Permutatively generating set for an IAPS

Definition
Let $$(G,\Phi)$$ be an IAPS of groups with the structure of a permutative APS. A permutatively generating set for $$G$$ is a collection of elements $$g_i \in G_{n_i}$$ with the property that for any natural number $$m$$:

$$G_m$$ is generated by elements obtained by taking block concatenations of $$g_i$$s and the identity element, and the images of these under the action of permutations.

Examples

 * Any symmetric group on a finite set is generated by transpositions. This tells us that the transposition on a two-element set forms a permutatively generating set for the permutation IAPS.
 * The general linear group is generated by elementary matrices. This tells us that the elementary $$2 \times 2$$ matrices form a permutatively generating set for the GL IAPS.