Dyck Paths and Positroids from Unit Interval Orders
Anastasia Chavez, Felix Gotti · arXiv (Cornell University) · 2016
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 \emph{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 give a combinatorial description of the $f$-vectors of unit interval orders. This is joint work with Felix Gotti.