[Submitted on 25 Jul 2026 (v1), last revised 28 Jul 2026 (this version, v2)]

View PDF HTML (experimental)

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 $r=2$.

Submission history

From: Yuchen Yang [view email]
[v1] Sat, 25 Jul 2026 02:02:41 UTC (7 KB)
[v2] Tue, 28 Jul 2026 09:17:14 UTC (7 KB)