Parking functions, Fubini rankings, and Boolean intervals in the weak order of ${\mathfrak{S}}_n$

Abstract

Let ${\mathfrak{S}}_n$ denote the symmetric group and let $W({\mathfrak{S}}_n)$ denote the weak order of ${\mathfrak{S}}_n$. Through a surprising connection to a subset of parking functions, which we call unit Fubini rankings, we provide a complete characterization and enumeration for the total number of Boolean intervals in $W({\mathfrak{S}}_n)$ and the total number of Boolean intervals of rank $k$ in $W({\mathfrak{S}}_n)$. Furthermore, for any $\pi \in {\mathfrak{S}}_n$, we establish that the number of Boolean intervals in $W({\mathfrak{S}}_n)$ with minimal element $\pi$ is a product of Fibonacci numbers. We conclude with some directions for further study.

17 pages, 4 figures. To appear in Journal of Combinatorics