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Let $dd(I;n)$ denote the number of permutations of $[n]$ with double descentset $I$. 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 certainclasses of rim hooks. Let $\mathcal{R}_I(n)$ denote the set of all rim hooks oflength $n$ with double descent set $I$, so that any tableau of one of these rimhooks corresponds to a permutation with double descent set $I$. We present aformula for the size of $\mathcal{R}_I(n)$ when $I$ is a singleton set, and wealso present a formula for the size of $\mathcal{R}_I(n)$ when $I$ is the emptyset. We additionally present several conjectures about the asymptotics ofcertain ratios of $dd(I;n)$.

Enumerating Permutations and Rim Hooks Characterized by Double Descent Sets