A characterization of lp-spaces symmetrically finitely represented in symmetric sequence spaces

For a separable symmetric sequence space X of fundamental type we identify the set F(X) of all p is an element of [1, infinity] such that l(p) is block finitely represented in the unit vector basis {e(k)}(k=1)(infinity) of X in such a way that the unit basis vectors of l(p) (c(0) if p = infinity) correspond to pairwise disjoint blocks of {e(k)} with the same ordered distribution. It turns out that F(X) coincides with the set of approximate eigenvalues of the operator (x(k)) bar right arrow Sigma(8)(k=2) 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 F(X) for arbitrary Lorentz and separable Orlicz sequence spaces.