  Four point tree-level local S-matrices form a module over ring of polynomials
of mandelstam invariants s, t and u. The module of local analytic S-matrices
can be encoded in terms of a partition function which is enumerated using
plethystic techniques. In this paper, we enumerate the plethystic contribution
to local four point photon, graviton and gluon multi-particle partition
functions that encode parity violating $2 \rightarrow 2$ scattering. We
generalise the counting problem solved in \cite{ Chowdhury:2019kaq,
Chowdhury:2020ddc} to project out parity violating sectors, a subtle task in
even dimensions \cite{Henning:2017fpj}. We explicitly enumerate the parity odd
contributions to the multi-letter partition function for gauge fields,
gravitons and gluons and evaluate the resulting parity violating partition
functions in $D=4,6$. We also perform a large $D$ analysis to show that parity
violating local interactions do not contribute to four particle scattering in
higher dimensions ($D\geq 8$). Our computations and observations for photons,
gravitons and gluons agree with the transformation properties of these
S-matrices previously conjectured in \cite{Chowdhury:2019kaq,
Chowdhury:2020ddc}

High Energy Physics - Theory