  We consider a large family of equivalence relations on permutations in Sn
that generalise those discovered by Knuth in his study of the
Robinson-Schensted correspondence. In our most general setting, two
permutations are equivalent if one can be obtained from the other by a sequence
of pattern-replacing moves of prescribed form; however, we limit our focus to
patterns where two elements are transposed, subject to the constraint that a
third element of a suitable type be in a suitable position. For various
instances of the problem, we compute the number of equivalence classes,
determine how many n-permutations are equivalent to the identity permutation,
or characterise this equivalence class. Although our results feature familiar
integer sequences (e.g., Catalan, Fibonacci, and Tribonacci numbers) and
special classes of permutations (layered, connected, and 123-avoiding), some of
the sequences that arise appear to be new.
