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We study a family of equivalence relations on $S_n$, the group ofpermutations on $n$ letters, created in a manner similar to that of the Knuthrelation and the forgotten relation. For our purposes, two permutations are inthe same equivalence class if one can be reached from the other through aseries of pattern-replacements using patterns whose order permutations are inthe same part of a predetermined partition of $S_c$. In particular, we areinterested in the number of classes created in $S_n$ by each relation and incharacterizing these classes.  Imposing the condition that the partition of $S_c$ has one nontrivial partcontaining the cyclic shifts of a single permutation, we find enumerations forthe number of nontrivial classes. When the permutation is the identity, we areable to compare the sizes of these classes and connect parts of the problem toYoung tableaux and Catalan lattice paths.  Imposing the condition that the partition has one nontrivial part containingall of the permutations in $S_c$ beginning with 1, we both enumerate andcharacterize the classes in $S_n$. We do the same for the partition that hastwo nontrivial parts, one containing all of the permutations in $S_c$ beginningwith 1, and one containing all of the permutations in $S_c$ ending with 1.

Equivalence Classes in $S_n$ for Three Families of Pattern-Replacement Relations