2015 USAMO 第 4 题

Steve is piling $m\geq 1$ indistinguishable stones on the squares of an $n\times n$ grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform stone moves, defined as follows. Consider any four grid squares, which are corners of a rectangle, i.e. in positions $(i, k), (i, l), (j, k), (j, l)$ for some $1\leq i, j, k, l\leq n$, such that $i<j$ and $k<l$. A stone move consists of either removing one stone from each of $(i, k)$ and $(j, l)$ and moving them to $(i, l)$ and $(j, k)$ respectively,j or removing one stone from each of $(i, l)$ and $(j, k)$ and moving them to $(i, k)$ and $(j, l)$ respectively.

Two ways of piling the stones are equivalent if they can be obtained from one another by a sequence of stone moves.

How many different non-equivalent ways can Steve pile the stones on the grid?

史蒂夫正在 $n\times n$ 网格的方格上堆放 $m\geq 1$ 无法区分的石头。每个方格可以放置任意高度的石块。当他以某种方式完成石头堆积后，他就可以执行石头移动，定义如下。考虑任意四个网格正方形，它们是矩形的角，即在位置 $(i, k), (i, l), (j, k), (j, l)$ 中，对于某些 $1\leq i, j, k, l\leq n$，使得 $i<j$ 和 $k<l$。移动石子包括从 $(i, k)$ 和 $(j, l)$ 中取出一颗石子，并将它们分别移动到 $(i, l)$ 和 $(j, k)$j，或者从 $(i, l)$ 和 $(j, k)$ 中取出一颗石子，并将它们分别移动到 $(i, k)$ 和 $(j, l)$。

如果可以通过一系列石头移动从彼此获得石头，则两种堆积石头的方法是等效的。

史蒂夫可以用多少种不同的非等效方式将石头堆放在网格上？