The paper is concerned with the asymptotic analysis of a family of Boltzmann (multiplicative) distributions over the set $\check{\varLambda}^{q}$ of strict integer partitions (i.e., with unequal parts) into perfect $q$-th powers. A combinatorial link is provided via a suitable conditioning by fixing the partition weight (the sum of parts) and length (the number of parts), leading to uniform distribution on the corresponding subspaces of partitions. The Boltzmann measure is calibrated through the hyper-parameters $\langle N\rangle$ and $\langle M\rangle$ controlling the expected weight and length, respectively. We study ``short'' partitions, where the parameter $\langle M\rangle$ is either fixed or grows slower than for typical plain (unconstrained) partitions. For this model, we obtain a variety of limit theorems including the asymptotics of the cumulative cardinality in the case of fixed $\langle M\rangle$ and a limit shape result in the case of slow growth of $\langle M\rangle$. In both cases, we also characterize the joint distribution of the weight and length, as well as the growth of the smallest and largest parts. Using these results we construct suitable sampling algorithms and analyse their performance.

翻译：---- “短”幂部分的玻尔兹曼分布：极限定理和采样 Translated abstract: 本文关注Boltzmann（乘法）分布在绝对值不同的完美 $q$ 次幂正整数（即不等部分）的集合 $ \check {\varLambda}^q $ 上的渐近分析。通过通过将分区重量（部分的总和）和长度（部分的数量）固定为一定值的适当调整，提供了一种组合链的联接，导致在相应的分区子空间上的均匀分布。Boltzmann度量通过控制期望的重量和长度的超参数 $ \langle N\rangle $ 和 $ \langle M\rangle $ 进行校准。我们研究“短”分区，其中参数 $ \langle M\rangle $ 要么固定，要么增长速度比典型的普通（非约束性）分区要慢。对于这个模型，我们得到了多种极限定理，包括在固定 $ \langle M\rangle $ 的情况下的累积基数的渐近行为以及在 $ \langle M\rangle $ 缓慢增长的情况下的极限形状结果。在这两种情况下，我们还表征了重量和长度的联合分布，以及最小和最大部分的增长。利用这些结果，我们构建了合适的采样算法并分析了它们的性能。