  Number of zeros seen by a particle around small clusters of other particles
is encoded in the root partition, and partly characterizes the correlations in
fractional quantum Hall trial wavefunctions. We explore a generalization
wherein we consider the counting of zeros seen by a cluster of particles on
another cluster. Numbers of such zeros between clusters in the Laughlin
wavefunctions are fully determined by the root partition. However, such a
counting is unclear for general Jain states where a polynomial expansion is
difficult. Here we consider the simplest state beyond the Laughlin
wavefunction, namely a state containing a single quasiparticle of the Laughlin
state. We show numerically and analytically that in the trial wavefunction for
the quasiparticle of the Laughlin state, counting of zeros seen by a cluster on
another cluster depends on the relative dimensions of the two clusters. We
further ask if the patterns in the counting of zeros extend, in at least an
approximate sense, to wavefunctions beyond the trial states. Using numerical
computations in systems up to $N=9$, we present results for the statistical
distribution of zeros around particle clusters at the center of an FQH droplet
in the ground state of a Hamiltonian that is perturbed away from the $V_1$
interaction (short-range repulsion). Evolution of this distribution with the
strength of the perturbation shows that the counting of zeros is altered by
even a weak perturbation away from the parent Hamiltonian, though the
perturbations do not change the phase of the system.
