Summing multi-norms defined by Orlicz spaces and symmetric sequence space

We develop the notion of the $(X_1,X_2)$-summing power-norm based on a~Banach space $E$, where $X_1$ and $X_2$ are symmetric sequence spaces. We study the particular case when $X_1$ and $X_2$ are Orlicz spaces ${\ell }_{\mathrm{\Phi }}$ and ${\ell }_{\mathrm{\Psi }}$ respectively and analyze under which conditions the $(\mathrm{\Phi },\mathrm{\Psi })$-summing power-norm becomes a~multinorm. In the case when $E$ is also a~symmetric sequence space $L$, we compute the precise value of $\Vert ({\delta }_1,\cdots ,{\delta }_n){\Vert }_n^{(X_1,X_2)}$ where $({\delta }_k)$ stands for the canonical basis of $L$, extending known results for the $(p,q)$-summing power-norm based on the space ${\ell }_r$ which corresponds to $X_1={\ell }_p$, $X_2={\ell }_q$, and $E={\ell }_r$.