AI 中文总结
针对斯坦利关于差分偏序集的问题4，通过构造局部有限1 - 差分偏序集，利用瓦格纳冠反射及商系数区分不同扩展选择，对角化得到所需偏序集，给出否定回答并得出多个非有理商级数。
AI 中文摘要
在1988年关于差分偏序集的论文的问题4中，斯坦利询问任意差分偏序集的加权k - 多重链级数是否总是其秩级数的k次幂的有理倍数。我们给出否定回答。对于k = 2，我们构造了一个局部有限的1 - 差分偏序集P，使得$M_{P,2}(q)/F_P(q)^2$在任何特征为零的域上都不是有理的。构造使用了瓦格纳的冠反射。在每个足够高的秩上有两个有效的一阶扩展，具有相同所需的格拉姆矩阵且新秩大小相差一。我们表明受该秩影响的第一个商系数区分了这两种选择。依次在它们之间选择并针对有理幂级数进行对角化产生所需的偏序集。相同的系数论证还产生了连续多个不同的商级数，其中连续多个是非有理的。
英文摘要
In Problem 4 of his 1988 paper on differential posets, Stanley asked whether the weighted $k$-multichain series of an arbitrary differential poset is always a rational multiple of the $k$th power of its rank series. We answer this question in the negative. Already for $k=2$, we construct a locally finite $1$-differential poset $P$ for which $M_{P,2}(q)/F_P(q)^2$ is not rational over any field of characteristic zero. The construction uses Wagner's crown reflection. At each sufficiently high rank there are two valid one-rank extensions with the same required Gram matrix and with new rank sizes differing by one. We show that the first quotient coefficient affected by that rank distinguishes the two choices. Choosing successively between them and diagonalizing against the rational power series produces the required poset. The same coefficient argument also yields continuum many distinct quotient series, of which continuum many are nonrational.
