  We carry out a systematic study of primary operators in the conformal field
theory of a free Weyl fermion. Using SO(4,2) characters we develop counting
formulas for primaries constructed using a fixed number of fermion fields. By
specializing to particular classes of primaries, we derive very explicit
formulas for the generating functions for the number of primaries in these
classes. We present a duality map between primary operators in the fermion
field theory and polynomial functions. This allows us to construct the
primaries that were counted. Next we show that these classes of primary fields
correspond to polynomial functions on certain permutation orbifolds. These
orbifolds have palindromic Hilbert series.
