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 $r=2$.

Disclosure

“24 3 where mr (n) denotes the minimum possible size of rank n among infinite r-differential posets. It neither determines mr (4) exactly nor addresses r = 1 or r = 2. Statement on AI-assisted discovery and human verification The counterexample presented in this paper was initially generated by the TARS agent system through an autonomous mathematical search and was subsequently examined and independently verified by the human”

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Classification

Substantial mathematical content or result generation

Multiplier

10

Verified

2026-08-23T19:05:13.909040Z

Structural counts