A characterization of $\ell^p$-spaces symmetrically finitely represented in symmetric sequence spaces

For a separable symmetric sequence space $X$ of fundamental type we identify the set ${\mathcal F}(X)$ of all $p\in [1,\infty]$ such that $\ell^p$ is block finitely represented in the unit vector basis $\{e_k\}_{k=1}^\infty$ of $X$ in such a way that the unit basis vectors of $\ell^p$ ($c_0$ if $p=\infty$) correspond to pairwise disjoint blocks of $\{e_k\}$ with the same ordered distribution. It turns out that ${\mathcal F}(X)$ coincides with the set of approximate eigenvalues of the operator $(x_k)\mapsto \sum_{k=2}^\infty x_{[k/2]}e_k$ in $X$. In turn, we establish that the latter set is the interval $[2^{\alpha_X},2^{\beta_X}]$, where $\alpha_X$ and $\beta_X$ are the Boyd indices of $X$. As an application, we find the set ${\mathcal F}(X)$ for arbitrary Lorentz and separable sequence Orlicz spaces.