Melou and Dulucq enumerated paths in the Hasse diagram of the following poset: the underlying set is that of all compositions, and a composition \mu covers another composition \lambda if \mu can be obtained from \lambda by adding 1 to one of the parts of \lambda, or by inserting a part of size 1 into \lambda. We employ the methods they developed in order to study the same problem for the following poset: the underlying set is the same, but \mu covers \lambda if \mu can be obtained from \lambda by adding 1 to one of the parts of \lambda, or by inserting a part of size 1 at the left or at the right of \lambda. This poset is of interest because of its relation to non-commutative term orders.