Interpolations between Bose and Fermi statistics

A comparative study of various interpolations between the grand canonical partition functions of Bose and Fermi statistics is undertaken. This includes a new interpolation inspired by the microscopic structure of Haldaneʹs g=1/m statistics. Explicit expresions for the corresponding canonical partition functions and the counting formulae are given. The question as to which of these statistics can be considered as generalized permutation group statistics is also examined. It is found that only one of the statistics considered falls into this category. The canonical partition function of this particular statistics turns out to be a Jack polynomial which in turn enables one to view this statistics as a generalization of statistics based on the permutation group.