  A Fishburn permutation is a permutation which avoids the bivincular pattern
$(231, \{1\}, \{1\})$, while an ascent sequence is a sequence of nonnegative
integers in which each entry is less than or equal to one more than the number
of ascents to its left. Fishburn permutations and ascent sequences are linked
by a bijection $g$ of Bousquet-M\'elou, Claesson, Dukes, and Kitaev. We write
$F_n(\sigma_1,\ldots,\sigma_k)$ to denote the set of Fishburn permutations of
length $n$ which avoid each of $\sigma_1,\ldots,\sigma_k$ and we write
$A_n(\alpha_1,\ldots,\alpha_k)$ to denote the set of ascent sequences which
avoid each of $\alpha_1,\ldots,\alpha_k$. We settle a conjecture of Gil and
Weiner by showing that $g$ restricts to a bijection between $F_n(3412)$ and
$A_n(201)$. Building on work of Gil and Weiner, we use elementary techniques to
enumerate $F_n(123)$ with respect to inversion number and number of
left-to-right maxima, obtaining expressions in terms of $q$-binomial
coefficients, and to enumerate $F_n(123,\sigma)$ for all $\sigma$. We use
generating tree techniques to study the generating functions for $F_n(321,
1423)$, $F_n(321,3124)$, and $F_n(321,2143)$ with respect to inversion number
and number of left-to-right maxima. We use these results to show
$|F_n(321,1423)| = |F_n(321,3124)| = F_{n+2} - n - 1$, where $F_n$ is a
Fibonacci number, and $|F_n(321,2143)| = 2^{n-1}$. We conclude with a variety
of conjectures and open problems.
