无单元素划分与无循环相邻对划分之间的最优精化兼容双射
An optimal refinement-compatible bijection between singleton-free partitions and partitions without cyclic adjacencies
AI总结：
本文针对Bernhart的论断，构造了无单元素划分与无循环相邻对划分之间兼容精化序的最优直接单轮双射，该双射还可推广至非交叉划分对应类。
AI中文摘要：
众所周知，集合[n]的无单元素划分数等于集合[n]中没有块包含两个循环相邻元素i、i+1（模n）的划分数。Bernhart指出这两类划分之间可能不存在简单双射，尽管Callan后来构造了一种算法双射，证明了单元素和相邻对的等分布性，但其构造需经过多轮交换。Chen和Wang的研究表明，Bernhart的论断可能仍有一定合理性。本文针对该论断，给出了这两类划分之间的直接“单轮”双射。与Callan的双射不同，我们的映射与划分上的精化序高度兼容：正向仅分解块，反向仅合并块，仅当n>2为偶数时存在一对例外情况。我们进一步发现该例外不可避免，从而确立了该双射相对于精化序的最优性。这些操作的特定局部形式——仅拆分出单元素块，且仅将单元素与其循环相邻元素所在的块合并——还确保该构造无需修改即可限制为非交叉划分对应类之间的双射。
英文摘要：
It is well known that the number of partitions of $[n]$ without singletons equals the number of partitions of $[n]$ in which no block contains two cyclically adjacent elements $i,i+1\pmod{n}$. Bernhart remarked that there might be no simple bijection between these two classes. Although Callan later constructed an algorithmic bijection proving the stronger equidistribution of singletons and adjacencies, his construction proceeds through multiple rounds of exchanges. Therefore, Bernhart's remark may still retain some validity, as suggested by Chen and Wang. In this article, we address this remark by giving a direct ``one-round'' bijection between the two classes. Unlike Callan's bijection, our map is closely compatible with the refinement order on partitions: in one direction it only decomposes blocks, while its inverse only merges blocks, with a single exceptional pair when $n>2$ is even. We further observe that this exception is unavoidable, establishing the optimality of the bijection with respect to the refinement order. The specific local form of these operations --- splitting off only singleton blocks and merging a singleton only with the block containing its cyclic neighbor --- also ensures that the construction restricts, without modification, to a bijection between the corresponding classes of noncrossing partitions.
