Let G be a finite group. A collection $$\Pi =\{H_1,\dots ,{H_r}\}$$ of subgroups G, where $$r>1$$ , is said non-trivial partition if every non-identity element belongs to one and only $$H_i$$ for some $$1\leqslant i\leqslant r$$ . We call group that does not admit any partition-free In this paper, we study whose all proper non-cyclic partitions.

We introduce the multiset partition algebra $${\cal M}{{\cal P}_k}\left(\xi \right)$$ over polynomial ring F[ξ], where F is a field of characteristic 0 and k positive integer. When ξ specialized to integer n, we establish Schur—Weyl duality between actions resulting P}_k}\left(n symmetric group Sn on Symk(Fn). The construction generalizes any vector λ non-negative integers yielding P}_\lambda}\...