Stanley’s rational-factor question for differential posets
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For a locally finite differential poset P, must its multichain generating function M_(P,k)(q) differ from F_P(q)^k by a rational factor? The proposed failure already concerns 1-differential posets and k=2.
Why it matters and what counts as progress
Why it matters: A nonrational quotient would show that multichain enumeration in differential posets need not have the proposed finite rational correction.
What counts as progress: Prove rationality under additional structure or construct a poset with a demonstrably nonrational quotient over characteristic-zero fields.
How the problem developed
- 1988 · Open question [1]
Richard P. Stanley Asked whether each k-multichain generating series of a differential poset is a rational multiple of the kth power of its rank-generating series.
- 28 Jul 2026 · Counterexample [2]
Dai and coauthors Constructed a locally finite 1-differential poset for which the two-multichain quotient is nonrational over every characteristic-zero field.
AI contribution
A Negative Answer to Stanley's Problem 4 on Differential Posets
Dai and coauthors construct a locally finite 1-differential poset P for which the two-multichain quotient M_(P,2)(q)/F_P(q)² is nonrational over every characteristic-zero field. This refutes Stanley’s proposed rational-factor property already at k=2.
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