Statistical aspects of weighted balls in boxes problems with an emphasis on population models

Abstract We investigate the following problems, including the classical integer partitioning question: In how many ways can a population of size n be split into k parts when the parts consist of different kinds of trees under various labelling schemes of their constitutive atoms (nodes)? What are the sizes of the parts or clusters? The answers strongly depend on whether the particles (balls) and the parts (boxes) are distinguishable or not. This weighted partitioning setup has weights that are independent of the box numbers, either in a combinatorial or probabilistic sense. This class of problems also possibly shows, for specific alternative choices of power-law cluster weights, a ‘Bose–Einstein-like’ phase transition from a condensed to a dilute phase. When weights depend on the box numbers, we run into classical Bose–Einstein versus Maxwell–Boltzmann occupancy problems, which we briefly revisit for comparison and completeness. Our contribution reveals how sensitive occupancy issues are to the notion of distinguishability.