A Sequence Representation of the Dyck Path Poset

Abstract

In this paper, a sequence representation of Dyck paths is presented, which yields a sequence representation of the Dyck path poset \({D}\) ordered by pattern containment. This representation makes it clear that the Dyck path poset \({D}\) takes the composition poset investigated by Sagan and Vatter as a subposet, and that the pattern containment order on Dyck paths exactly agrees with a generalized subword order also presented by Sagan and Vatter. As applications of the representation, we describe the Möbius function of \({D}\) and establish the Möbius inverse of the rank function of \({D}\) in terms of Dyck sequences. In the end, a Sperner and unimodal subposet of \({D}\) is given.