  There are two bijections from unit interval orders on $n$ elements to Dyck
paths from $(0,0)$ to $(n,n)$. One is to consider the pairs of incomparable
elements, which form the set of boxes between some Dyck path and the diagonal.
Another is to find a particular part listing (in the sense of Guay-Paquet)
which yields an isomorphic poset, and to interpret the part listing as the area
sequence of a Dyck path. Matherne, Morales, and Selover conjectured that, for
any unit interval order, these two Dyck paths are related by Haglund's
well-known zeta bijection. In this paper we prove their conjecture.
