Pattern Avoidance in "Flattened" Partitions
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.
