Dyck Paths and Positroids from Unit Interval Orders

It is well known that the number of non-isomorphic unit interval orders on \([n]\) equals the \(n\)-th Catalan number. Using work of Skandera and Reed and work of Postnikov, we show that each unit interval order on \([n]\) naturally induces a rank \(n\) positroid on \([2n]\). We call the positroids produced in this fashion unit interval positroids. We characterize the unit interval positroids by describing their associated decorated permutations, showing that each one must be a \(2n\)-cycle encoding a Dyck path of length \(2n\). We also provide recipes to read the decorated permutation of a unit interval positroid \(P\) from both the antiadjacency matrix and the interval representation of the unit interval order inducing \(P\). Using our characterization of the decorated permutation, we describe the Le-diagrams corresponding to unit interval positroids. In addition, we give a necessary and sufficient condition for two Grassmann cells parameterized by unit interval positroids to be adjacent inside the Grassmann cell complex. Finally, we propose a potential approach to find the \(f\)-vector of a unit interval order.