Standard paths in another composition poset

Abstract

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 μcovers another composition λif μcan be obtained from λby adding 1 to one of the parts of λ, or by inserting a part of size 1 into λ. We employ the methods they developed in order to study the same problem for the following poset: the underlying set is the same, but μcovers λif μcan be obtained from λby adding 1 to one of the parts of λ, or by inserting a part of size 1 at the left or at the right of λ. This poset is of interest because of its relation to non-commutative term orders.