Dyck paths and positroids from unit interval orders

摘要

AbstractIt is well known that the number of non-isomorphic unit interval orders on[n]equals then-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 ranknpositroid on[2n]. We call the positroids produced in this fashionunit 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 positroidPfrom both the antiadjacency matrix and the interval representation of the unit interval order inducingP. 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 u