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

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 Yr, the r-fold Cartesian power of Young's lattice. We disprove the resulting universal coefficientwise lower bound. For every r≥ 3, we construct an infinite r-differential poset P(r) satisfying lvert P(r)4rvert=lvert (Yr)4rvert-lfloor r/3rfloor. For r=3, the construction replaces thirteen rank-four lower-cover blocks of Y3 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.