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：本文研究了一类定义在严格整数分划（即各部分互异）集合 $\check{\varLambda}^{q}$ 上的玻尔兹曼（乘法）分布的渐近分析，其中各部分为完美 $q$ 次幂。通过固定分划权（各部分之和）和长度（部分数）进行适当条件化，建立了组合学联系，从而在相应的分划子空间上得到均匀分布。玻尔兹曼测度通过控制期望权重的超参数 $\langle N\rangle$ 和控制期望长度的超参数 $\langle M\rangle$ 进行校准。我们研究了“短”分划，其中参数 $\langle M\rangle$ 或固定不变，或增长速度慢于典型普通（无约束）分划。针对该模型，我们获得了一系列极限定理，包括 $\langle M\rangle$ 固定时累积基数的渐近性，以及 $\langle M\rangle$ 缓慢增长时的极限形状结果。在这两种情形中，我们还刻画了权重与长度的联合分布，以及最小部分和最大部分的增长特性。利用这些结果，我们构建了相应的采样算法并分析了其性能。