An Explicit Counterexample to Stanley's Rankwise Lower-Bound Conjecture for Differential Posets

Abstract: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 $r=2$.

Combinatorics (math.CO); Artificial Intelligence (cs.AI)

arXiv:2607.22988 [math.CO]

(or arXiv:2607.22988v2 [math.CO] for this version)

https://doi.org/10.48550/arXiv.2607.22988 Focus to learn more arXiv-issued DOI via DataCite

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.