Dyck Paths and Positroids from Unit Interval Orders

TL;DR

This paper establishes a connection between unit interval orders, Dyck paths, and positroids, providing characterizations, recipes, and conditions that deepen understanding of their combinatorial and geometric structures.

Contribution

It introduces the concept of unit interval positroids, characterizes them via Dyck paths and decorated permutations, and explores their associated Grassmann cells and potential f-vector computations.

Findings

Unit interval positroids correspond to Dyck paths encoded as 2n-cycles.

Decorated permutations of these positroids can be derived from adjacency matrices and interval representations.

Conditions for adjacency of Grassmann cells parameterized by these positroids are established.

Abstract

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.