Patterns and coincidences abound in mathematics, and especially in com-

binatorics.

An enumerative coincidence occurs whenever the answer, or

sequence of answers, to one enumerative problem is the same as that to

another – particularly when there is no reason to suspect this might be so.

The objective of this thesis is to investigate enumerative coincidences in

permutation classes from first principles, the ultimate goal being to pro-

vide general methods that can describe enumerative coincidences in most

combinatorial classes.