A Negative Answer to Stanley's Problem 4 on Differential Posets

Abstract

In Problem 4 of his 1988 paper on differential posets, Stanley asked whether the weighted $k$-multichain series of an arbitrary differential poset is always a rational multiple of the $k$th power of its rank series. We answer this question in the negative. Already for $k=2$, we construct a locally finite $1$-differential poset $P$ for which $M_{P,2}(q)/F_P(q{)}^2$ is not rational over any field of characteristic zero. The construction uses Wagner's crown reflection. At each sufficiently high rank there are two valid one-rank extensions with the same required Gram matrix and with new rank sizes differing by one. We show that the first quotient coefficient affected by that rank distinguishes the two choices. Choosing successively between them and diagonalizing against the rational power series produces the required poset. The same coefficient argument also yields continuum many distinct quotient series, of which continuum many are nonrational.