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

TL;DR

This paper explores the structure of Boolean intervals in the weak order of the symmetric group, revealing connections to parking functions and Fibonacci numbers, and providing enumeration formulas for these intervals.

Contribution

It introduces unit Fubini rankings, characterizes and counts Boolean intervals in the weak order, and links their enumeration to Fibonacci products, advancing understanding of symmetric group order structures.

Findings

Complete enumeration of Boolean intervals in $W(rak{S}_n)$

Characterization of Boolean intervals via parking functions

Number of intervals with minimal element $pi$ is a Fibonacci product

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.