  We study a family of equivalence relations on $S_n$, the group of
permutations on $n$ letters, created in a manner similar to that of the Knuth
relation and the forgotten relation. For our purposes, two permutations are in
the same equivalence class if one can be reached from the other through a
series of pattern-replacements using patterns whose order permutations are in
the same part of a predetermined partition of $S_c$.
  When the partition is of $S_3$ and has one nontrivial part and that part is
of size greater than two, we provide formulas for the number of classes created
in each previously unsolved case. When the partition is of $S_3$ and has two
nontrivial parts, each of size two (as do the Knuth and forgotten relations),
we enumerate the classes for $13$ of the $14$ unresolved cases. In two of these
cases, enumerations arise which are the same as those yielded by the Knuth and
forgotten relations. The reasons for this phenomenon are still largely a
mystery.
