  To flatten a set partition (with apologies to Mathematica) means to form a
permutation by erasing the dividers between its blocks. Of course, the result
depends on how the blocks are listed. For the usual listing--increasing entries
in each block and blocks arranged in increasing order of their first
entries--we count the partitions of [n] whose flattening avoids a single
3-letter pattern. Five counting sequences arise: a null sequence, the powers of
2, the Fibonacci numbers, the Catalan numbers, and the binomial transform of
the Catalan numbers.
