Minimum rank sizes of r-differential posets
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Does the product Yʳ of r copies of Young’s lattice minimize every rank size among r-differential posets? Stanley’s proposal compares each rank separately, and the recorded counterexample target is rank four for every r≥3.
Why it matters and what counts as progress
Why it matters: Rankwise minimality predicts how small an infinite differential poset can be compared with the canonical Young-lattice product.
What counts as progress: Construct an infinite differential poset with a smaller rank or prove a sharp replacement lower bound.
How the problem developed
- 1988 · Question [1]
Richard P. Stanley Asked for the least possible size of a fixed rank in an r-differential poset and suggested that the corresponding rank of Y^r is minimal.
- 25 Jul 2026 · Counterexample [2]
Xinan Dai, Yuchen Yang, Wenhao Deng, Yingdong Shi and Tailin Wu; the TARS agent system generated the counterexample, which the authors independently verified Constructed, for every r ≥ 3, an infinite r-differential poset whose fourth rank is floor(r/3) smaller than that of Y^r, disproving the proposed universal lower bound.
AI contribution
An Explicit Counterexample to Stanley's Rankwise Lower-Bound Conjecture for Differential Posets
For each integer r≥3, Dai and coauthors construct an infinite r-differential poset that agrees with Yʳ through rank three but has rank-four size |(Yʳ)₄|−⌊r/3⌋. At r=3 the first sizes are 1, 3, 9, 22, 50, compared with 51 for Y³.
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