Enumerating Permutations and Rim Hooks Characterized by Double Descent Sets

TL;DR

This paper develops recursive formulas and estimation methods for counting permutations and rim hooks characterized by double descent sets, providing explicit formulas for singleton and empty sets, and exploring asymptotic conjectures.

Contribution

It introduces new recursive formulas and enumeration techniques for permutations and rim hooks with specific double descent sets, including explicit formulas for singleton and empty sets.

Findings

Recursive formula for $dd(I;n)$ when $I$ is a singleton

Enumeration formulas for rim hooks with singleton and empty double descent sets

Conjectures on the asymptotic behavior of ratios of $dd(I;n)$

Abstract

Let $dd(I;n)$ denote the number of permutations of $[n]$ with double descent set $I$. For singleton sets $I$, we present a recursive formula for $dd(I;n)$ and a method to estimate $dd(I;n)$. We also discuss the enumeration of certain classes of rim hooks. Let $\mathcal{R}_I(n)$ denote the set of all rim hooks of length $n$ with double descent set $I$, so that any tableau of one of these rim hooks corresponds to a permutation with double descent set $I$. We present a formula for the size of $\mathcal{R}_I(n)$ when $I$ is a singleton set, and we also present a formula for the size of $\mathcal{R}_I(n)$ when $I$ is the empty set. We additionally present several conjectures about the asymptotics of certain ratios of $dd(I;n)$.