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In this paper, we consider the family of pattern-replacement equivalencerelations referred to as the "indices and values adjacent" case. Each suchequivalence is determined by a partition $P$ of a subset of $S_c$ for some $c$.In 2010, Linton, Propp, Roby, and West posed a number of open problems in thearea of pattern-replacement equivalences. Five, in particular, have remainedunsolved until now, the enumeration of equivalence classes under the $\{123,132\}$-equivalence, under the $\{123, 321\}$-equivalence, under the $\{123,132, 213\}$ equivalence, and under the $\{123, 132, 213, 321\}$-equivalence. Wefind formulas for three of the five equivalences and systems of representativesfor the equivalence classes of the other two. We generalize our results to holdfor all replacement partitions of $S_3$, as well as for an infinite family ofother replacement partitions. In addition, we characterize the equivalenceclasses in $S_n$ under the $S_c$-equivalence, finding a generalization ofStanley's results on the $\{12, 21\}$-equivalence.  To do this, we introduce a notion of confluence that often allows one to finda representative element in each equivalence class under a given equivalencerelation. Using an inclusion-exclusion argument, we are able to use this tocount the equivalence classes under equivalence relations satisfying certainconditions.

New Results on Doubly Adjacent Pattern-Replacement Equivalences