Wavefunctions and counting formulas for quasiholes of clustered quantum Hall states on a sphere

The quasiholes of the Read-Rezayi clustered quantum Hall states are considered, for any number of particles and quasiholes on a sphere, and for any degree k of clustering. A set of trial wavefunctions, that are zero-energy eigenstates of a k+1-body interaction, and so are symmetric polynomials that vanish when any k+1 particle coordinates are equal, is obtained explicitly and proved to be both complete and linearly independent. Formulas for the number of states are obtained, without the use of (but in agreement with) conformal field theory, and extended to give the number of states for each angular momentum. An interesting recursive structure emerges in the states that relates those for k to those for k-1. It is pointed out that the same numbers of zero-energy states can be proved to occur in certain one-dimensional models that have recently been obtained as limits of the two-dimensional k+1-body interaction Hamiltonians, using results from the combinatorial literature.