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 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 [ \card{P^{(r)}_4} =\card{(Y^r)_4}-\left\lfloor\frac r3\right\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.