  The present paper produces examples of Gauss diagram formulae for virtual
knot invariants which have no analogue in the classical knot case. These
combinatorial formulae contain additional information about how a subdiagram is
embedded in a virtual knot diagram. The additional information comes from the
second author's recently discovered notion of parity. For a parity of flat
virtual knots, the new combinatorial formulae are Kauffman finite-type
invariants. However, many of the combinatorial formulae possess exotic
properties. It is shown that there exists an integer valued virtualization
invariant combinatorial formula of order n for every n (i.e. it is stable under
the map which changes the direction of one arrow but preserves the sign).
Hence, it is not of Goussarov-Polyak-Viro finite-type. Moreover, every
homogeneous Polyak-Viro combinatorial formula admits a decomposition into an
"even" part and an "odd" part. For the Gaussian parity, neither part of the
formula is of GPV finite-type when it is nonconstant on the set of classical
knots. In addition, eleven new non-trivial combinatorial formulae of order 2
are presented which are not of GPV finite-type.
