  Inversion sequences of length $n$, $\mathbf{I}_n$, are integer sequences
$(e_1, \ldots, e_n)$ with $0 \leq e_i < n$ for each $i$. The study of patterns
in inversion sequences was initiated recently by Mansour-Shattuck and
Corteel-Martinez-Savage-Weselcouch through a systematic study of inversion
sequences avoiding words of length 3. We continue this investigation by
generalizing the notion of a pattern to a fixed triple of binary relations
$(\rho_1,\rho_2,\rho_3)$ and consider the set
$\mathbf{I}_n(\rho_1,\rho_2,\rho_3)$ consisting of those $e \in \mathbf{I}_n$
with no $i < j < k$ such that $e_i \rho_1 e_j$, $e_j \rho_2 e_k$, and $e_i
\rho_3 e_k$. We show that "avoiding a triple of relations" can characterize
inversion sequences with a variety of monotonicity or unimodality conditions,
or with multiplicity constraints on the elements. We uncover several
interesting enumeration results and relate pattern avoiding inversion sequences
to familiar combinatorial families. We highlight open questions about the
relationship between pattern avoiding inversion sequences and families such as
plane permutations and Baxter permutations. For several combinatorial
sequences, pattern avoiding inversion sequences provide a simpler
interpretation than otherwise known.
