Improvements to exact Boltzmann sampling using probabilistic divide-and-conquer and the recursive method

We demonstrate an approach for exact sampling of certain discrete combinatorial distributions, which is a hybrid of exact Boltzmann sampling and the recursive method, using probabilistic divide-and-conquer (PDC). The approach specializes to exact Boltzmann sampling in the trivial setting, and specializes to PDC deterministic second half in the first non-trivial application. A large class of examples is given for which this method broadly applies, and several examples are worked out explicitly.