I shall report on recent joint work with James Smith (\url{https://doi.org/10.1016/j.jmaa.2025.129235}) in which we study the lattice of closed ideals of bounded operators on two families of Banach spaces: the Baernstein spaces $B_p$ for $1<p<\infty$ and the $p$-convexified Schreier spaces $S_p$ for $1\le p<\infty$. Our main conclusion is that there are $2^{\mathfrak{c}}$ many closed ideals that lie between the ideals of compact and strictly singular operators on each of these spaces, and also $2^{\mathfrak{c}}$ many closed ideals that contain projections of infinite rank.
Counterparts of results of Gasparis and Leung using a numerical index to distinguish the isomorphism types of subspaces spanned by subsequences of the unit vector basis for the classical Schreier space $S_1$ play a key role in the proofs, as does the Johnson--Schechtman technique for constructing $2^{\mathfrak{c}}$ many closed ideals of operators on a Banach space.
I intend to begin from first principles, without assuming any prior familiarity with the Schreier sets or the associated Banach spaces mentioned in the title, with emphasis on the combinatorial aspects of our work, notably through the above-mentioned index of Gasparis and Leung.