  Correlations of two or more particles have been an essential tool for
understanding the hydrodynamic behavior of the quark-gluon plasma created in
ultra-relativistic nuclear collisions. In this paper, we extend that framework
to introduce a mathematical construction of multi-particle correlators that
utilize correlations between arbitrary numbers of particles of interest (e.g.
particles selected for their strangeness, heavy flavor, and conserved charges)
and inclusive reference particles to estimate the azimuthal anisotropies of
rare probes. To estimate the fluctuations and correlations in the azimuthal
anisotropies of these particle of interest, we use these correlators in a
system of cumulants, raw moments, and central moments. We finally introduce two
classes of observables that can compare the fluctuations in the azimuthal
anisotropies of particles of interest with reference particles at each order.
