Assuming that a plane partition of the positive integer n is chosen uniformly at random from the set of all such partitions, we propose a general asymptotic scheme for the computation of expectations of various plane partition statistics as n becomes large. The generating functions that arise in this study are of the form Q(x)F(x), where $$Q(x)=\prod _{j=1}^\infty (1-x^j)^{-j}$$Q(x)=∏j=1∞(1-xj)-j is the generating function for the number of plane partitions. We show how asymptotics of such expectations can be obtained directly from the asymptotic expansion of the function F(x) around $$x=1$$x=1. The representation of a plane partition as a solid diagram of volume n allows interpretations of these statistics in terms of its dimensions and shape. As an application of our main result, we obtain the asymptotic behavior of the expected values of the largest part, the number of columns, the number of rows (that is, the three dimensions of the solid diagram) and the trace (the number of cubes in the wall on the main diagonal of the solid diagram). Our results are similar to those of Grabner et al. (Comb Probab Comput 23:1057–1086, 2014) related to linear integer partition statistics. We base our study on the Hayman’s method for admissible power series.

中文翻译：

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平面划分统计期望的渐近分析

假设从所有这些分区的集合中随机均匀地选择一个正整数 n 的平面分区，我们提出了一种通用渐近方案，用于计算随着 n 变大的各种平面分区统计量的期望。本研究中出现的生成函数的形式为 Q(x)F(x)，其中 $$Q(x)=\prod _{j=1}^\infty (1-x^j)^{- j}$$Q(x)=∏j=1∞(1-xj)-j 是平面分区数的生成函数。我们展示了如何从函数 F(x) 在 $$x=1$$x=1 附近的渐近展开直接获得这种期望的渐近性。将平面分区表示为体积 n 的实体图，可以根据其尺寸和形状来解释这些统计数据。作为我们主要结果的应用，我们得到最大部分的期望值、列数、行数（即实心图的三个维度）和迹线（主对角线上墙上的立方体的数量）的渐近行为实线图）。我们的结果与 Grabner 等人的结果相似。(Comb Probab Comput 23:1057–1086, 2014) 与线性整数分区统计相关。我们的研究基于海曼的可容许幂级数方法。