AI 中文总结
该研究引入回文间隙分拆，推导其生成函数与计数公式，构造相关双射，还将其与Ferrers图的自补性关联，解答了Keith的部分问题并区分了可补与自补分拆。
AI 中文摘要
我们引入并研究“回文间隙分拆”：即连续部分之间的差序列为回文的分拆。我们分别处理奇数个部分和偶数个部分的分拆，推导了该类分拆的生成函数，并得到了跟踪部分数量的双变量细化形式。进一步地，我们给出了将n分拆为恰好r个部分的回文间隙分拆数量的显式闭式公式。随后，我们构造了两个显式双射：一个是有2m+1个部分的回文间隙分拆与至多有m+1个部分的普通分拆之间的双射，另一个是有2m个部分的回文间隙分拆与至多有m+1个部分的普通分拆之间的双射；在这些双射中，回文间隙分拆的权重由其像的最大部分或两个最大部分编码，将这两个双射复合可得到奇数个部分和偶数个部分的回文间隙分拆类之间的显式双射。最后，我们证明一个分拆是回文间隙分拆当且仅当将其按左对齐行绘制的Ferrers图，在其自然关联的矩形内经180°旋转后是自补的，这将回文间隙分拆与相对于该矩形的自补分拆等同起来。这为Keith关于可补分拆和自补分拆计数问题给出了部分解答：我们给出了自补情形的显式生成函数，并在n=15时展示了一个可补但非自补的分拆，证实了这两个概念确实是不同的。
英文摘要
We introduce and study palindromic-gap partitions: partitions whose sequence of successive differences between consecutive parts is a palindrome. We derive the generating function for this family, treating partitions with an odd and an even number of parts separately, and obtain a bivariate refinement tracking the number of parts. Specializing further, we give explicit closed formulas for the number of palindromic-gap partitions of $n$ into exactly $r$ parts. We then construct explicit bijections between palindromic-gap partitions with $2m+1$, respectively $2m$, parts and ordinary partitions with at most $m+1$ parts, in which the weight of a palindromic-gap partition is encoded by the largest part, or the two largest parts, of its image; composing these yields an explicit bijection between the odd and even families themselves. Finally, we show that a partition is palindromic-gap if and only if its Ferrers diagram, drawn with left-justified rows, is self-complementary under a $180^\circ$ rotation inside its own naturally associated rectangle, identifying palindromic-gap partitions with self-complementary partitions relative to this rectangle. This yields a partial answer to a question of Keith on the enumeration of complementable and self-complementary partitions: we give explicit generating functions for the self-complementary case and exhibit, at $n=15$, a partition that is complementable but not self-complementary, confirming that the two notions are genuinely distinct.
Comments20 pages
