Proof of a Conjecture of Matherne, Morales, and Selover on Encodings of Unit Interval Orders

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 as forming the set of boxes between a Dyck path and the diagonal. Another is to find a particular part listing (in the sense of Guay-Paquet, Rowland and Morales) 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.