发表机构
Fudan University; Westlake University; ShanghaiTech University（复旦大学; 西湖大学; 上海科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对斯坦利关于r - 差分偏序集固定秩最小基数的猜想，本文构造反例。对$r\geq 3$，构建无限r - 差分偏序集$P^{(r)}$，满足特定等式，如$r = 3$时改变下覆盖块得到新秩序列，经反射扩展得无限差分偏序集，未涉及$r = 1$和$r = 2$。
AI 中文摘要
1988年斯坦利在关于差分偏序集的论文的问题6中，询问了r - 差分偏序集固定秩的最小可能基数，并认为最小值应由杨格的r重笛卡尔幂$Y^r$达到。我们反驳了由此产生的通用系数下界。对于每个$r\geq 3$，我们构造了一个无限的r - 差分偏序集$P^{(r)}$，满足$\card{P^{(r)}_4} =\card{(Y^r)_4}-\left\lfloor\frac r3\right\rfloor$。对于$r = 3$，构造将$Y^3$的13个四阶下覆盖块替换为12个具有相同点和对关联重数的块，产生初始秩序列$1,3,9,22,50$而不是$1,3,9,22,51$。然后通过反射扩展得到一个无限差分偏序集。该构造未涉及$r = 1$和$r = 2$的情况。
英文摘要
In Problem 6 of his 1988 paper on differential posets, Stanley asked for the least possible cardinality of a fixed rank of an $r$-differential poset and suggested that the minimum should be attained by $Y^r$, the $r$-fold Cartesian power of Young's lattice. We disprove the resulting universal coefficientwise lower bound. For every $r\geq 3$, we construct an infinite $r$-differential poset $P^{(r)}$ satisfying $\lvert P^{(r)}_4\rvert=\lvert (Y^r)_4\rvert-\lfloor r/3\rfloor$. For $r=3$, the construction replaces thirteen rank-four lower-cover blocks of $Y^3$ by twelve blocks with the same point and pair incidence multiplicities, producing the initial rank sequence $1,3,9,22,50$ instead of $1,3,9,22,51$. A reflection extension then yields an infinite differential poset. The construction does not address the cases $r=1$ and $r=2$.
