Under what circumstances might every extension of a combinatorial structure
contain more copies of another one than the original did? This property, which
we call prolificity, holds universally in some cases (e.g., finite linear
orders) and only trivially in others (e.g., permutations). Integer
compositions, or equivalently layered permutations, provide a middle ground. In
that setting, there are prolific compositions for a given pattern if and only
if that pattern begins and ends with 1. For each pattern, there is an easily
constructed automaton that recognises prolific compositions for that pattern.
Some instances where there is a unique minimal prolific composition for a
pattern are classified
