Quasi-differential posets and cover functions of distributive lattices. II: A problem in Stanley's \textit{Enumerative combinatorics} (Q1423493)

A distributive lattice \(L\) with \(0\) is finitary if every interval is finite. A function \(f:N_0 \rightarrow N_0\) is a cover function for \(L\) if every element with \(n\) lower covers has \(f(n)\) upper covers. All non-decreasing cover functions have been characterized by the author in Part I [J. Comb. Theory, Ser. A 90, 123--147 (2000; Zbl 0951.06002)], settling a 1975 conjecture of Richard P.~Stanley. In this paper, all finitary distributive lattices with cover functions are characterized. A problem in Stanley's Enumerative combinatorics is thus solved.