I'm interested in expressions that are invariant under the exchange of raw moments and cumulants. This is trivially true of all expressions written only in terms of first order moments but nontrivial expressions exist for higher orders. A couple of examples are as follows:
$\kappa _{2,0}\kappa_{0,1}{}^2-\kappa _{0,2} \kappa _{1,0}{}^2=\mu'_{2,0}\mu'_{0,1}{}^2-\mu'_{0,2} \mu'_{1,0}{}^2$
$\kappa _{2,0}\kappa_{0,1}{}^2+\kappa _{0,2} \kappa _{1,0}{}^2-2\kappa_{1,1} \kappa_{1,0}\kappa_{0,1}=\mu'_{2,0}\mu'_{0,1}{}^2+\mu'_{0,2} \mu'_{1,0}{}^2-2\mu'_{1,1} \mu'_{1,0}\mu'_{0,1}$
Does anybody know if these have a specific name or know anything else about them? I'm looking for any material I can find in order to learn more about them. Thanks!