We provide an ordinal invariant, with values in $\omega_1+1 = \omega_1\cup \{ \omega_1\}$, associated with an analytic P-ideal that gives a stratification of the class of all analytic P-ideals, with the value of the invariant equal to $\omega$ equivalent to being summable-like and the value of the invariant equal to $\omega_1$ equivalent to being density-like. We prove that the ordinal invariant interacts well with Tukey reduction. We find an analytic P-ideal ${\mathcal T}^*$ for which the value of the invariant is neither $\omega$ nor $\omega_1$; in fact, we compute it to be $\omega^\omega$. The ideal ${\mathcal T}^*$ comes from the Banach space originally defined by Tsirelson.
This is joint work with Stevo Todorcevic.