推翻了关于微分序集秩最小基数的猜想，给出了反例。
An Explicit Counterexample to Stanley's Rankwise Lower-Bound Conjecture for Differential Posets
- 构造无限r-微分序集，使第4秩大小小于原猜想下界。
- 当r=3时，第4秩元素数为50而非51，减少1个。
- 适用于组合数学与序结构研究者，尤其关注序集极值问题。
在1988年关于微分序集的论文中，斯坦利提出了一个问题：对于r-微分序集，固定秩的最小基数是多少？他推测该最小值由Y^r（Young格的r重笛卡尔积）实现。本文推翻了这一普遍系数下界猜想。对每个r≥3，构造了一个无限r-微分序集P^{(r)}，满足|P^{(r)}_4| = |(Y^r)_4| - ⌊r/3⌋。当r=3时，通过将Y^3中十三个第4秩下覆盖块替换为十二个具有相同点对重数的块，得到初始秩序列1,3,9,22,50，而非原期望的1,3,9,22,51。随后通过反射扩展得到无限微分序集。该构造未涉及r=1和r=2的情况。
原文摘要 · Abstract (English)
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 $r=2$.
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