证 明 题（数学·2024年·罗马尼亚）

Let n be a positive integer. Initially, a bishop is placed in each square of the top row of a 2ⁿ×2ⁿ chessboard; those bishops are numbered from 1 to 2ⁿ ,from left to right. A jump is a simultaneous move made by all bishops such that the following conditions are satisfied:

each bishop moves diagonally, in a straight line, some number of squares, and

at the end of the jump, the bishops all stand in different squares of the same row.

Find the total number of permutations σ of the numbers 1,2,⋯,2ⁿ with the following property: There exists a sequence of jumps such that all bishops end up on the bottom row arranged in the order σ(1),σ(2),⋯,σ(2ⁿ ), from left to right.

【译】设n是正整数.最开始在一个2ⁿ×2ⁿ的方格棋盘上的第一行的每个小方格内均放置一枚“象”，这些“象”从左到右依次编号：1,2,⋯,2ⁿ.

定义一次“跳跃”操作为同时移动所有的“象”并满足如下条件：

每一枚“象”可沿对角线方向移动任意方格；

在这次“跳跃”操作结束时，所有的“象”恰在同一行的不同方格.

求满足下列条件的数1,2,⋯,2ⁿ的排列σ的总个数：存在一系列的“跳跃”操作，使得结束时所有的“象”都在棋盘的最后一行，并且从左到右编号依次为：σ(1),σ(2),⋯,σ(2ⁿ ).

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