Symmetric finite representability of $$\ell ^p$$-spaces in rearrangement invariant spaces on [0, 1]

For a separable rearrangement invariant space X on [0, 1] of fundamental type we identify the set of all $$p\in [1,\infty ]$$ such that $$\ell ^p$$ is finitely represented in X in such a way that the unit basis vectors of $$\ell ^p$$ ( $$c_0$$ if $$p=\infty $$ ) correspond to pairwise disjoint and equimeasurable functions. This can be treated as a follow up of a paper by the first-named author related to separable rearrangement invariant spaces on $$(0,\infty )$$ .