  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 $S_c$. In particular, we are
interested in the number of classes created in $S_n$ by each relation and in
characterizing these classes.
  Imposing the condition that the partition of $S_c$ has one nontrivial part
containing the cyclic shifts of a single permutation, we find enumerations for
the number of nontrivial classes. When the permutation is the identity, we are
able to compare the sizes of these classes and connect parts of the problem to
Young tableaux and Catalan lattice paths.
  Imposing the condition that the partition has one nontrivial part containing
all of the permutations in $S_c$ beginning with 1, we both enumerate and
characterize the classes in $S_n$. We do the same for the partition that has
two nontrivial parts, one containing all of the permutations in $S_c$ beginning
with 1, and one containing all of the permutations in $S_c$ ending with 1.
