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Dyck Paths and Positroids from Unit Interval Orders

Banff International Research Station for Mathematical Innovation and Discovery

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.

Mathematics; Combinatorics; Group theory and generalizations

video/mp4

Author affiliation: University of California - Berkeley

2017-11-13

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/63586

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/