to bottom, so that the double sum of its elements equals $n$. Together with the largest part size $X_n$, consider also the counts of the non-zero rows and columns of the matrix of a plane partition. It turns out that these three quantities measure the sizes of the corresponding solid diagram of a plane partition in the $3D$ space. (The solid diagram is a heap of $n$ unit cubes placed in the first octant of a coordinate system in a $3D$ space whose columns composed by stacked cubes have non-increasing heights along the $x$- and $y$-axis; the height of this heap along the $z$-axis is just $X_n$, the largest part size.) Similarly to Ferrers diagrams for linear integer partitions, the three sizes of this heap appear to be identically distributed for all $n\ge 1$ (for more details, see \[19; p. 371\]). Their joint limiting distribution was found in \[17\]. The marginal limiting distributions (including the limiting distribution of $X_n$) were obtained in \[14\]. For more details on various properties of plane partitions and their applications to combinatorics and analysis of algorithms, we refer the reader to \[2; Chap. 11\], \[16; Chap. 11\] and \[19; Chap. 7\].

Our study is also 