  Multidimensional permutations, or $d$-permutations, are represented by their
diagrams on $[n]^d$ such that there exists exactly one point per hyperplane
$x_i$ that satisfies $x_i= j$ for $i \in [d]$ and $j \in [n]$. Bonichon and
Morel previously enumerated $3$-permutations avoiding small patterns, and we
extend their results by first proving four conjectures, which exhaustively
enumerate $3$-permutations avoiding any two fixed patterns of size $3$. We
further provide a enumerative result relating $3$-permutation avoidance classes
with their respective recurrence relations. In particular, we show a recurrence
relation for $3$-permutations avoiding the patterns $132$ and $213$, which
contributes a new sequence to the OEIS database. We then extend our results to
completely enumerate $3$-permutations avoiding three patterns of size $3$.
