  The study of pattern avoidance in permutations, and specifically in flattened
partitions is an active area of current research. In this paper, we count the
number of distinct flattened partitions over [n] avoiding a single pattern, as
well as a pair of two patterns. Several counting sequences, namely Catalan
numbers, powers of two, Fibonacci numbers and Motzkin numbers arise. We also
consider other combinatorial statistics, namely runs and inversions, and
establish some bijections in situations where the statistics coincide.
