Jellyfish partition categories

For each positive integer $n$, we introduce a monoidal category $\mathcal{J}\mathcal{P}(n)$ using a generalization of partition diagrams. When the characteristic of the ground field is either 0 or at least $n$, we show $\mathcal{J}\mathcal{P}(n)$ is monoidally equivalent to the full subcategory of $\mathrm{Rep}(A_n)$ whose objects are tensor powers of the natural $n$-dimensional permutation representation of the alternating group $A_n$.