Let R be a ommutative ring with unity and F a finite free R-module. For a nonnegative integer r, there exists a natural filtration of$S_r(S_2F)$ such that its associated graded module is isomorphic to $Sigma_{{lambda}{epsilon}{ au}_r};L_{lambda}F$, where ${Gamma}_{gamma}$ set of partitions such that $$mid${lambda}$mid$-2r,{{widetilde}{lambda}}-{{widetilde}{lambda}}_1},...,{{widetilde}{lambda}}_k},;each;{{widetilde}{lambda}}_t}$</TEX>,is even. We call such filtrations plethysm formulas. We extend the above plethysm formula to the version of chain complexes. By plethysm formula we mean the composition of universally free functors. $Let{emptyset}:G->F$ be a morphism of finite free R-modules. We construct the natural decomposition of $S_{r}(S_2{emptyset})$,up to filtrations, whose associated graded complex is isomorphic to ${Sigma}_{{lambda}{varepsilon}{ au}}_r};L_{lambda}{emptyset}$.