Generalized clustering conditions of Jack polynomials at negative Jack parameter α

We present several conjectures on the behavior and clustering properties of Jack polynomials at a negative parameter α=- k+1 r-1, with partitions that violate the (k,r,N) - admissibility rule of [Feigin [Int. Math. Res. Notices 23, 1223 (2002)]. We find that the "highest weight" Jack polynomials of specific partitions represent the minimum degree polynomials in N variables that vanish when s distinct clusters of k+1 particles are formed, where s and k are positive integers. Explicit counting formulas are conjectured. The generalized clustering conditions are useful in a forthcoming description of fractional quantum Hall quasiparticles.